PHAM Huyen

This thesis aims at providing theoretical tools to support the development and management of intermittent renewable energies in short-term electricity markets.In the first part, we develop an exploitable equilibrium model for price formation in intraday electricity markets. To this end, we propose a non-cooperative game between several generators interacting in the market and facing intermittent renewable generation. Using game theory and stochastic control theory, we derive explicit optimal strategies for these generators and a closed-form equilibrium price for different information structures and player characteristics. Our model is able to reproduce and explain the main stylized facts of the intraday market such as the specific time dependence of volatility and the correlation between price and renewable generation forecasts.In the second part, we study dynamic probabilistic forecasts as diffusion processes. We propose several stochastic differential equation models to capture the dynamic evolution of the uncertainty associated with a forecast, derive the associated predictive densities and calibrate the model on real weather data. We then apply it to the problem of a wind producer receiving sequential updates of probabilistic wind speed forecasts, which are used to predict its production, and make buying or selling decisions on the market. We show to what extent this method can be advantageous compared to the use of point forecasts in decision-making processes. Finally, in the last part, we propose to study the properties of aggregated shallow neural networks. We explore the PAC-Bayesian framework as an alternative to the classical empirical risk minimization approach. We focus on Gaussian priors and derive non-asymptotic risk bounds for aggregate neural networks. This analysis also provides a theoretical basis for parameter tuning and offers new perspectives for applications of aggregate neural networks to practical high-dimensional problems, which are increasingly present in energy-related decision processes involving renewable generation or storage.

No summary available.

This paper presents machine learning techniques and deep reinforcement learningbased algorithms for the efficient resolution of nonlinear partial differential equations and dynamic optimization problems arising in investment decisions and derivative pricing in financial engineering. We survey recent results in the literature, present new developments, notably in the fully nonlinear case, and compare the different schemes illustrated by numerical tests on various financial applications. We conclude by highlighting some future research directions.

Machine learning methods for solving nonlinear partial differential equations (PDEs) are hot topical issues, and different algorithms proposed in the literature show efficient numerical approximation in high dimension. In this paper, we introduce a class of PDEs that are invariant to permutations, and called symmetric PDEs. Such problems are widespread, ranging from cosmology to quantum mechanics, and option pricing/hedging in multi-asset market with exchangeable payoff. Our main application comes actually from the particles approximation of mean-field control problems. We design deep learning algorithms based on certain types of neural networks, named PointNet and DeepSet (and their associated derivative networks), for computing simultaneously an approximation of the solution and its gradient to symmetric PDEs. We illustrate the performance and accuracy of the PointNet/DeepSet networks compared to classical feedforward ones, and provide several numerical results of our algorithm for the examples of a mean-field systemic risk, mean-variance problem and a min/max linear quadratic McKean-Vlasov control problem.

We establish existence and uniqueness for infinite-dimensional Riccati equations taking values in the Banach space $L^1(\mu \otimes \mu)$ for certain signed matrix measures $\mu$ which are not necessarily finite. Such equations can be seen as the infinite-dimensional analogue of matrix Riccati equations, and they appear in the linear-quadratic control theory of stochastic Volterra equations.

We prove a rate of convergence of order 1/N for the N-particle approximation of a second-order partial differential equation in the space of probability measures, like the Master equation or Bellman equation of mean-field control problem under common noise. The proof relies on backward stochastic differential equations techniques.

This thesis is divided into three parts. In the first part, we apply Principal-Agent theory to some market microstructure problems. First, we develop an incentive policy to improve the quality of market liquidity in the context of market-making activity in a bed and a dark pool managed by the same exchange. We then adapt this incentive design to the regulation of market-making activity when several market-makers compete on a platform. We also propose a form of incentive based on the choice of asymmetric tick sizes for buying and selling an asset. We then address the issue of designing a derivatives market, using a quantization method to select the options listed on the platform, and Principal-Agent theory to create incentives for an options market-maker. Finally, we develop an incentive mechanism robust to the model specification to increase investment in green bonds.The second part of this thesis is devoted to high-dimensional options market-making. The second part of this paper is devoted to the market-making of high-dimensional options. Assuming constant Greeks, we first propose a model to deal with long-maturity options. Then we propose an approximation of the value function to handle non-constant Greeks and short maturity options. Finally, we develop a model for the high frequency dynamics of the implied volatility surface. Using multidimensional Hawkes processes, we show how this model can reproduce many stylized facts such as the skew, the smile and the term structure of the surface.The last part of this thesis is devoted to optimal trading problems in high dimension. First, we develop a model for optimal trading of stocks listed on several platforms. For a large number of platforms, we use a deep reinforcement learning method to compute the optimal trader controls. Then, we propose a methodology to solve trading problems in an approximately optimal way without using stochastic control theory. We present a model in which an agent exhibits approximately optimal behavior if it uses the gradient of the macroscopic trajectory as a short-term signal. Finally, we present two new developments on the optimal execution literature. First, we show that we can obtain an analytical solution to the Almgren-Chriss execution problem with geometric Brownian motion and quadratic penalty. Second, we propose an application of the latent order book model to the optimal execution problem of a portfolio of assets, in the context of liquidity stress tests.

We study the Bellman equation in the Wasserstein space arising in the study of mean field control problems, namely stochastic optimal control problems for McKean-Vlasov diffusion processes. Using the standard notion of viscosity solution à la Crandall-Lions extended to our Wasserstein setting, we prove a comparison result under general conditions, which coupled with the dynamic programming principle, implies that the value function is the unique viscosity solution of the Master Bellman equation. This is the first uniqueness result in such a second-order context. The classical arguments used in the standard cases of equations in finite-dimensional spaces or in infinite-dimensional separable Hilbert spaces do not extend to the present framework, due to the awkward nature of the underlying Wasserstein space. The adopted strategy is based on finite-dimensional approximations of the value function obtained in terms of the related cooperative n-player game, and on the construction of a smooth gauge-type function, built starting from a regularization of a sharpe estimate of the Wasserstein metric. such a gauge-type function is used to generate maxima/minima through a suitable extension of the Borwein-Preiss generalization of Ekeland's variational principle on the Wasserstein space.

This thesis is divided into four parts that can be read independently. In this manuscript, we make some contributions to the theoretical study and to the applications in finance of optimal quantization. In the first part, we recall the theoretical foundations of optimal quantization as well as the classical numerical methods to construct optimal quantizers. The second part focuses on the numerical integration problem in dimension 1, which arises when one wishes to compute numerically expectations, such as in the valuation of derivatives. We recall the existing strong and weak error results and extend the results of the second order convergences to other classes of less regular functions. In a second part, we present a weak error result in dimension 1 and a second development in higher dimension for a product quantizer. In the third part, we focus on a first numerical application. We introduce a stationary Heston model in which the initial condition of the volatility is assumed to be random with the stationary distribution of the EDS of the CIR governing the volatility. This variant of the original Heston model produces a more pronounced implied volatility smile for European options on short maturities than the standard model. We then develop a numerical method based on recursive quantization produced for the evaluation of Bermudian and barrier options. The fourth and last part deals with a second numerical application, the valuation of Bermudian options on exchange rates in a 3-factor model. These products are known in the markets as PRDCs. We propose two schemes to evaluate this type of options, both based on optimal product quantization and establish a priori error estimates.

We propose a microstructural modeling framework for studying optimal market making policies in a FIFO (first in first out) limit order book (LOB). In this context, the limit orders, market orders, and cancel orders arrivals in the LOB are modeled as Cox point processes with intensities that only depend on the state of the LOB. These are high-dimensional models which are realistic from a micro-structure point of view and have been recently developed in the literature. In this context, we consider a market maker who stands ready to buy and sell stock on a regular and continuous basis at a publicly quoted price, and identifies the strategies that maximize her P&L penalized by her inventory. We apply the theory of Markov Decision Processes and dynamic programming method to characterize analytically the solutions to our optimal market making problem. The second part of the paper deals with the numerical aspect of the high-dimensional trading problem. We use a control randomization method combined with quantization method to compute the optimal strategies. Several computational tests are performed on simulated data to illustrate the efficiency of the computed optimal strategy. In particular, we simulated an order book with constant/ symmet-ric/ asymmetrical/ state dependent intensities, and compared the computed optimal strategy with naive strategies.

This paper studies a variation of the continuous-time mean-variance portfolio selection where a tracking-error penalization is added to the mean-variance criterion. The tracking error term penalizes the distance between the allocation controls and a refe\-rence portfolio with same wealth and fixed weights. Such consideration is motivated as fo\-llows: (i) On the one hand, it is a way to robustify the mean-variance allocation in case of misspecified parameters, by ``fitting" it to a reference portfolio that can be agnostic to market parameters. (ii) On the other hand, it is a procedure to track a benchmark and improve the Sharpe ratio of the resulting portfolio by considering a mean-variance criterion in the objective function. This problem is formulated as a McKean-Vlasov control problem. We provide explicit solutions for the optimal portfolio strategy and asymptotic expansions of the portfolio strategy and efficient frontier for small values of the tracking error parameter. Finally, we compare the Sharpe ratios obtained by the standard mean-variance allocation and the penalized one for four different reference portfolios: equal-weights, minimum-variance, equal risk contributions and shrinking portfolio. This comparison is done on a simulated misspecified model, and on a backtest performed with historical data. Our results show that in most cases, the penalized portfolio outperforms in terms of Sharpe ratio both the standard mean-variance and the reference portfolio.

This paper studies a variation of the continuous-time mean-variance portfolio selection where a tracking-error penalization is added to the mean-variance criterion. The tracking error term penalizes the distance between the allocation controls and a refe\-rence portfolio with same wealth and fixed weights. Such consideration is motivated as fo\-llows: (i) On the one hand, it is a way to robustify the mean-variance allocation in case of misspecified parameters, by ``fitting" it to a reference portfolio that can be agnostic to market parameters. (ii) On the other hand, it is a procedure to track a benchmark and improve the Sharpe ratio of the resulting portfolio by considering a mean-variance criterion in the objective function. This problem is formulated as a McKean-Vlasov control problem. We provide explicit solutions for the optimal portfolio strategy and asymptotic expansions of the portfolio strategy and efficient frontier for small values of the tracking error parameter. Finally, we compare the Sharpe ratios obtained by the standard mean-variance allocation and the penalized one for four different reference portfolios: equal-weights, minimum-variance, equal risk contributions and shrinking portfolio. This comparison is done on a simulated misspecified model, and on a backtest performed with historical data. Our results show that in most cases, the penalized portfolio outperforms in terms of Sharpe ratio both the standard mean-variance and the reference portfolio.

This paper concerns portfolio selection with multiple assets under rough covariance matrix. We investigate the continuous-time Markowitz mean-variance problem for a multivariate class of affine and quadratic Volterra models. In this incomplete non-Markovian and non-semimartingale market framework with unbounded random coefficients, the optimal portfolio strategy is expressed by means of a Riccati backward stochastic differential equation (BSDE). In the case of affine Volterra models, we derive explicit solutions to this BSDE in terms of multi-dimensional Riccati-Volterra equations. This framework includes multivariate rough Heston models and extends the results of \cite{han2019mean}. In the quadratic case, we obtain new analytic formulae for the the Riccati BSDE and we establish their link with infinite dimensional Riccati equations. This covers rough Stein-Stein and Wishart type covariance models. Numerical results on a two dimensional rough Stein-Stein model illustrate the impact of rough volatilities and stochastic correlations on the optimal Markowitz strategy. In particular for positively correlated assets, we find that the optimal strategy in our model is a `buy rough sell smooth' one.

This paper concerns portfolio selection with multiple assets under rough covariance matrix. We investigate the continuous-time Markowitz mean-variance problem for a multivariate class of affine and quadratic Volterra models. In this incomplete non-Markovian and non-semimartingale market framework with unbounded random coefficients, the optimal portfolio strategy is expressed by means of a Riccati backward stochastic differential equation (BSDE). In the case of affine Volterra models, we derive explicit solutions to this BSDE in terms of multi-dimensional Riccati-Volterra equations. This framework includes multivariate rough Heston models and extends the results of \cite{han2019mean}. In the quadratic case, we obtain new analytic formulae for the the Riccati BSDE and we establish their link with infinite dimensional Riccati equations. This covers rough Stein-Stein and Wishart type covariance models. Numerical results on a two dimensional rough Stein-Stein model illustrate the impact of rough volatilities and stochastic correlations on the optimal Markowitz strategy. In particular for positively correlated assets, we find that the optimal strategy in our model is a `buy rough sell smooth' one.

