TOUZI Nizar

This thesis is composed of two independent parts. The first part focuses on the application of the Principal-Agent problem (c.f. Cvitanic & Zhang (2013) and Cvitanic. et al. (2018)) for solving modeling problems in energy markets. The second one deals with the joint laws of a martingale and its current maximum.We first focus on the electricity capacity market, and in particular capacity remuneration mechanisms. Given the increasing share of renewable energies in the electricity production, "classical" power plants (e.g. gas or coal) are less and less used, which makes them unprofitable and not economically viable. However, their closure would expose consumers to the risk of a blackout in the event of a peak in electricity demand, since electricity cannot be stored. Thus, generation capacity must always be maintained above demand, which requires a "capacity payment mechanism" to remunerate seldom used power plants, which can be understood as an insurance to be paid against electricity blackouts.We then address the issue of incentives for decarbonization. The objective is to propose a model of an instrument that can be used by a public agent (the state) to encourage the different sectors to reduce their carbon emissions in a context of moral hazard (where the state does not observe the effort of the actors and therefore cannot know whether a decrease in emissions comes from a decrease in production and consumption or from a management effort. The second part (independent) is motivated by model calibration and arbitrage on a financial market with barrier options. It presents a result on the joint laws of a martingale and its current maximum. We consider a family of probabilities in dimension 2, and we give necessary and sufficient conditions ensuring the existence of a martingale such that its marginal laws coupled with those of its current maximum coincide with the given probabilities.We follow the methodology of Hirsch and Roynette (2012) based on a martingale construction by DHS associated with a well-posed Fokker-Planck PDE verified by the given marginal laws under regularity assumptions, then in a general framework with regularization and boundary crossing.

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In this thesis, we use tools from stochastic optimal control and stochastic and convex optimization to develop mechanisms to drive energy storage systems to manage the production uncertainty of intermittent energy sources (solar and wind).First, we introduce a mechanism in which a consumer commits to follow a consumption profile on the grid, and then controls its storage systems to follow this profile in real time. We model this situation by a mean-field control problem, for which we obtain theoretical and numerical results. Then, we introduce a problem of controlling a large number of thermal storage units subject to a common noise and providing services to the network. We show that this control problem can be replaced by a stochastic differential Stackelberg problem. This allows a decentralized control scheme with performance guarantees, while preserving the privacy of the consumers' data and limiting the telecommunication requirements. Next, we develop a Newton method for stochastic control problems. We show that the Newton step can be computed by solving Stochastic Retrograde Differential Equations, then we propose an appropriate linear search method, and prove the global convergence of the obtained Newton method in a suitable space. Its numerical performance is illustrated on a problem of controlling a large number of batteries providing services to the network. Finally, we study the extension of the "Alternating Current Optimal Power Flow" problem to the stochastic multistage case in order to control an electrical network equipped with storage systems. For this problem, we give realistic and verifiable a priori conditions guaranteeing the absence of relaxation jumps, as well as an a posteriori bound on the latter. In the broader framework of non-convex multistage problems with a generic structure, we also establish a priori bounds on the duality jump, based on results related to the Shapley-Folkman Theorem.

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.

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This thesis is composed of two independent parts, the first one grouping two distinct problems. In the first part, we first study the Principal-Agent problem in degenerate systems, which arise naturally in partially observed environments where the Agent and the Principal observe only a part of the system. We present an approach based on the stochastic maximum principle, which aims to extend existing work that uses the principle of dynamic programming in non-degenerate systems. First, we solve the Principal problem in an extended contract set given by the first-order condition of the Agent problem in the form of a path-dependent stochastic differential equation (EDSPR). Then we use the sufficient condition of the Agent problem to verify that the obtained optimal contract is implementable. A parallel study is devoted to the existence and uniqueness of the solution of path-dependent EDSPRs in Chapter IV. We extend the decoupling field method to cases where the coefficients of the equations can depend on the trajectory of the forward process. We also prove a stability property for such EDSPRs. Finally, we study the moral hazard problem with several Principals. The Agent can only work for one Principal at a time and thus faces an optimal switching problem. Using the randomization method we show that the Agent's value function and its optimal effort are given by an Itô process. This representation helps us to solve the Principal problem when there are infinitely many Principals in equilibrium according to a mean-field game. We justify the mean-field formulation by a chaos propagation argument.The second part of this thesis consists of chapters V and VI. The motivation of this work is to give a rigorous theoretical foundation for the convergence of gradient descent type algorithms which are often used in the solution of non-convex problems such as the calibration of a neural network. For non-convex problems of the hidden layer neural network type, the key idea is to transform the problem into a convex problem by raising it in the space of measurements. We show that the corresponding energy function admits a unique minimizer which can be characterized by a first order condition using the derivation in the space of measures in the sense of Lions. We then present an analysis of the long term behavior of the Langevin mean-field dynamics, which has a gradient flow structure in the 2-Wasserstein metric. We show that the marginal law flow induced by the mean-field Langevin dynamics converges to a stationary law using La Salle's invariance principle, which is the minimizer of the energy function.In the case of deep neural networks, we model them using a continuous-time optimal control problem. We first give the first order condition using Pontryagin's principle, which will then help us to introduce the system of mean-field Langevin equations, whose invariant measure corresponds to the minimizer of the optimal control problem. Finally, with the reflection coupling method we show that the marginal law of the mean-field Langevin system converges to the invariant measure with an exponential speed.