This paper studies a variation of the continuous-time mean-variance portfolio selection where a tracking-error penalization is added to the mean-variance criterion. The tracking error term penalizes the distance between the allocation controls and a refe\-rence portfolio with same wealth and fixed weights. Such consideration is motivated as fo\-llows: (i) On the one hand, it is a way to robustify the mean-variance allocation in case of misspecified parameters, by ``fitting" it to a reference portfolio that can be agnostic to market parameters. (ii) On the other hand, it is a procedure to track a benchmark and improve the Sharpe ratio of the resulting portfolio by considering a mean-variance criterion in the objective function. This problem is formulated as a McKean-Vlasov control problem. We provide explicit solutions for the optimal portfolio strategy and asymptotic expansions of the portfolio strategy and efficient frontier for small values of the tracking error parameter. Finally, we compare the Sharpe ratios obtained by the standard mean-variance allocation and the penalized one for four different reference portfolios: equal-weights, minimum-variance, equal risk contributions and shrinking portfolio. This comparison is done on a simulated misspecified model, and on a backtest performed with historical data. Our results show that in most cases, the penalized portfolio outperforms in terms of Sharpe ratio both the standard mean-variance and the reference portfolio.

We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate simultaneously the solution and its gradient by deep neural networks. These approximations are performed at each time step from the minimization of loss functions defined recursively by backward induction. The methodology is extended to variational inequalities arising in optimal stopping problems. We analyze the convergence of the deep learning schemes and provide error estimates in terms of the universal approximation of neural networks. Numerical results show that our algorithms give very good results till dimension 50 (and certainly above), for both PDEs and variational inequalities problems. For the PDEs resolution, our results are very similar to those obtained by the recent method in \cite{weinan2017deep} when the latter converges to the right solution or does not diverge. Numerical tests indicate that the proposed methods are not stuck in poor local minima as it can be the case with the algorithm designed in \cite{weinan2017deep}, and no divergence is experienced. The only limitation seems to be due to the inability of the considered deep neural networks to represent a solution with a too complex structure in high dimension.

We develop multistep machine learning schemes for solving nonlinear partial differential equations (PDEs) in high dimension. The method is based on probabilistic representation of PDEs by backward stochastic differential equations (BSDEs) and its iterated time discretization. In the case of semilinear PDEs, our algorithm estimates simultaneously by backward induction the solution and its gradient by neural networks through sequential minimizations of suitable quadratic loss functions that are performed by stochastic gradient descent. The approach is extended to the more challenging case of fully nonlinear PDEs, and we propose different approximations of the Hessian of the solution to the PDE, i.e., the $\Gamma$-component of the BSDE, by combining Malliavin weights and neural networks. Extensive numerical tests are carried out with various examples of semilinear PDEs including viscous Burgers equation and examples of fully nonlinear PDEs like Hamilton-Jacobi-Bellman equations arising in portfolio selection problems with stochastic volatilities, or Monge-Ampère equations in dimension up to 15. The performance and accuracy of our numerical results are compared with some other recent machine learning algorithms in the literature, see \cite{HJE17}, \cite{HPW19}, \cite{BEJ19}, \cite{BBCJN19} and \cite{phawar19}. Furthermore, we provide a rigorous approximation error analysis of the deep backward multistep scheme as well as the deep splitting method for semilinear PDEs, which yields convergence rate in terms of the number of neurons for shallow neural networks.

No summary available.

This thesis is formulated in three parts with eight chapters and presents a research theme dealing with controlled processes/particles/interacting agents.In the first part of the thesis, we focus our attention on the study of interacting controlled processes representing a cooperative equilibrium, also known as Pareto equilibrium. A cooperative equilibrium can be seen as a situation where there is no way to improve the preference criterion of one agent without lowering the preference criterion of at least one other agent. It is now well known that this type of optimization problem is related, when the number of agents goes to infinity, to McKean-Vlasov optimal control. In the first three chapters of this thesis, we provide a precise mathematical answer to the link between these two optimization problems in different frameworks improving the existing literature, in particular by taking into account the control law while allowing a common noise situation.After studying the behavior of cooperative equilibria, we conclude the first part where we spend time in the analysis of the limit problem i.e. McKean-Vlasov optimal control, through the establishment of the dynamic programming principle (DPP) for this stochastic control problem.The second part of this thesis is devoted to the study of interacting controlled processes now representing a Nash equilibrium, also known as competitive equilibrium. A Nash equilibrium situation in a game is one in which no one has anything to gain by unilaterally leaving his own position. Since the pioneering work of Larsy - Lions and Huang - Malhamé - Caines, the behavior of Nash equilibria when the number of agents reaches infinity has been intensively studied and the associated limit game is known as Mean Field Games (MFG). In this second part, we first analyze the convergence of competitive equilibria to MFGs in a framework with the control law and with volatility control, then, the question of the existence of the MFG equilibrium in this context is studied.Finally, the last part, which consists of only one chapter, is devoted to some numerical methods for solving the limit problem i.e. McKean - Vlasov optimal control. Inspired by the proof of convergence of the cooperative equilibrium, we give a numerical algorithm to solve the McKean-Vlasov optimal control problem and prove its convergence. Then, we implement our algorithm using neural networks and test its efficiency on some application examples, namely mean-variance portfolio selection, the interbank systemic risk model and optimal liquidation with market impact.

This thesis is a study of various optimal portfolio allocation problems where the rate of appreciation, called drift, of the Brownian motion of asset dynamics is uncertain. We consider an investor with a belief about drift in the form of a probability distribution, called a priori. Uncertainty about drift is taken into account by a Bayesian learning approach that updates the a priori probability distribution of drift. The thesis is divided into two independent parts. The first part contains two chapters: the first one develops the theoretical results, and the second one contains a detailed application of these results on market data. The first part of the thesis is devoted to the Markowitz portfolio selection problem in the multidimensional case with drift uncertainty. This uncertainty is modeled via an arbitrary a priori law that is updated using Bayesian filtering. We first transform the Bayesian Markowitz problem into a standard auxiliary control problem for which dynamic programming is applied. Then, we show the existence and uniqueness of a regular solution to the associated semi-linear partial differential equation (PDE). In the case of an a priori Gaussian distribution, the multidimensional solution is explicitly computed. Moreover, we study the quantitative impact of learning from the progressively observed data, by comparing the strategy that updates the drift estimate, called learning strategy, to the one that keeps it constant, called nonlearning strategy. Finally, we analyze the sensitivity of the learning gain, called information value, to different parameters. We then illustrate the theory with a detailed application of the previous results to historical market data. We highlight the robustness of the added value of learning by comparing the optimal learning and non-learning strategies in different investment universes: indices of different asset classes, currencies and smart beta strategies. The second part deals with a discrete time portfolio optimization problem. Here, the investor's objective is to maximize the expected utility of the terminal wealth of a portfolio of risky assets, assuming an uncertain drift and a maximum drawdown constraint satisfied. In this section, we formulate the problem in the general case, and we numerically solve the Gaussian case with the constant relative risk aversion (CRRA) utility function, via a deep learning algorithm. Finally, we study the sensitivity of the strategy to the degree of uncertainty surrounding the drift estimate and empirically illustrate the convergence of the unlearned strategy to a constrained Merton problem, without short selling.

We study the optimal control of path-dependent McKean-Vlasov equations valued in Hilbert spaces motivated by non Markovian mean-field models driven by stochastic PDEs. We first establish the well-posedness of the state equation, and then we prove the dynamic programming principle (DPP) in such a general framework. The crucial law invariance property of the value function V is rigorously obtained, which means that V can be viewed as a function on the Wasserstein space of probability measures on the set of continuous functions valued in Hilbert space. We then define a notion of pathwise measure derivative, which extends the Wasserstein derivative due to Lions [41], and prove a related functional Itô formula in the spirit of Dupire [24] and Wu and Zhang [51]. The Master Bellman equation is derived from the DPP by means of a suitable notion of viscosity solution. We provide different formulations and simplifications of such a Bellman equation notably in the special case when there is no dependence on the law of the control.

We formulate an equilibrium model of intraday trading in electricity markets. Agents face balancing constraints between their customers consumption plus intraday sales and their production plus intraday purchases. They have continuously updated forecast of their customers consumption at maturity with decreasing volatility error. Forecasts are prone to idiosyncratic noise as well as common noise (weather). Agents production capacities are subject to independent random outages, which are each modelled by a Markov chain. The equilibrium price is defined as the price that minimises trading cost plus imbalance cost of each agent and satisfies the usual market clearing condition. Existence and uniqueness of the equilibrium are proved, and we show that the equilibrium price and the optimal trading strategies are martingales. The main economic insights are the following. (i) When there is no uncertainty on generation, it is shown that the market price is a convex combination of forecasted marginal cost of each agent, with deterministic weights. Furthermore, the equilibrium market price follows Almgren and Chriss's model and we identify the fundamental part as well as the permanent market impact. It turns out that heterogeneity across agents is a necessary condition for the Samuelson's effect to hold. (ii) When there is production uncertainty, the price volatility becomes stochastic but converges to the case without production uncertainty when the number of agents increases to infinity. Further, on a two-agent case, we show that the potential outages of a low marginal cost producer reduces her sales position.

This thesis proposes models and methods to study risk control in large financial systems. In the first part, we propose a structural approach: we consider a financial system represented as a network of institutions connected to each other by strategic interactions that are sources of funding but also by interactions that expose them to default contagion risk. The novelty of our approach lies in the fact that these two types of interactions interfere. We propose new notions of equilibrium for these systems and study the optimal connectivity of the network and the associated systemic risk. In a second part, we introduce systemic risk measures defined by backward stochastic differential equations directed by mean-field operators and study associated optimal stopping problems. The last part deals with optimal portfolio liquidation issues.

The thesis deals with numerical schemes for Markovian decision problems (MDPs), partial differential equations (PDEs), backward stochastic differential equations (SRs), as well as reflected backward stochastic differential equations (SRDEs). The thesis is divided into three parts.The first part deals with numerical methods for solving MDPs, based on quantization and local or global regression. A market-making problem is proposed: it is solved theoretically by rewriting it as an MDP. and numerically by using the new algorithm. In a second step, a Markovian embedding method is proposed to reduce McKean-Vlasov type probabilities with partial information to MDPs. This method is implemented on three different McKean-Vlasov type problems with partial information, which are then numerically solved using numerical methods based on regression and quantization.In the second part, new algorithms are proposed to solve MDPs in high dimension. The latter are based on neural networks, which have proven in practice to be the best for learning high dimensional functions. The consistency of the proposed algorithms is proved, and they are tested on many stochastic control problems, which allows to illustrate their performances.In the third part, we focus on methods based on neural networks to solve PDEs, EDSRs and reflected EDSRs. The convergence of the proposed algorithms is proved and they are compared to other recent algorithms of the literature on some examples, which allows to illustrate their very good performances.

No summary available.

We develop an exhaustive study of Markov decision process (MDP) under mean field interaction both on states and actions in the presence of common noise, and when optimization is performed over open-loop controls on infinite horizon. Such model, called CMKV-MDP for conditional McKean-Vlasov MDP, arises and is obtained here rigorously with a rate of convergence as the asymptotic problem of N-cooperative agents controlled by a social planner/influencer that observes the environment noises but not necessarily the individual states of the agents. We highlight the crucial role of relaxed controls and randomization hypothesis for this class of models with respect to classical MDP theory. We prove the correspondence between CMKV-MDP and a general lifted MDP on the space of probability measures, and establish the dynamic programming Bellman fixed point equation satisfied by the value function, as well as the existence of-optimal randomized feedback controls. The arguments of proof involve an original measurable optimal coupling for the Wasserstein distance. This provides a procedure for learning strategies in a large population of interacting collaborative agents. MSC Classification: 90C40, 49L20.

We establish existence and uniqueness for infinite dimensional Riccati equations taking values in the Banach space L 1 (µ ⊗ µ) for certain signed matrix measures µ which are not necessarily finite. Such equations can be seen as the infinite dimensional analogue of matrix Riccati equations and they appear in the Linear-Quadratic control theory of stochastic Volterra equations.

We provide an exhaustive treatment of Linear-Quadratic control problems for a class of stochastic Volterra equations of convolution type, whose kernels are Laplace transforms of certain signed matrix measures which are not necessarily finite. These equations are in general neither Markovian nor semimartingales, and include the fractional Brownian motion with Hurst index smaller than $1/2$ as a special case. We establish the correspondence of the initial problem with a possibly infinite dimensional Markovian one in a Banach space, which allows us to identify the Markovian controlled state variables. Using a refined martingale verification argument combined with a squares completion technique, we prove that the value function is of linear quadratic form in these state variables with a linear optimal feedback control, depending on non-standard Banach space valued Riccati equations. Furthermore, we show that the value function of the stochastic Volterra optimization problem can be approximated by that of conventional finite dimensional Markovian Linear--Quadratic problems, which is of crucial importance for numerical implementation.

In this thesis, we make some contributions to the theoretical and numerical study of some stochastic control problems, as well as their applications to financial mathematics and financial risk management. These applications concern problems of valuation and weak hedging of financial products, as well as regulatory issues. We propose numerical methods to efficiently compute these quantities for which no explicit formula exists. Finally, we study backward stochastic differential equations related to new switching problems with cost uncertainty.

We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, through a sequence of learning problems obtained from the minimization of suitable quadratic loss functions and training simulations. This methodology extends to the fully non-linear case the approach recently proposed in [HPW19] for semi-linear PDEs. Numerical tests illustrate the performance and accuracy of our method on several examples in high dimension with nonlinearity on the Hessian term including a linear quadratic control problem with control on the diffusion coefficient.

We propose a microstructural modeling framework for studying optimal market making policies in a FIFO (first in first out) limit order book (LOB). In this context, the limit orders, market orders, and cancel orders arrivals in the LOB are modeled as Cox point processes with intensities that only depend on the state of the LOB. These are high-dimensional models which are realistic from a micro-structure point of view and have been recently developed in the literature. In this context, we consider a market maker who stands ready to buy and sell stock on a regular and continuous basis at a publicly quoted price, and identifies the strategies that maximize her P&L penalized by her inventory. We apply the theory of Markov Decision Processes and dynamic programming method to characterize analytically the solutions to our optimal market making problem. The second part of the paper deals with the numerical aspect of the high-dimensional trading problem. We use a control randomization method combined with quantization method to compute the optimal strategies. Several computational tests are performed on simulated data to illustrate the efficiency of the computed optimal strategy. In particular, we simulated an order book with constant/ symmet-ric/ asymmetrical/ state dependent intensities, and compared the computed optimal strategy with naive strategies.

This paper presents several numerical applications of deep learning-based algorithms that have been introduced in [HPBL18]. Numerical and comparative tests using TensorFlow illustrate the performance of our different algorithms, namely control learning by performance iteration (algorithms NNcontPI and ClassifPI), control learning by hybrid iteration (algorithms Hybrid-Now and Hybrid-LaterQ), on the 100-dimensional nonlinear PDEs examples from [EHJ17] and on quadratic backward stochastic differential equations as in [CR16]. We also performed tests on low-dimension control problems such as an option hedging problem in finance, as well as energy storage problems arising in the valuation of gas storage and in microgrid management. Numerical results and comparisons to quantization-type algorithms Qknn, as an efficient algorithm to numerically solve low-dimensional control problems, are also provided. and some corresponding codes are available on https://github.com/comeh/.

We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate simultaneously the solution and its gradient by deep neural networks. These approximations are performed at each time step from the minimization of loss functions defined recursively by backward induction. The methodology is extended to variational inequalities arising in optimal stopping problems. We analyze the convergence of the deep learning schemes and provide error estimates in terms of the universal approximation of neural networks. Numerical results show that our algorithms give very good results till dimension 50 (and certainly above), for both PDEs and variational inequalities problems. For the PDEs resolution, our results are very similar to those obtained by the recent method in \cite{weinan2017deep} when the latter converges to the right solution or does not diverge. Numerical tests indicate that the proposed methods are not stuck in poor local minima as it can be the case with the algorithm designed in \cite{weinan2017deep}, and no divergence is experienced. The only limitation seems to be due to the inability of the considered deep neural networks to represent a solution with a too complex structure in high dimension.

We study the Markowitz portfolio selection problem with unknown drift vector in the multidimensional framework. The prior belief on the uncertain expected rate of return is modeled by an arbitrary probability law, and a Bayesian approach from filtering theory is used to learn the posterior distribution about the drift given the observed market data of the assets. The Bayesian Markowitz problem is then embedded into an auxiliary standard control problem that we characterize by a dynamic programming method and prove the existence and uniqueness of a smooth solution to the related semi-linear partial differential equation (PDE). The optimal Markowitz portfolio strategy is explicitly computed in the case of a Gaussian prior distribution. Finally, we measure the quantitative impact of learning, updating the strategy from observed data, compared to non-learning, using a constant drift in an uncertain context, and analyze the sensitivity of the value of information w.r.t. various relevant parameters of our model.

We consider a multi-player stochastic differential game with linear McKean-Vlasov dynamics and quadratic cost functional depending on the variance and mean of the state and control actions of the players in open-loop form. Finite and infinite horizon problems with possibly some random coefficients as well as common noise are addressed. We propose a simple direct approach based on weak martingale optimality principle together with a fixed point argument in the space of controls for solving this game problem. The Nash equilibria are characterized in terms of systems of Riccati ordinary differential equations and linear mean-field backward stochastic differential equations: existence and uniqueness conditions are provided for such systems. Finally, we illustrate our results on a toy example.

We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in [4], we prove a corresponding randomized dynamic programming principle (DPP) for the value function, which is obtained from a flow property of an associated filter process. This DPP is the key step towards our main result: a characterization of the value function of the partial observation control problem as the unique viscosity solution to the corresponding dynamic programming Hamilton-Jacobi-Bellman (HJB) equation. The latter is formulated as a new, fully non linear partial differential equation on the Wasserstein space of probability measures. An important feature of our approach is that it does not require any non-degeneracy condition on the diffusion coefficient, and no condition is imposed to guarantee existence of a density for the filter process solution to the controlled Zakai equation, as usually done for the separated problem. Finally, we give an explicit solution to our HJB equation in the case of a partially observed non Gaussian linear quadratic model.

We address a class of McKean-Vlasov (MKV) control problems with common noise, called polynomial conditional MKV, and extending the known class of linear quadratic stochastic MKV control problems. We show how this polynomial class can be reduced by suitable Markov embedding to finite-dimensional stochastic control problems, and provide a discussion and comparison of three probabilistic numerical methods for solving the reduced control problem: quantization, regression by control randomization, and regress later methods. Our numerical results are illustrated on various examples from portfolio selection and liquidation under drift uncertainty, and a model of interbank systemic risk with partial observation.

We study zero-sum stochastic differential games where the state dynamics of the two players is governed by a generalized McKean-Vlasov (or mean-field) stochastic differential equation in which the distribution of both state and controls of each player appears in the drift and diffusion coefficients, as well as in the running and terminal payoff functions. We prove the dynamic programming principle (DPP) in this general setting, which also includes the control case with only one player, where it is the first time that DPP is proved for open-loop controls. We also show that the upper and lower value functions are viscosity solutions to a corresponding upper and lower Master Bellman-Isaacs equation. Our results extend the seminal work of Fleming and Souganidis [15] to the McKean-Vlasov setting.

This paper focuses on a dynamic multi-asset mean-variance portfolio selection problem under model uncertainty. We develop a continuous time framework for taking into account ambiguity aversion about both expected return rates and correlation matrix of the assets, and for studying the join effects on portfolio diversification. The dynamic setting allows us to consider time varying ambiguity sets, which include the cases where the drift and correlation are estimated on a rolling window of historical data or when the investor takes into account learning on the ambiguity. In this context, we prove a general separation principle for the associated robust control problem, which allows us to reduce the determination of the optimal dynamic strategy to the parametric computation of the minimal risk premium function. Our results provide a justification for under-diversification, as documented in empirical studies and in the static models [16], [34]. Furthermore, we explicitly quantify the degree of under-diversification in terms of correlation bounds and Sharpe ratios proximities, and emphasize the different features induced by drift and correlation ambiguity. In particular, we show that an investor with a poor confidence in the expected return estimation does not hold any risky asset, and on the other hand, trades only one risky asset when the level of ambiguity on correlation matrix is large. We also provide a complete picture of the diversification for the optimal robust portfolio in the three-asset case JEL Classification: G11, C61 MSC Classification: 91G10, 91G80, 60H30.

In this thesis, we study several financial mathematics problems related to the valuation of derivatives. Through different asymptotic approaches, we develop methods to compute accurate approximations of the price of certain types of options in cases where no explicit formula exists.In the first chapter, we focus on the valuation of options whose payoff depends on the trajectory of the underlying by Monte Carlo methods, when the underlying is modeled by an affine process with stochastic volatility. We prove a principle of large trajectory deviations in long time, which we use to compute, using Varadhan's lemma, an asymptotically optimal change of measure, allowing to significantly reduce the variance of the Monte-Carlo estimator of option prices.The second chapter considers the valuation by Monte-Carlo methods of options depending on multiple underlyings, such as basket options, in Wishart's stochastic volatility model, which generalizes the Heston model. Following the same approach as in the previous chapter, we prove that the process vérifie a principle of large deviations in long time, which we use to significantly reduce the variance of the Monte Carlo estimator of option prices, through an asymptotically optimal change of measure. In parallel, we use the principle of large deviations to characterize the long-time behavior of the Black-Scholes implied volatility of basket options.In the third chapter, we study the valuation of realized variance options, when the spot volatility is modeled by a constant volatility diffusion process. We use recent asymptotic results on the densities of hypo-elliptic diffusions to compute an expansion of the realized variance density, which we integrate to obtain the expansion of the option price, and then their Black-Scholes implied volatility.The final chapter is devoted to the valuation of interest rate derivatives in the Lévy model of the Libor market, which generalizes the classical Libor market model (log-normal) by adding jumps. By writing the former as a perturbation of the latter and using the Feynman-Kac representation, we explicitly compute the asymptotic expansion of the price of interest rate derivatives, in particular, caplets and swaptions.

This paper studies a robust continuous-time Markowitz portfolio selection pro\-blem where the model uncertainty carries on the covariance matrix of multiple risky assets. This problem is formulated into a min-max mean-variance problem over a set of non-dominated probability measures that is solved by a McKean-Vlasov dynamic programming approach, which allows us to characterize the solution in terms of a Bellman-Isaacs equation in the Wasserstein space of probability measures. We provide explicit solutions for the optimal robust portfolio strategies and illustrate our results in the case of uncertain volatilities and ambiguous correlation between two risky assets. We then derive the robust efficient frontier in closed-form, and obtain a lower bound for the Sharpe ratio of any robust efficient portfolio strategy. Finally, we compare the performance of Sharpe ratios for a robust investor and for an investor with a misspecified model. MSC Classification: 91G10, 91G80, 60H30.

We consider a multi-player stochastic differential game with linear McKean-Vlasov dynamics and quadratic cost functional depending on the variance and mean of the state and control actions of the players in open-loop form. Finite and infinite horizon problems with possibly some random coefficients as well as common noise are addressed. We propose a simple direct approach based on weak martingale optimality principle together with a fixed point argument in the space of controls for solving this game problem. The Nash equilibria are characterized in terms of systems of Riccati ordinary differential equations and linear mean-field backward stochastic differential equations: existence and uniqueness conditions are provided for such systems. Finally, we illustrate our results on a toy example.

We propose a simple and original approach for solving linear-quadratic mean-field stochastic control problems. We study both finite-horizon and infinite-horizon pro\-blems, and allow notably some coefficients to be stochastic. Extension to the common noise case is also addressed. Our method is based on a suitable version of the martingale formulation for verification theorems in control theory. The optimal control involves the solution to a system of Riccati ordinary differential equations and to a linear mean-field backward stochastic differential equation.