This thesis is organized in three parts. The first part examines the relationship between microscopic and macroscopic market dynamics by focusing on the properties of volatility. In the second part, we focus on the stochastic optimal control of point processes. Finally, in the third part, we study two market design problems. We start this thesis by studying the links between the no-arbitrage principle and the volatility irregularity. Using a scaling method, we show that we can effectively connect these two notions by analyzing the market impact of metaorders. More precisely, we model the market order flow using linear Hawkes processes. We then show that the no-arbitrage principle and the existence of a non-trivial market impact imply that volatility is rough and more precisely that it follows a rough Heston model. We then examine a class of microscopic models where the order flow is a quadratic Hawkes process. The objective is to extend the rough Heston model to continuous models allowing to reproduce the Zumbach effect. Finally, we use one of these models, the quadratic rough Heston model, for the joint calibration of the SPX and VIX volatility slicks. Motivated by the intensive use of point processes in the first part, we are interested in the stochastic control of point processes in the second part. Our objective is to provide theoretical results for applications in finance. We start by considering the case of Hawkes process control. We prove the existence of a solution and then propose a method to apply this control in practice. We then examine the scaling limits of stochastic control problems in the context of population dynamics models. More precisely, we consider a sequence of models of discrete population dynamics which converge to a model for a continuous population. For each model we consider a control problem. We prove that the sequence of optimal controls associated to the discrete models converges to the optimal control associated to the continuous model. This result is based on the continuity, with respect to different parameters, of the solution of a backward-looking schostatic differential equation.In the last part we consider two market design problems. First, we examine the question of the organization of a liquid derivatives market. Focusing on an options market, we propose a two-step method that can be easily applied in practice. The first step is to select the options that will be listed on the market. For this purpose, we use a quantization algorithm that allows us to select the options most in demand by investors. We then propose a pricing incentive method to encourage market makers to offer attractive prices. We formalize this problem as a principal-agent problem that we solve explicitly. Finally, we find the optimal duration of an auction for markets organized in sequential auctions, the case of zero duration corresponding to the case of a continuous double auction. We use a model where the market takers are in competition and we consider that the optimal duration is the one corresponding to the most efficient price discovery process. After proving the existence of a Nash equilibrium for the competition between market takers, we apply our results on market data. For most assets, the optimal duration is between 2 and 10 minutes.

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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.

We address the mechanism design problem of an exchange setting suitable make-take fees to attract liquidity on its platform. Using a principal-agent approach, we provide the optimal compensation scheme of a market maker in quasi-explicit form. This contract depends essentially on the market maker inventory trajectory and on the volatility of the asset. We also provide the optimal quotes that should be displayed by the market maker. The simplicity of our formulas allows us to analyze in details the effects of optimal contracting with an exchange, compared to a situation without contract. We show in particular that it improves liquidity and reduces trading costs for investors. We extend our study to an oligopoly of symmetric exchanges and we study the impact of such common agency policy on the system.

This thesis is motivated by the study of the stability of the martingale optimal transport problem, and is naturally organized into two parts. In the first part, we exhibit a new family of martingale couplings between two one-dimensional probability measures μ and ν comparable in convex order. In particular, this family contains the inverse transform martingale coupling, which is explicit in terms of quantile functions of the marginals. The integral M_1(μ,ν) of |x-y| against each of these couplings is increased by twice the Wasserstein distance W_1(μ,ν) between μ and ν. We show a similar inequality when |x-y| and W_1 are respectively replaced by |x-y|^ρ and the product of W_ρ by the centered moment of order ρ of the second marginal raised to the exponent ρ-1, for any ρ∈[1,+∞[. We then study the generalization of this new stability inequality to the higher dimension. Finally, we establish a strong connection between our new family of martingale couplings and the projection of a coupling between two comparable given marginals in convex order onto the set of martingale couplings between these same marginals. This last projection is taken with respect to the adapted Wasserstein distance, which majors the usual Wasserstein distance and thus induces a finer topology better suited for financial modeling, since it takes into account the temporal structure of martingales. In the second part, we prove that any martingale coupling whose marginals are approximated by comparable probability measures in convex order can itself be approximated by martingale couplings in the sense of the adapted Wasserstein distance. We then discuss various applications of this result. In particular, we strengthen a stability result for the weak optimal transport problem and establish a stability result for the weak optimal martingale transport problem. We derive stability with respect to the marginals of the over-replication price of VIX futures contracts.