We study the Markowitz portfolio selection problem with unknown drift vector in the multidimensional framework. The prior belief on the uncertain expected rate of return is modeled by an arbitrary probability law, and a Bayesian approach from filtering theory is used to learn the posterior distribution about the drift given the observed market data of the assets. The Bayesian Markowitz problem is then embedded into an auxiliary standard control problem that we characterize by a dynamic programming method and prove the existence and uniqueness of a smooth solution to the related semi-linear partial differential equation (PDE). The optimal Markowitz portfolio strategy is explicitly computed in the case of a Gaussian prior distribution. Finally, we measure the quantitative impact of learning, updating the strategy from observed data, compared to non-learning, using a constant drift in an uncertain context, and analyze the sensitivity of the value of information w.r.t. various relevant parameters of our model.

We consider the stochastic optimal control problem of McKean-Vlasov stochastic differential equation where the coefficients may depend upon the joint law of the state and control. By using feedback controls, we reformulate the problem into a deterministic control problem with only the marginal distribution of the process as controlled state variable, and prove that dynamic programming principle holds in its general form. Then, by relying on the notion of differentiability with respect to pro\-bability measures recently introduced by P.L. Lions in [32], and a special Itô formula for flows of probability measures, we derive the (dynamic programming) Bellman equation for mean-field stochastic control problem, and prove a veri\-fication theorem in our McKean-Vlasov framework. We give explicit solutions to the Bellman equation for the linear quadratic mean-field control problem, with applications to the mean-variance portfolio selection and a systemic risk model. We also consider a notion of lifted visc-sity solutions for the Bellman equation, and show the viscosity property and uniqueness of the value function to the McKean-Vlasov control problem. Finally, we consider the case of McKean-Vlasov control problem with open-loop controls and discuss the associated dynamic programming equation that we compare with the case of closed-loop controls.

No summary available.

No summary available.

We study zero-sum stochastic differential games where the state dynamics of the two players is governed by a generalized McKean-Vlasov (or mean-field) stochastic differential equation in which the distribution of both state and controls of each player appears in the drift and diffusion coefficients, as well as in the running and terminal payoff functions. We prove the dynamic programming principle (DPP) in this general setting, which also includes the control case with only one player, where it is the first time that DPP is proved for open-loop controls. We also show that the upper and lower value functions are viscosity solutions to a corresponding upper and lower Master Bellman-Isaacs equation. Our results extend the seminal work of Fleming and Souganidis [15] to the McKean-Vlasov setting.

We propose a simple and original approach for solving linear-quadratic mean-field stochastic control problems. We study both finite-horizon and infinite-horizon pro\-blems, and allow notably some coefficients to be stochastic. Extension to the common noise case is also addressed. Our method is based on a suitable version of the martingale formulation for verification theorems in control theory. The optimal control involves the solution to a system of Riccati ordinary differential equations and to a linear mean-field backward stochastic differential equation.

This thesis is composed of three chapters that deal with impulse control problems. In the first chapter, we introduce a general framework for impulse control with uncertainty. Knowing an a priori law on unknown parameters, we explain how it should evolve and integrate it to the optimal control problem. We characterize the solution through a quasivariational parabolic equation that can be solved numerically and give examples of applications to finance. In the second chapter, we introduce an impulse control problem with uncertainty in an actuarial setting. An (re)insurer faces natural catastrophes and can issue CAT bonds to reduce the risk taken. We again characterize the optimal control problem through a numerically solvable quasi-variational parabolic equation and give some application examples. In the last chapter, we propose a model of the price through a completely endogenous order book. We solve impulse optimal control problems (order placement) of rational economic agents that we gather on a same market.

This thesis studies the optimal control of McKean-Vlasov type dynamics and its applications in financial mathematics. The thesis contains two parts. In the first part, we develop the dynamic programming method for solving McKean-Vlasov type stochastic control problems. By using the appropriate admissible controls, we can reformulate the value function in terms of the law (resp. the conditional law) of the process as the only state variable and obtain the flow property of the law (resp. the conditional law) of the process, which allow us to obtain the principle of dynamic programming in all generality. Then we obtain the corresponding Bellman equation, based on the notion of differentiability with respect to probability measures introduced by P.L. Lions [Lio12] and the Itô formula for the probability stream. Finally we show the viscosity property and the uniqueness of the value function of the Bellman equation. In the first chapter, we summarize some useful results from differential calculus and stochastic analysis on the Wasserstein space. In the second chapter, we consider stochastic optimal control of nonlinear mean-field systems in discrete time. The third chapter studies the stochastic optimal control problem of McKean-Vlasov type EDS without common noise in continuous time where the coefficients can depend on the joint state and control law, and finally in the last chapter of this part we are interested in the optimal control of McKean-Vlasov type stochastic dynamics in the presence of common noise in continuous time. In the second part, we propose a robust portfolio allocation model allowing for uncertainty in the expected return and the correlation matrix of multiple assets, in a continuous time mean-variance framework. This problem is formulated as a mean-field differential game. We then show a separation principle for the associated problem. Our explicit results provide a quantitative justification for underdiversification, as shown in empirical studies.

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming (DP). Differently from the classical approximate DP approach, we first approximate the optimal policy by means of neural networks in the spirit of deep reinforcement learning, and then the value function by Monte Carlo regression. This is achieved in the DP recursion by performance or hybrid iteration, and regress now or later/quantization methods from numerical probabilities. We provide a theoretical justification of these algorithms. Consistency and rate of convergence for the control and value function estimates are analyzed and expressed in terms of the universal approximation error of the neural networks. Numerical results on various applications are presented in a companion paper [2] and illustrate the performance of our algorithms.

We address a class of McKean-Vlasov (MKV) control problems with common noise, called polynomial conditional MKV, and extending the known class of linear quadratic stochastic MKV control problems. We show how this polynomial class can be reduced by suitable Markov embedding to finite-dimensional stochastic control problems, and provide a discussion and comparison of three probabilistic numerical methods for solving the reduced control problem: quantization, regression by control randomization, and regress later methods. Our numerical results are illustrated on various examples from portfolio selection and liquidation under drift uncertainty, and a model of interbank systemic risk with partial observation.

Finite-horizon stochastic optimal control problems are a class of optimal control problems involving stochastic processes considered over a bounded time interval. Like many optimal control problems, these problems are solved using the principle of dynamic programming which induces a partial differential equation (PDE) called Hamilton-Jacobi-Bellman equation. Methods based on the discretization of the space in the form of a grid, probabilistic methods or more recently max-plus methods can then be used to solve this equation. However, the first type of method fails when a high dimensional space is considered because of the curse of the dimension while the second type of method has so far only allowed to solve problems where the non-linearity of the partial differential equation with respect to the Hessian is not too strong. As for the third type of method, it leads to an explosion of the complexity of the value function. In this thesis, we introduce two new probabilistic schemes to enlarge the class of problems that can be solved by probabilistic methods. One is adapted to PDEs with bounded coefficients while the other can be applied to PDEs with bounded or unbounded coefficients. We prove the convergence of both probabilistic schemes and obtain estimates of the convergence error in the case of PDEs with bounded coefficients. We also give some results on the behavior of the second scheme in the case of PDEs with unbounded coefficients. Then, we introduce a completely new method for solving finite horizon stochastic optimal control problems which we call the max-plus probabilistic method. It allows to use the nonlinear character of max-plus methods in a probabilistic context while controlling the complexity of the value function. An application to the computation of the over-replication price of an option in an uncertain correlation model is given in the case of a 2 and 5 dimensional space.

This thesis focuses on valuation and financial strategies for hedging risks in energy markets. These markets present particularities that distinguish them from standard financial markets, notably illiquidity and incompleteness. Illiquidity is reflected in high transaction costs and constraints on volumes traded. Incompleteness is the inability to perfectly replicate derivatives. We focus on different aspects of market incompleteness. The first part deals with valuation in Lévy models. We obtain an approximate formula for the indifference price and we measure the minimum premium to be brought over the Black-Scholes model. The second part concerns the valuation of spread options in the presence of stochastic correlation. Spread options deal with the price difference between two underlying assets -- for example gas and electricity -- and are widely used in the energy markets. We propose an efficient numerical procedure to calculate the price of these options. Finally, the third part deals with the valuation of a product with an exogenous risk for which forecasts exist. We propose an optimal dynamic strategy in the presence of volume risk, and apply it to the valuation of wind farms. In addition, a section is devoted to asymptotic optimal strategies in the presence of transaction costs.

No summary available.

The classical theory of derivatives valuation is based on the absence of transaction costs and infinite liquidity. However, these assumptions are no longer true in the real market, especially when the transaction is large and the assets illiquid. The first part of this thesis focuses on proposing a model that incorporates both the transaction cost and the impact on the price of the underlying asset. We start by deriving the continuous time asset dynamics as the limit of the discrete time dynamics. Under the constraint of a zero position on the asset at the beginning and at maturity, we obtain a quasi-linear equation for the price of the derivative, in the sense of viscosity. We offer the perfect hedging strategy when the equation admits a regular solution. As for the hedging of a covered European option under the gamma constraint, the dynamic program principle used previously is no longer valid. Following the techniques of the stochastic target and the partial differential equation, we show that the price of the over-replication has become a viscosity solution of a nonlinear equation of parabolic type. We also construct the ε-optimal strategy, and propose a numerical scheme.The second part of this thesis is devoted to studies on a new numerical scheme of EDSR, based on the branching process. We first approximate the Lipschitzian generator by a sequence of local polynomials, and then apply the Picard iteration. Each Picard iteration can be represented in terms of a branching process. We demonstrate the convergence of our scheme on the infinite time horizon. A concrete example is discussed at the end in order to illustrate the performance of our algorithm.

No summary available.

We develop and implement a method for maximum likelihood estimation of a regime-switching stochastic volatility model. Our model uses a continuous time stochastic process for the stock dynamics with the instantaneous variance driven by a Cox-Ingersoll-Ross (CIR) process and each parameter modulated by a hidden Markov chain. We propose an extension of the EM algorithm through the Baum-Welch implementation to estimate our model and filter the hidden state of the Markov chain while using the VIX index to invert the latent volatility state. Using Monte Carlo simulations, we test the convergence of our algorithm and compare it with an approximate likelihood procedure where the volatility state is replaced by the VIX index. We found that our method is more accurate than the approximate procedure. Then, we apply Fourier methods to derive a semi-analytical expression of S&P 500 and VIX option prices, which we calibrate to market data. We show that the model is sufficiently rich to encapsulate important features of the joint dynamics of the stock and the volatility and to consistently fit option market prices.

We develop and implement a method for maximum likelihood estimation of a regime-switching stochastic volatility model. Our model uses a continuous time stochastic process for the stock dynamics with the instantaneous variance driven by a Cox-Ingersoll-Ross (CIR) process and each parameter modulated by a hidden Markov chain. We propose an extension of the EM algorithm through the Baum-Welch implementation to estimate our model and filter the hidden state of the Markov chain while using the VIX index to invert the latent volatility state. Using Monte Carlo simulations, we test the convergence of our algorithm and compare it with an approximate likelihood procedure where the volatility state is replaced by the VIX index. We found that our method is more accurate than the approximate procedure. Then, we apply Fourier methods to derive a semi-analytical expression of S&P 500 and VIX option prices, which we calibrate to market data. We show that the model is sufficiently rich to encapsulate important features of the joint dynamics of the stock and the volatility and to consistently fit option market prices.

We consider the stochastic optimal control problem of McKean-Vlasov stochastic differential equation where the coefficients may depend upon the joint law of the state and control. By using feedback controls, we reformulate the problem into a deterministic control problem with only the marginal distribution of the process as controlled state variable, and prove that dynamic programming principle holds in its general form. Then, by relying on the notion of differentiability with respect to pro\-bability measures recently introduced by P.L. Lions in [32], and a special Itô formula for flows of probability measures, we derive the (dynamic programming) Bellman equation for mean-field stochastic control problem, and prove a veri\-fication theorem in our McKean-Vlasov framework. We give explicit solutions to the Bellman equation for the linear quadratic mean-field control problem, with applications to the mean-variance portfolio selection and a systemic risk model. We also consider a notion of lifted visc-sity solutions for the Bellman equation, and show the viscosity property and uniqueness of the value function to the McKean-Vlasov control problem. Finally, we consider the case of McKean-Vlasov control problem with open-loop controls and discuss the associated dynamic programming equation that we compare with the case of closed-loop controls.

This paper analyses the interaction between centralised carbon emissive technologies and distributed intermittent non-emissive technologies. In our model, there is a representative consumer who can satisfy her electricity demand by investing in distributed generation (solar panels) and by buying power from a centralised firm at a price the firm sets. Distributed generation is intermittent and induces an externality cost to the consumer. The firm provides non-random electricity generation subject to a carbon tax and to transmission costs. The objective of the consumer is to satisfy her demand while minimising investment costs, payments to the firm and intermittency costs. The objective of the firm is to satisfy the consumer's residual demand while minimising investment costs, demand deviation costs, and maximising the payments from the consumer. We formulate the investment decisions as McKean-Vlasov control problems with stochastic coefficients. We provide explicit, price model-free solutions to the optimal decision problems faced by each player, the solution of the Pareto optimum, and the Stackelberg equilibrium where the firm is the leader. We find that, from the social planner's point of view, the carbon tax or transmission costs are necessary to justify a positive share of distributed capacity in the long-term, whatever the respective investment costs of both technologies are. The Stackelberg equilibrium is far from the Pareto equilibrium and leads to an over-investment in distributed energy and to a much higher price for centralised energy.

This paper studies a robust continuous-time Markowitz portfolio selection pro\-blem where the model uncertainty carries on the covariance matrix of multiple risky assets. This problem is formulated into a min-max mean-variance problem over a set of non-dominated probability measures that is solved by a McKean-Vlasov dynamic programming approach, which allows us to characterize the solution in terms of a Bellman-Isaacs equation in the Wasserstein space of probability measures. We provide explicit solutions for the optimal robust portfolio strategies and illustrate our results in the case of uncertain volatilities and ambiguous correlation between two risky assets. We then derive the robust efficient frontier in closed-form, and obtain a lower bound for the Sharpe ratio of any robust efficient portfolio strategy. Finally, we compare the performance of Sharpe ratios for a robust investor and for an investor with a misspecified model. MSC Classification: 91G10, 91G80, 60H30.

We consider the optimal control problem for a linear conditional McKean-Vlasov equation with quadratic cost functional. The coefficients of the system and the weigh-ting matrices in the cost functional are allowed to be adapted processes with respect to the common noise filtration. Semi closed-loop strategies are introduced, and following the dynamic programming approach in [32], we solve the problem and characterize time-consistent optimal control by means of a system of decoupled backward stochastic Riccati differential equations. We present several financial applications with explicit solutions, and revisit in particular optimal tracking problems with price impact, and the conditional mean-variance portfolio selection in incomplete market model.

We analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities. These equations are related to robust switching control problems introduced in [3]. We prove that, as time horizon goes to infinity (resp. discount factor goes to zero) the long run average solution to the parabolic system (resp. the limiting discounted solution to the elliptic system) is characterized by a solution of a nonlinear system of ergodic variational inequalities. Our results hold under a dissipativity condition and without any non degeneracy assumption on the diffusion term. Our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities.

We provide a stochastic representation for a general class of viscous Hamilton-Jacobi (HJ) equations, which has convexity and superlinear nonlinearity in its gradient term, via a type of backward stochastic differential equation (BSDE) with constraint in the martingale part. We compare our result with the classical representation in terms of (super)quadratic BSDE, and show in particular that existence of a solution to the viscous HJ equation can be obtained under more general growth assumptions on the coefficients, including both unbounded diffusion coefficient and terminal data.

This thesis deals with the mathematical modeling of sovereign risk and its applications.In the first chapter, motivated by the Eurozone sovereign debt crisis, we propose a model of sovereign default risk. This model takes into account both the movement of sovereign creditworthiness and the impact of critical political events, adding an idiosyncratic credit risk. We focus on the probabilities of default occurring on the dates of critical political events, for which we obtain analytical formulas in a Markovian framework, where we carefully deal with some unusual features, among them the CEV model when the elasticity parameter β >1. We explicitly determine the compensating process of the defect and show that the intensity process does not exist, which contrasts our model with classical approaches. In the second chapter, by examining some hybrid models from the literature, we consider a class of random times with discontinuous conditional distributions for which the classical assumptions of filtrations magnification are not satisfied. We extend the density approach to a more general setting, where Jacod's assumption relaxes, afin order to deal with such random times in the universe of progressive magnification of filtrations. We also study classical problems: the computation of the compensator, the decomposition of the Azema surmartingale, and the characterization of martingales. The decomposition of martingales and semimartingales in the extended filtration affirms that the H' hypothesis remains valid in this generalized setting. In the third chapter, we present applications of the models proposed in the previous chapters. The most important application of the sovereign default model and the generalized density approach is the valuation of securities subject to default risk. The results explain the large negative jumps in the actuarial yield of the Greek long-term bond during the sovereign debt crisis. Greece's creditworthiness tends to worsen over the filter years and the bond yield has negative jumps during critical political events. In particular, the size of a jump depends on the severity of an exogenous shock, the time since the last political event, and the value of the recovery. The generalized density approach also makes it possible to model simultaneous defaults which, although rare, have a severe impact on the market.

The main objective of this thesis is to provide new theoretical results concerning the performance of investments based on stochastic models. To do so, we consider the optimal investment strategy in the framework of a risky asset model with constant volatility and a hidden Ornstein Uhlenbeck process. In the first chapter, we present the context and the objectives of this study. We present, also, the different methods used, as well as the main results obtained. In the second chapter, we focus on the feasibility of trend calibration. We answer this question with analytical results and numerical simulations. We close this chapter by also quantifing the impact of a calibration error on the trend estimate and exploit the results to detect its sign. In the third chapter, we assume that the agent is able to calibrate the trend well and we study the impact that the non-observability of the trend has on the performance of the optimal strategy. To do so, we consider the case of a logarithmic utility and an observed or unobserved trend. In each of the two cases, we explain the asymptotic limit of the expectation and the variance of the logarithmic return as a function of the signal-to-noise ratio and the speed of reversion to the mean of the trend. We conclude this study by showing that the asymptotic Sharpe ratio of the optimal strategy with partial observations cannot exceed 2/(3^1.5)∗100% of the asymptotic Sharpe ratio of the optimal strategy with complete information. The fourth chapter studies the robustness of the optimal strategy with calibration error and compares its performance to a technical analysis strategy. To do so, we characterize, analytically, the asymptotic expectation of the logarithmic return of each of these two strategies. We show, through our theoretical results and numerical simulations, that a technical analysis strategy is more robust than the poorly calibrated optimal strategy.

We study the large time behavior of solutions to fully nonlinear parabolic equations of Hamilton-Jacobi-Bellman type arising typically in stochastic control theory with control both on drift and diffusion coefficients. We prove that, as time horizon goes to infinity, the long run average solution is characterized by a nonlinear ergodic equation. Our results hold under dissipativity conditions, and without any nondegeneracy assumption on the diffusion term. Our approach uses mainly probabilistic arguments relying on new backward SDE representation for nonlinear parabolic, elliptic and ergodic equations.

We consider a unifying framework for stochastic control problem including the following features: partial observation, path-dependence (both with respect to the state and the control), and without any non-degeneracy condition on the stochastic differential equation (SDE) for the controlled state process, driven by a Wiener process. In this context, we develop a general methodology, refereed to as the randomization method, studied in [23] for classical Markovian control under full observation, and consisting basically in replacing the control by an exogenous process independent of the driving noise of the SDE. Our first main result is to prove the equivalence between the primal control problem and the randomized control problem where optimization is performed over change of equivalent probability measures affecting the characteristics of the exogenous process. The randomized problem turns out to be associated by duality and separation argument to a backward SDE, which leads to the so-called randomized dynamic programming principle and randomized equation in terms of the path-dependent filter, and then characterizes the value function of the primal problem. In particular, classical optimal control problems with partial observation affected by non-degenerate Gaussian noise fall within the scope of our framework, and are treated by means of an associated backward SDE.

We consider the optimal control problem for a linear conditional McKean-Vlasov equation with quadratic cost functional. The coefficients of the system and the weigh-ting matrices in the cost functional are allowed to be adapted processes with respect to the common noise filtration. Semi closed-loop strategies are introduced, and following the dynamic programming approach in [32], we solve the problem and characterize time-consistent optimal control by means of a system of decoupled backward stochastic Riccati differential equations. We present several financial applications with explicit solutions, and revisit in particular optimal tracking problems with price impact, and the conditional mean-variance portfolio selection in incomplete market model.

We consider the stochastic optimal control problem of nonlinear mean-field systems in discrete time. We reformulate the problem into a deterministic control problem with marginal distribution as controlled state variable, and prove that dynamic programming principle holds in its general form. We apply our method for solving explicitly the mean-variance portfolio selection and the multivariate linear-quadratic McKean-Vlasov control problem.

We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in [4], we prove a corresponding randomized dynamic programming principle (DPP) for the value function, which is obtained from a flow property of an associated filter process. This DPP is the key step towards our main result: a characterization of the value function of the partial observation control problem as the unique viscosity solution to the corresponding dynamic programming Hamilton-Jacobi-Bellman (HJB) equation. The latter is formulated as a new, fully non linear partial differential equation on the Wasserstein space of probability measures. An important feature of our approach is that it does not require any non-degeneracy condition on the diffusion coefficient, and no condition is imposed to guarantee existence of a density for the filter process solution to the controlled Zakai equation, as usually done for the separated problem. Finally, we give an explicit solution to our HJB equation in the case of a partially observed non Gaussian linear quadratic model.

We study the optimal control of general stochastic McKean-Vlasov equation. Such problem is motivated originally from the asymptotic formulation of cooperative equilibrium for a large population of particles (players) in mean-field interaction under common noise. Our first main result is to state a dynamic programming principle for the value function in the Wasserstein space of probability measures, which is proved from a flow property of the conditional law of the controlled state process. Next, by relying on the notion of differentiability with respect to probability measures due to P.L. Lions [32], and Itô's formula along a flow of conditional measures, we derive the dynamic programming Hamilton-Jacobi-Bellman equation, and prove the viscosity property together with a uniqueness result for the value function. Finally, we solve explicitly the linear-quadratic stochastic McKean-Vlasov control problem and give an application to an interbank systemic risk model with common noise.

No summary available.

This thesis presents three main research topics, the first two being independent and the last one indicating the relation of the first two problems in a concrete case.In the first part we focus on the martingale optimal transport problem in Skorokhod space, whose first goal is to study systematically the tension of martingale transport schemes. We first focus on the upper semicontinuity of the primal problem with respect to the marginal distributions. Using the S-topology introduced by Jakubowski, we derive the upper semicontinuity and show the first duality. We also give two dual problems concerning the robust overcoverage of an exotic option, and we establish the corresponding dualities, by adapting the principle of dynamic programming and the discretization argument initiated by Dolinsky and Soner.The second part of this thesis deals with the optimal Skorokhod folding problem. We first formulate this optimization problem in terms of probability measures on an extended space and its dual problems. Using the classical duality. convex approach and the optimal stopping theory, we obtain the duality results. We also relate these results to martingale optimal transport in the space of continuous functions, from which the corresponding dualities are derived for a particular class of payment functions. Next, we provide an alternative proof of the monotonicity principle established by Beiglbock, Cox and Huesmann, which allows us to characterize optimizers by their geometric support. We show at the end a stability result which contains two parts: the stability of the optimization problem with respect to the target marginals and the connection with another problem of the optimal folding.The last part concerns the application of stochastic control to the martingale optimal transport with the local time dependent payoff function, and to the Skorokhod folding. For the case of one marginal, we find the optimizers for the primal and dual problems via the Vallois solutions, and consequently show the optimality of the Vallois solutions, which includes the optimal martingale transport and the optimal Skorokhod folding. For the case of two marginals, we obtain a generalization of the Vallois solution. Finally, a special case of several marginals is studied, where the stopping times given by Vallois are well ordered.

We consider a robust switching control problem. The controller only observes the evolution of the state process, and thus uses feedback (closed-loop) switching strategies, a non standard class of switching controls introduced in this paper. The adverse player (nature) chooses open-loop controls that represent the so-called Knightian uncertainty, i.e., misspecifications of the model. The (half) game switcher versus nature is then formulated as a two-step (robust) optimization problem. We develop the stochastic Perron method in this framework, and prove that it produces a viscosity sub and supersolution to a system of Hamilton-Jacobi-Bellman (HJB) variational inequalities, which envelope the value function. Together with a comparison principle, this characterizes the value function of the game as the unique viscosity solution to the HJB equation, and shows as a byproduct the dynamic programming principle for robust feedback switching control problem.

We analyze a stochastic optimal control problem, where the state process follows a McKean-Vlasov dynamics and the diffusion coefficient can be degenerate. We prove that its value function V admits a nonlinear Feynman-Kac representation in terms of a class of forward-backward stochastic differential equations, with an autonomous forward process. We exploit this probabilistic representation to rigorously prove the dynamic programming principle (DPP) for V. The Feynman-Kac representation we obtain has an important role beyond its intermediary role in obtaining our main result: in fact it would be useful in developing probabilistic numerical schemes for V. The DPP is important in obtaining a characterization of the value function as a solution of a non-linear partial differential equation (the so-called Hamilton-Jacobi-Belman equation), in this case on the Wasserstein space of measures. We should note that the usual way of solving these equations is through the Pontryagin maximum principle, which requires some convexity assumptions. There were attempts in using the dynamic programming approach before, but these works assumed a priori that the controls were of Markovian feedback type, which helps write the problem only in terms of the distribution of the state process (and the control problem becomes a deterministic problem). In this paper, we will consider open-loop controls and derive the dynamic programming principle in this most general case. In order to obtain the Feynman-Kac representation and the randomized dynamic programming principle, we implement the so-called randomization method, which consists in formulating a new McKean-Vlasov control problem, expressed in weak form taking the supremum over a family of equivalent probability measures. One of the main results of the paper is the proof that this latter control problem has the same value function V of the original control problem.

This thesis explores theoretically and empirically the implications of the joint stock/option dynamics on various issues related to options trading. First, we study the joint dynamics between an option on a stock and an option on the market index. The CAPM model provides an adequate mathematical framework for this study because it allows to model the joint dynamics of a stock and its market index. Moving to option prices, we show that beta and idiosyncratic volatility, parameters of the model, allow us to characterize the relationship between the implied volatility surfaces of the stock and the index. We then turn to the estimation of the beta parameter under the risk-neutral probability using option prices. This measure, called implied beta, represents the information contained in the option prices about the realization of the beta parameter in the future.For this reason, we try to see, if implied beta has any predictive power of the future beta.By conducting an empirical study, we conclude that implied beta does not improve the predictive ability compared to the historical beta which is computed through the linear regression of the stock returns on the index returns. Better yet, we note that the oscillation of the implied beta around the future beta can lead to arbitrage opportunities, and we propose an arbitrage strategy that allows to monetize this gap. On the other hand, we show that the implied beta estimator could be used to hedge options on the stock using index instruments, this hedging concerns notably the volatility risk and also the delta risk. In the second part of our work, we are interested in the problem of market making on options. In this study, we assume that the model of the underlying's dynamics under the risk-neutral probability could be misspecified, which reflects a mismatch between the implied distribution of the underlying and its historical distribution.First, we consider the case of a risk-neutral market maker who aims to maximize the expectation of his future wealth. Using a stochastic optimal control approach, we determine the optimal call and put prices on the option and interpret the effect of price inefficiency on the optimal strategy. In a second step, we consider that the market maker is risk averse and therefore tries to reduce the uncertainty associated with his inventory. By solving an optimization problem based on a mean-variance criterion, we obtain analytical approximations of the optimal buying and selling prices. We also show the effects of inventory and price inefficiency on the optimal strategy. We then turn to the market making of options in a higher dimension. Thus, following the same reasoning, we present a framework for the market making of two options with different underlyings with the constraint of variance reduction related to the inventory risk held by the market maker. In the last part of our work, we study the joint dynamics between the implied volatility at the currency and the underlying, and we try to establish the link between these joint dynamics and the implied skew. We are interested in an indicator called "Skew Stickiness Ratio" which has been introduced in the recent literature. This indicator measures the sensitivity of the implied volatility of the currency to the movements of the underlying. We propose a method that allows us to estimate the value of this indicator under the risk-neutral probability without the need to admit assumptions on the dynamics of the underlying. [.].

We consider a non-Markovian optimal stopping problem on finite horizon. We prove that the value process can be represented by means of a backward stochastic differential equation (BSDE), defined on an enlarged probability space, containing a stochastic integral having a one-jump point process as integrator and an (unknown) process with a sign constraint as integrand. This provides an alternative representation with respect to the classical one given by a reflected BSDE. The connection between the two BSDEs is also clarified. Finally, we prove that the value of the optimal stopping problem is the same as the value of an auxiliary optimization problem where the intensity of the point process is controlled.

This thesis consists of two parts that can be read independently. In the first part of the thesis, three uses of backward-looking stochastic differential equations are presented. The first chapter is an application of these equations to the mean-variance hedging problem in an incomplete market where multiple defaults can occur. We make a conditional density assumption on the default times. We then decompose the value function into a sequence of value functions between two consecutive defaults and prove the quadratic form of each of them. Finally, we illustrate our results in a particular case with 2 fault times following independent exponential laws. The next two chapters are extensions of the paper [75]. The second chapter is the study of a class of backward stochastic differential equations with negative jumps and upper barrier. The existence and uniqueness of a minimal solution are proved by double penalization under regularity assumptions on the barrier. This method allows to solve the case where the diffusion coefficient is degenerate. We also show, in an adapted Markovian framework, the link between our class of backward equations and nonlinear variational inequalities. In particular, our backward equation representation yields a Feynman-Kac type formula for partial differential equations associated with stochastic differential games of the controller and zero-sum stopper type, where the control affects both the volatility drift terms. Moreover, we obtain a dual formula for the minimum solution set of the backward equation, which gives a new representation of the controller and zero-sum stochastic differential games. The third chapter is related to model uncertainty, where uncertainty affects both volatility and intensity. These stochastic control problems are associated with integro-differential partial differential equations such that the jump part is characterized by the measure lambda(a,. ) depending on a parameter a. We do not assume that the family lambda(a,. ) is dominated. We obtain a Feynman-Kac type nonlinear formula to the value function associated with these control problems. For this, we introduce a class of backward stochastic differential equations with jump and a partially constrained diffusive part. Here also the case where the diffusion coefficient is degenerate is solved In the second part of the thesis, a conditional asset-liability management problem is solved We first obtain the definition domain of the value function associated with the problem by identifying the minimal wealth for which there is an admissible investment strategy allowing to satisfy the constraint at maturity. This minimal wealth is identified as a viscosity solution of a PDE. We also show that its Fenschel-Legendre transform is a viscosity solution of another PDE, which allows us to obtain a numerical scheme with a faster convergence. We then identify the value function related to the problem of interest as a viscosity solution of a PDE on its domain of definition. Finally, we numerically solve the problem by presenting graphs of the minimum richness, the value function of the problem and the optimal strategy.

This survey reviews portfolio selection problem for long-term horizon. We consider two objectives: (i) maximize the probability for outperforming a target growth rate of wealth process (ii) minimize the probability of falling below a target growth rate. We study the asymptotic behavior of these criteria formulated as large deviations control pro\-blems, that we solve by duality method leading to ergodic risk-sensitive portfolio optimization problems. Special emphasis is placed on linear factor models where explicit solutions are obtained.

We introduce a new model for describing the fluctuations of a tick-by-tick single asset price. Our model is based on Markov renewal processes. We consider a point process associated to the timestamps of the price jumps, and marks associated to price increments. By modeling the marks with a suitable Markov chain, we can reproduce the strong mean-reversion of price returns known as microstructure noise. Moreover, by using Markov renewal processes, we can model the presence of spikes in intensity of market activity, i.e. the volatility clustering, and consider dependence between price increments and jump times. We also provide simple parametric and nonparametric statistical procedures for the estimation of our model. We obtain closed-form formula for the mean signature plot, and show the diffusive behavior of our model at large scale limit. We illustrate our results by numerical simulations, and that our model is consistent with empirical data on the Euribor future.

We propose a new probabilistic numerical scheme for fully nonlinear equation of Hamilton-Jacobi-Bellman (HJB) type associated to stochastic control problem, which is based on the Feynman-Kac representation in [12] by means of control randomization and backward stochastic differential equation with nonpositive jumps. We study a discrete time approximation for the minimal solution to this class of BSDE when the time step goes to zero, which provides both an approximation for the value function and for an optimal control in feedback form. We obtained a convergence rate without any ellipticity condition on the controlled diffusion coefficient. Explicit implementable scheme based on Monte-Carlo simulations and empirical regressions, associated error analysis, and numerical experiments are performed in the companion paper [13].

The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in stochastic analysis and the emergence of the theory of backward stochastic differential equations (BSDEs), which can be viewed as nonlinear Feynman-Kac formulas. We review in this paper the main ideas and results in this area, and present implications of these probabilistic representations for the numerical resolution of nonlinear PDEs, together with some applications to stochastic control problems and model uncertainty in finance.

No summary available.

We establish explicit socially optimal rules for an irreversible investment decision with time-to-build and uncertainty. Assuming a price sensitive demand function with a random intercept, we provide comparative statics and economic interpretations for three models of demand (arithmetic Brownian, geometric Brownian, and the Cox-Ingersoll-Ross). Committed capacity, that is, the installed capacity plus the investment in the pipeline, must never drop below the best predictor of future demand, minus two biases. The discounting bias takes into account the fact that investment is paid upfront for future use. the precautionary bias multiplies a type of risk aversion index by the local volatility. Relying on the analytical forms, we discuss in detail the economic effects.

In this thesis, we deal with the modeling of asset prices in a limit order book and the application of optimal control techniques to algorithmic trading, in particular market making. For assets with small ticks, we develop a market making algorithm in a limit order book where the order arrivals follow a Poisson distribution with mean that decreases exponentially with the distance of the order from the mid-price. Thanks to asymptotic development techniques, we obtain explicit results for a very large class of price models, for which we assume to know only the first two moments. For large-ticket assets, we propose a new model based on a semi-Markovian process, with which we replicate known market phenomena such as mean reversion, large-scale Brownian behavior, and the dependence of the variance estimator on the observation frequency. In this environment, we describe a market making algorithm using optimal control techniques and asymptotic development, reducing the numerical part to the minimum. Finally, we improve the previous model by using VLMC (Variable Length Markov Chains), which allow us to describe the long memory of the price, and, even if we abandon explicit formulas, allow us to obtain interesting applications to algorithmic trading.

We consider the stochastic optimal control problem of nonlinear mean-field systems in discrete time. We reformulate the problem into a deterministic control problem with marginal distribution as controlled state variable, and prove that dynamic programming principle holds in its general form. We apply our method for solving explicitly the mean-variance portfolio selection and the multivariate linear-quadratic McKean-Vlasov control problem.

We aim to provide a Feynman-Kac type representation for Hamilton-Jacobi-Bellman equation, in terms of Forward Backward Stochastic Differential Equation (FBSDE) with a simulatable forward process. For this purpose, we introduce a class of BSDE where the jumps component of the solution is subject to a partial nonpositive constraint. Existence and approximation of a unique minimal solution is proved by a penalization method under mild assumptions. We then show how minimal solution to this BSDE class provides a new probabilistic representation for nonlinear integro-partial differential equations (IPDEs) of Hamilton-Jacobi-Bellman (HJB) type, when considering a regime switching forward SDE in a Markovian framework. Moreover, we state a dual formula of this BSDE minimal solution involving equivalent change of probability measures. This gives in particular an original representation for value functions of stochastic control problems including controlled diffusion coefficient.

We consider the problem of optimal trading for a power producer in the context of intraday electricity markets. The aim is to minimize the imbalance cost induced by the random residual demand in electricity, i.e. the consumption from the clients minus the production from renewable energy. For a simple linear price impact model and a quadratic criterion, we explicitly obtain approximate optimal strategies in the intraday market and thermal power generation, and exhibit some remarkable properties of the trading rate. Furthermore, we study the case when there are jumps on the demand forecast and on the intraday price, typically due to error in the prediction of wind power generation. Finally, we solve the problem when taking into account delay constraints in thermal power production.

This thesis deals with the study of mathematical models of price movements in electricity markets, from the point of view of process statistics and stochastic optimal control. In the first part, we estimate the volatility components of a multidimensional diffusion process representing the evolution of prices on the electricity futures market. Its dynamics is driven by two Brownian motions. We seek to perform the estimation efficiently in terms of speed of convergence, and limit variance with respect to the parametric part of these components. This requires an extension of the usual definition of efficiency in the Cramér-Rao sense. Our estimation methods are based on the realized quadratic variation of the observed process. In the second part, we add model error terms to the observations of the previous model, to overcome the problem of overdetermination that arises when the dimension of the observed process is greater than two. The estimation techniques are still based on the realized quadratic variation, and we propose other tools to continue estimating the volatility components with the optimal speed in the presence of the error terms. Numerical tests are used to highlight the presence of such errors in our data. Finally, in the last part we solve the problem of a producer who intervenes on the intraday electricity market in order to compensate for the costs related to the random returns of his production units. Through his actions, he has an impact on the market. The prices and its anticipation of the demand of its consumers are modeled by a jumping diffusion. The tools of stochastic optimal control allow us to determine his strategy in an approximate problem. We give conditions for this strategy to be very close to optimality in the initial problem, and illustrate it numerically.

We consider the problem of optimal trading for a power producer in the context of intraday electricity markets. The aim is to minimize the imbalance cost induced by the random residual demand in electricity, i.e. the consumption from the clients minus the production from renewable energy. For a simple linear price impact model and a quadratic criterion, we explicitly obtain approximate optimal strategies in the intraday market and thermal power generation, and exhibit some remarkable properties of the trading rate. Furthermore, we study the case when there are jumps on the demand forecast and on the intraday price, typically due to error in the prediction of wind power generation. Finally, we solve the problem when taking into account delay constraints in thermal power production.

We study a class of reflected backward stochastic differential equations with nonpositive jumps and upper barrier. Existence and uniqueness of a minimal solution is proved by a double pena\-lization approach under regularity assumptions on the obstacle. In a suitable regime switching diffusion framework, we show the connection between our class of BSDEs and fully nonlinear variational inequalities. Our BSDE representation provides in particular a Feynman-Kac type formula for PDEs associated to general zero-sum stochastic differential controller-and-stopper games, where control affect both drift and diffusion term, and the diffusion coefficient can be degenerate. Moreover, we state a dual game formula of this BSDE minimal solution involving equivalent change of probability measures, and discount processes. This gives in particular a new representation for zero-sum stochastic differential controller-and-stopper games.

This thesis was realized within the company Invivoo. The main objective was to find investment strategies: to have a high gain and a low risk. The research work was mainly focused on this last point. In this sense, we wanted to generalize a model fidèle to the reality of the financiers' markets, both for low and high frequency data and, at very high frequency, variation by variation.

High Frequency finance has recently evolved from statistical modeling and analysis of financial data – where the initial goal was to reproduce stylized facts and develop appropriate inference tools – toward trading optimization, where an agent seeks to execute an order (or a series of orders) in a stochastic environment that may react to the trading algorithm of the agent (market impact, invoentory). This context poses new scientific challenges addressed by the minisymposium OPSTAHF.

This thesis deals with the numerical solution of general stochastic control problems, with notable applications for electricity markets. We first propose a structural model for the price of electricity, allowing for price spikes well above the marginal fuel price under strained market conditions. This model aliows to price and partially hedge electricity derivatives, using fuel forwards as hedging instruments. Then, we propose an algorithm, which combines Monte-Carlo simulations with local basis regressions, to solve general optimal switching problems. A comprehensive rate of convergence of the method is provided. Moreover, we manage to make the algorithm parcimonious in memory (and hence suitable for high dimensional problems) by generalizing to this framework a memory reduction method that avoids the storage of the sample paths. We illustrate this on the problem of investments in new power plants (our structural power price model allowing the new plants to impact the price of electricity). Finally, we study more general stochastic control problems (the control can be continuous and impact the drift and volatility of the state process), the solutions of which belong to the class of fully nonlinear Hamilton-Jacobi-Bellman equations, and can be handled via constrained Backward Stochastic Differential Equations, for which we develop a backward algoritm based on control randomization and parametric optimizations. A rate of convergence between the constraPned BSDE and its discrete version is provided, as well as an estimate of the optimal control. This algorithm is then applied to the problem of superreplication of options under uncertain volatilities (and correlations).

We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [9] for representing fully nonlinear HJB equations. In particular, this allows us to numerically solve stochastic control problems with controlled volatility, possibly degenerate. Our backward scheme, based on least-squares regressions, takes advantage of high-dimensional properties of Monte-Carlo methods, and also provides a parametric estimate in feedback form for the optimal control. A partial analysis of the error of the scheme is provided, as well as numerical tests on the problem of superreplication of option with uncertain volatilities and/or correlations, including a detailed comparison with the numerical results from the alternative scheme proposed in [7].

In this paper, we present a probabilistic numerical algorithm combining dynamic programming, Monte Carlo simulations, and local basis regressions to solve nonstationary optimal multiple switching problems in infinite horizon. We provide the rate of convergence of the method in terms of the time step used to discretize the problem, of the regression basis used to approximate conditional expectations, and of the truncating time horizon. To make the method viable for problems in high dimension and long time horizon, we extend a memory reduction method to the general Euler scheme, so that, when performing the numerical resolution, the storage of the Monte Carlo simulation paths is not needed. Then, we apply this algorithm to a model of optimal investment in power plants in dimension eight, i.e., with two different technologies and six random factors.

We establish explicit socially optimal rules for an irreversible investment decision with time-to-build and uncertainty. Assuming a price sensitive demand function with a random intercept, we provide comparative statics and economic interpretations for three models of demand (arithmetic Brownian, geometric Brownian, and the Cox-Ingersoll-Ross). Committed capacity, that is, the installed capacity plus the investment in the pipeline, must never drop below the best predictor of future demand, minus two biases. The discounting bias takes into account the fact that investment is paid upfront for future use. the precautionary bias multiplies a type of risk aversion index by the local volatility. Relying on the analytical forms, we discuss in detail the economic effects.

The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in stochastic analysis and the emergence of the theory of backward stochastic differential equations (BSDEs), which can be viewed as nonlinear Feynman-Kac formulas. We review in this paper the main ideas and results in this area, and present implications of these probabilistic representations for the numerical resolution of nonlinear PDEs, together with some applications to stochastic control problems and model uncertainty in finance.

This survey reviews portfolio selection problem for long-term horizon. We consider two objectives: (i) maximize the probability for outperforming a target growth rate of wealth process (ii) minimize the probability of falling below a target growth rate. We study the asymptotic behavior of these criteria formulated as large deviations control pro\-blems, that we solve by duality method leading to ergodic risk-sensitive portfolio optimization problems. Special emphasis is placed on linear factor models where explicit solutions are obtained.

The objective of this thesis is to develop the theory of stochastic control and its applications in population dynamics. From a theoretical point of view, we present the study of stochastic control problems with finite horizon on diffusion, nonlinear branching and branch-diffusion processes. In each case, we reason by the dynamic programming method, taking care to carefully prove a conditioning argument analogous to the strong Markov property for controlled processes. The principle of dynamic programming then allows us to prove that the value function is a solution (regular or viscosity) of the corresponding Hamilton-Jacobi-Bellman equation. In the regular case, we also identify a Markovian optimal control by a verification theorem. From an application point of view, we are interested in the mathematical modeling of cancer and its therapeutic strategies. More precisely, we build a hybrid model of tumor growth that accounts for the fundamental role of acidity in the evolution of the disease. The targets of therapy are explicitly included as parameters of the model in order to use it as a support for the evaluation of therapeutic strategies.

We propose a framework to study optimal trading policies in a one-tick pro-rata limit order book, as typically arises in short-term interest rate futures contracts. The high-frequency trader has the choice to trade via market orders or limit orders, which are represented respectively by impulse controls and regular controls. We model and discuss the consequences of the two main features of this particular microstructure: first, the limit orders sent by the high frequency trader are only partially executed, and therefore she has no control on the executed quantity. For this purpose, cumulative executed volumes are modelled by compound Poisson processes. Second, the high frequency trader faces the overtrading risk, which is the risk of brutal variations in her inventory. The consequences of this risk are investigated in the context of optimal liquidation. The optimal trading problem is studied by stochastic control and dynamic progra\-mming methods, which lead to a characterization of the value function in terms of an integro quasi-variational inequality. We then provide the associated numerical resolution procedure, and convergence of this computational scheme is proved. Next, we examine several situations where we can on one hand simplify the numerical procedure by reducing the number of state variables, and on the other hand focus on specific cases of practical interest. We examine both a market making problem and a best execution problem in the case where the mid-price process is a martingale. We also detail a high frequency trading strategy in the case where a (predictive) directional information on the mid-price is available. Each of the resulting strategies are illustrated by numerical tests.

The objective of my thesis is to study mathematically a volatility breakout indicator widely used by practitioners in the trading room. The Bollinger Bands indicator belongs to the family of so-called technical analysis methods and is therefore based exclusively on the recent history of the price considered and a principle deduced from past market observations, independently of any mathematical model. My work consists in studying the performance of this indicator in a universe governed by stochastic differential equations (Black-Scholes) whose diffusion coefficient changes its value at an unknown and unobservable random time, for a practitioner wishing to maximize an objective function (for example, a certain expected utility of the portfolio value at a certain maturity). In the framework of the model, the Bollinger indicator can be interpreted as an estimator of the time of the next break. In the case of small volatilities, we show that the behavior of the density of the indicator depends on the volatility, which makes it possible to detect, for a large enough volatility ratio, the volatility regime in which the indicator's distribution is located. Also, in the case of high volatilities, we show by an approach via the Laplace transform, that the asymptotic behavior of the indicator's distribution tails depends on the volatility. This makes it possible to detect the change in the large volatilities. Then, we are interested in a comparative study between the Bollinger indicator and the classical estimator of the quadratic variation for the detection of change in volatility. Finally, we study the optimal portfolio management which is described by a non-standard stochastic problem in the sense that the admissible controls are constrained to be functionals of the observed prices. We solve this control problem by drawing on the work of Pham and Jiao to decompose the initial portfolio allocation problem into a post-breakdown management problem and a pre-breakdown problem, and each of these problems is solved by the dynamic programming method. Thus, a verification theorem is proved for this stochastic control problem.

No summary available.

We study a class of reflected backward stochastic differential equations with nonpositive jumps and upper barrier. Existence and uniqueness of a minimal solution is proved by a double pena\-lization approach under regularity assumptions on the obstacle. In a suitable regime switching diffusion framework, we show the connection between our class of BSDEs and fully nonlinear variational inequalities. Our BSDE representation provides in particular a Feynman-Kac type formula for PDEs associated to general zero-sum stochastic differential controller-and-stopper games, where control affect both drift and diffusion term, and the diffusion coefficient can be degenerate. Moreover, we state a dual game formula of this BSDE minimal solution involving equivalent change of probability measures, and discount processes. This gives in particular a new representation for zero-sum stochastic differential controller-and-stopper games.

We introduce a new model for describing the fluctuations of a tick-by-tick single asset price. Our model is based on Markov renewal processes. We consider a point process associated to the timestamps of the price jumps, and marks associated to price increments. By modeling the marks with a suitable Markov chain, we can reproduce the strong mean-reversion of price returns known as microstructure noise. Moreover, by using Markov renewal processes, we can model the presence of spikes in intensity of market activity, i.e. the volatility clustering, and consider dependence between price increments and jump times. We also provide simple parametric and nonparametric statistical procedures for the estimation of our model. We obtain closed-form formula for the mean signature plot, and show the diffusive behavior of our model at large scale limit. We illustrate our results by numerical simulations, and that our model is consistent with empirical data on the Euribor future.

We introduce a new model for describing the fluctuations of a tick-by-tick single asset price. Our model is based on Markov renewal processes. We consider a point process associated to the timestamps of the price jumps, and marks associated to price increments. By modeling the marks with a suitable Markov chain, we can reproduce the strong mean-reversion of price returns known as microstructure noise. Moreover, by using Markov renewal processes, we can model the presence of spikes in intensity of market activity, i.e. the volatility clustering, and consider dependence between price increments and jump times. We also provide simple parametric and nonparametric statistical procedures for the estimation of our model. We obtain closed-form formula for the mean signature plot, and show the diffusive behavior of our model at large scale limit. We illustrate our results by numerical simulations, and that our model is consistent with empirical data on the Euribor future.

We study an optimal high frequency trading problem within a market microstructure model aiming at a good compromise between accuracy and tractability. The stock price is modeled by a Markov Renewal Process (MRP), while market orders arrive in the limit order book via a point process correlated with the stock price, and taking into account the adverse selection risk. We apply stochastic control methods in this semi-Markov framework, and show how to reduce remarkably the complexity of the associated Hamilton-Jacobi-Bellman equation by suitable change of variables that exploits the specific symmetry of the problem. We then handle numerically the remaining part of the HJB equation, simplified into an integro-ordinary differential equation, by a bidimensional Euler scheme. Statistical procedures and numerical tests for computing the optimal limit order strategies illustrate our results.

We study optimal stochastic control problem for non-Markovian stochastic differential equations (SDEs) where the drift, diffusion coefficients, and gain functionals are path-dependent, and importantly we do not make any ellipticity assumption on the SDE. We develop a controls randomization approach, and prove that the value function can be reformulated under a family of dominated measures on an enlarged filtered probability space. This value function is then characterized by a backward SDE with nonpositive jumps under a single probability measure, which can be viewed as a path-dependent version of the Hamilton-Jacobi-Bellman equation, and an extension to $G$ expectation.

We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [9] for representing fully nonlinear HJB equations. In particular, this allows us to numerically solve stochastic control problems with controlled volatility, possibly degenerate. Our backward scheme, based on least-squares regressions, takes advantage of high-dimensional properties of Monte-Carlo methods, and also provides a parametric estimate in feedback form for the optimal control. A partial analysis of the error of the scheme is provided, as well as numerical tests on the problem of superreplication of option with uncertain volatilities and/or correlations, including a detailed comparison with the numerical results from the alternative scheme proposed in [7].

We propose a quantitative approach to some high frequency trading problematics. We are interested in several aspects of this field, from minimizing indirect trading costs to market making, and more generally in profit maximization strategies over a finite time horizon. We build an original framework that reflects specificities of high frequency trading, and especially the distinction between passive and active trading, thanks to mixed stochastic control methods. We carefully model high frequency market phonomena, and for each of them we propose calibration methods that are compatible with practical constraints of algorithmic trading.

This paper studies a {\it reversible} investment problem where a social planner aims to control its capacity production in order to fit optimally the random demand of a good. Our model allows for general diffusion dynamics on the demand as well as general cost functional. The resulting optimization problem leads to a degenerate two-dimensional bounded variation singular stochastic control problem, for which explicit solution is not available in general and the standard verification approach can not be applied a priori. We use a direct viscosity solutions approach for deriving some features of the optimal free boundary function, and for displaying the structure of the solution. In the quadratic cost case, we are able to prove a smooth-fit $C^2$ property, which gives rise to a full characterization of the optimal boundaries and value function.

In this thesis, we are interested in hedging derivatives in incomplete markets. The chosen approach can be seen as an extension of M. Schweizer's work on local minimization of quadratic risk. Indeed, while remaining within the framework of asset modeling by semimartingales, our method consists in replacing the quadratic risk criterion by a more general risk criterion, in the form of a convex functional of the local cost. We first obtain existence, uniqueness and characterization results for optimal strategies in a frictionless market, in discrete and continuous time. Then we explain these strategies in the framework of diffusion models with and without jumps. We also extend our method to the case where liquidity is no longer infinite. Finally, we show through numerical simulations the effects of the choice of the risk functional on the constitution of the optimal portfolio.

The work presented in this thesis was motivated by issues raised by the valuation of contracts traded in the gas market: gas storage and supply contracts. These contracts incorporate optionality and constraints, which makes their valuation difficult in a context of random commodity prices. The valuation of these contracts leads to complex stochastic control problems: optimal switching or impulse control and high dimensional stochastic control. The first part of this thesis is a relatively exhaustive review of the literature, putting in perspective the different existing valuation approaches. In a second part, we consider a numerical method for solving impulse control problems based on their representation as a solution of constrained jumping EDSRs. We propose a discrete time approximation using a penalty to handle the constraint and give a convergence rate of the introduced error. Combined with Monte Carlo techniques, this method has been numerically tested on three problems: optimal biomass management, evaluation of swing options and gas storage contracts. In a third part, we propose a method for the valuation of options whose payoff depends on moving averages of underlying prices. It uses a finite dimensional approximation of the dynamics of moving average processes, based on a truncated Laguerre series development. The numerical results provided include examples of gas swings with strike prices indexed to moving averages of oil prices.

This thesis is divided into two parts that may be read independently. The first part is about the mathematical modelling of liquidity risk. The aspect of illiquidity studied here is the constraint on the trading dates, meaning that in opposition to the classical models where investors may trade continuously, we assume that trading is only possible at discrete random times. We then use optimal control techniques (dynamic programming and Hamilton-Jacobi-Bellman equations) to identify the value functions and optimal investment strategies under these constraints. The first chapter focuses on a utility maximisation problem in finite horizon, in a framework inspired by energy markets. In the second chapter we study an illiquid market with regime-switching, and in the third chapter we consider a market in which the agent has the possibility to invest in a liquid asset and an illiquid asset which are correlated. In the second part we present probabilistic quantization methods to solve numerically an optimal switching problem. We first consider a discrete time approximation of our problem and prove a convergence rate. Then we propose two numerical quantization methods : a markovian approach where we quantize the gaussian in the Euler scheme, and, in the case where the underlying diffusion is not controlled, a marginal quantization approach inspired by numerical methods for the optimal stopping problem.

This thesis deals with the optimization of asset portfolios subject to default risk. The current crisis has allowed us to understand that it is important to take into account the risk of default to be able to give the real value of its portfolio. Indeed, due to the different exchanges of the financial market actors, the financial system has become a network of several connections which it is essential to identify in order to evaluate the risk of investing in a financial asset. In this thesis, we define a financial system with a finite number of connections and we propose a model of the dynamics of an asset in such a system by taking into account the connections between the different assets. The measurement of the correlation will be done through the jump intensity of the processes. Using Stochastic Differential Backward Equations (SDGE), we will derive the price of a contingent asset and take into account the model risk in order to better evaluate the optimal consumption and wealth if one invests in such a market.

We study the link between backward-looking SDEs and some stochastic optimization problems and their applications in finance. In the first part, we focus on the representation by EDSR of sequential stochastic optimization problems: impulse control and optimal switching. We introduce the notion of jump-constrained EDSR and show that it provides a representation of the solutions of Markovian impulse control problems. We then link this class of EDSRs to oblique reflection EDSRs and to the process values of optimal switching problems. In the second part we study the discretization of the EDSRs involved above. We introduce a discretization of the jump-constrained EDSRs using the penalized EDSR approximation for which we obtain convergence. We then study the discretization of obliquely reflected EDSRs. We obtain for the proposed scheme a convergence speed to the continuously reflected solution. Finally, in the third part, we study an optimal portfolio liquidation problem with risk and execution cost. We consider a financial market on which an agent must liquidate a position in a risky asset. The intervention of this agent affects the market price of this asset and leads to an execution cost modeling the liquidity risk. We characterize the value function of our problem as a minimal solution of a quasi-variational inequation in the sense of constrained viscosity.

We study some applications of stochastic control to option hedging in the presence of illiquidity. In the first part, we consider an option overhedging problem in a stochastic volatility model. The originality of this problem is that the asset used to hedge the volatility is illiquid and that the agent will have to make a finite amount of transactions. The second part concerns option hedging in the presence of uncertain volatility whose dynamics are not specified. We introduce a criterion for non-trivial option prices, allowing the agent to lose money for volatility realizations that he considers unlikely. Finally, in a third part, we study an impulse control problem for which the controls take effect with delay. This study applies in particular to the hedging of options on hedge funds, for which buy and sell orders are executed with delay. In each part, we characterize the value function of the problem as the unique viscosity solution of a partial differential equation. In the first and third parts, we introduce in a second chapter algorithms for numerically solving these PDEs by finite differences. The convergence of these algorithms is proved theoretically.

This thesis is composed of two independent parts that focus on the application of probability to the field of finance. The first part studies the regularity of the solutions of certain types of backward-looking stochastic differential equations (SRDEs) and reflexive differential equations, as well as numerical approximation schemes of these solutions. In finance, the main application is the pricing and hedging of American and gambling options, but our work is not limited to this framework. The proposed systematic method is based on the study of equations that are reflected only on a discrete time grid. In finance, these equations are interpreted as Bermuda options. In a general framework of multidimensional convex domains that can, under certain conditions, evolve randomly, we obtain convergence and regularity results for these discretely reflected equations that we extend to continuously reflected SDEs. The second part deals with a theoretical problem in mathematical finance. We deal with the valuation of American options in the framework of market models with proportional transaction costs, both for discrete and continuous time. We obtain an over-replication theorem for these contingent assets in the very general framework of ladlag option processes.

This thesis focuses on portfolio optimization in partial observations. The work is organized in three parts that analyze the following topics: Part 1. Portfolio optimization in partial observations in a diffusion model with jumps. Part 2. Indifference pricing in partial observations in a diffusion model with jumps. Part 3. Numerical approximation by quantization of discrete time control problems with partial observations and applications in finance. In the first two parts we consider the case of continuous time observations while in the third we analyze the case of discrete time observations.

We study some applications of stochastic control to real options and liquidity risk. More precisely, in the first part, we are interested in an optimal portfolio selection problem under a liquidity risk model, then in the second part, in two real options: a regime-switching problem and a coupled singular control and regime-switching problem for a dividend policy with reversible investment, and finally, in the last part, in the existence of an equilibrium in a competitive market under asymmetric information. In solving these problems, especially in the first two parts, stochastic control techniques will be used. The typical approach is to express the principle of dynamic programming related to each problem in order to obtain a PDE characterization of the value functions. By this approach, we show, in the liquidity risk problem and the two real options, that the corresponding value functions are the unique solution of the associated system of variational HJB inequalities. In each problem of the first two parts, the solutions, in particular the optimal controls, can be obtained either in an explicit way or by an iterative method.

We develop a numerical solution approach to grid filtering, using optimal quantization results for random variables. We implement two filter computation algorithms using 0-order and 1-order approximation techniques. We propose implementable versions of these algorithms and study the behavior of the error of the approximations as a function of the quantizer size based on the stationarity property of optimal quantizers. We position this grid approach in relation to the Monte Carlo particle approach through the comparison of the two methods and their experimentation on different state models. In a second part, we focus on the advantage of quantization for the preprocessing of offline data to develop a filtering algorithm by quantization of the observations (and the signal). The error is also studied and a convergence rate is established as a function of the quantizer size. Finally, the quantization of the filter as a random variable is studied in order to solve an American option pricing problem in a market with unobserved stochastic volatility. All results are illustrated through numerical examples.

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The present work consists of six chapters divided into three parts and has as a common feature probabilistic and stochastic control applications to the evaluation of incompletely running contingent assets. The first chapter deals with the optimal stopping time problem of a jump-controlled diffusion process and shows, generalizing the results of Lions (1983), that the value function is characterized as the unique viscosity solution of the Hamilton-Jacobi-Bellman equation from dynamic programming, with appropriate boundary conditions. The second chapter establishes existence and uniqueness results in the class of regular functions c#1#,#2 when the preceding integrodifferential operator is linear, i.e., when there is no control on the jump diffusion process. The second part focuses on the contingent asset coverage and valuation problem in an incomplete walk setting. Using quadratic optimization criteria, we determine in Chapter 3 the pricing and hedging strategy that best duplicates a given contingent asset, when the pricing processes are semi martingales. From an equilibrium pricing approach, Chapter 4, in a stochastic volatility model framework, gives necessary and sufficient conditions on an Arrow-Debreu price system given to be consistent with an additive multi-agent intertemporal equilibrium model. Finally, the third part deals with option pricing in a diffusion model with jumps. Using the results of part 1, we study the regularity of the price of a European option (chapter 5) and that of an American option (chapter 6) and characterize them as solutions of parabolic integrodifferential equations of the second order, with appropriate boundary conditions. We establish some properties of an option, related to the incompleteness of the walk caused by the presence of jumps.