We address the mechanism design problem of an exchange setting suitable make-take fees to attract liquidity on its platform. Using a principal-agent approach, we provide the optimal compensation scheme of a market maker in quasi-explicit form. This contract depends essentially on the market maker inventory trajectory and on the volatility of the asset. We also provide the optimal quotes that should be displayed by the market maker. The simplicity of our formulas allows us to analyze in details the effects of optimal contracting with an exchange, compared to a situation without contract. We show in particular that it improves liquidity and reduces trading costs for investors. We extend our study to an oligopoly of symmetric exchanges and we study the impact of such common agency policy on the system.

In this paper we present a variational calculus approach to Principal-Agent problem with a lump-sum payment on finite horizon in degenerate stochastic systems, such as filtered partially observed linear systems. Our work extends the existing methodologies in the Principal-Agent literature using dynamic programming and BSDE representation of the contracts in the non-degenerate controlled stochastic systems. We first solve the Principal's problem in an enlarged set of contracts defined by a forward-backward SDE system given by the first order condition of the Agent's problem using variational calculus. Then we use the sufficient condition of the Agent's problem to verify that the optimal contract that we obtain by solving the Principal's problem is indeed implementable (i.e. belonging to the admissible contract set). Importantly we consider the control problem in a weak formulation. Finally, we give explicit solution of the Principal-Agent problem in partially observed linear systems and extend our results to some mean field interacting Agents case.

Backward stochastic differential equations extend the martingale representation theorem to the nonlinear setting. This can be seen as path-dependent counterpart of the extension from the heat equation to fully nonlinear parabolic equations in the Markov setting. This paper extends such a nonlinear representation to the context where the random variable of interest is measurable with respect to the information at a finite stopping time. We provide a complete wellposedness theory which covers the semilinear case (backward SDE), the semilinear case with obstacle (reflected backward SDE), and the fully nonlinear case (second order backward SDE).

Martingale transport plans on the line are known from Beiglbock & Juillet to have an irreducible decomposition on a (at most) countable union of intervals. We provide an extension of this decomposition for martingale transport plans in R^d, d larger than one. Our decomposition is a partition of R^d consisting of a possibly uncountable family of relatively open convex components, with the required measurability so that the disintegration is well-defined. We justify the relevance of our decomposition by proving the existence of a martingale transport plan filling these components. We also deduce from this decomposition a characterization of the structure of polar sets with respect to all martingale transport plans.

The theory of backward SDEs extends the predictable representation property of Brownian motion to the nonlinear framework, thus providing a path-dependent analog of fully nonlinear parabolic PDEs. In this paper, we consider backward SDEs, their reflected version, and their second-order extension, in the context where the final data and the generator satisfy L1-type of integrability condition. Our main objective is to provide the corresponding existence and uniqueness results for general Lipschitz generators. The uniqueness holds in the so-called Doob class of processes, simultaneously under an appropriate class of measures. We emphasize that the previous literature only deals with backward SDEs, and requires either that the generator is separable in (y,z), see Peng [Pen97], or strictly sublinear in the gradient variable z, see [BDHPS03], or that the final data satisfies an LlnL-integrability condition, see [HT18]. We by-pass these conditions by defining L1-integrability under the nonlinear expectation operator induced by the previously mentioned class of measures.

This thesis consists of two parts that can be read independently. In the first part of the thesis, we study hedging and option pricing problems related to a risk measure. Our main approach is the use of an asymmetric risk function and an asymptotic framework in which we obtain optimal solutions through nonlinear partial differential equations (PDEs).In the first chapter, we focus on the valuation and hedging of European options. We consider the problem of optimizing the residual risk generated by a discrete-time hedge in the presence of an asymmetric risk criterion. Instead of analyzing the asymptotic behavior of the solution of the associated discrete problem, we study the asymmetric residual risk measure integrated in a Markovian framework. In this context, we show the existence of this asymptotic risk measure. We then describe an asymptotically optimal hedging strategy via the solution of a totally nonlinear PDE. The second chapter applies this hedging method to the problem of valuing the output of a power plant. Since the power plant generates maintenance costs whether it is on or off, we are interested in reducing the risk associated with the uncertain revenues of this power plant by hedging with futures contracts. In the second part of the thesis, we consider several control problems related to economics and finance.The third chapter is dedicated to the study of a class of McKean-Vlasov (MKV) type problem with common noise, called conditional polynomial MKV. We reduce this polynomial class by Markov folding to finite dimensional control problems.We compare three different probabilistic techniques for numerically solving the reduced problem: quantization, control randomization regression, and delayed regression. We provide many numerical examples, such as portfolio selection with uncertainty about an underlying trend.In the fourth chapter, we solve dynamic programming equations associated with financial valuations in the energy market. We consider that a calibrated model for the underlyings is not available and that a small sample obtained from historical data is accessible.Moreover, in this context, we assume that futures contracts are often governed by hidden factors modeled by Markov processes. We propose a non-intrusive method to solve these equations through empirical regression techniques using only the historical log price of observable futures contracts.

This thesis aims at understanding several aspects of the roughness of volatility observed universally on financial assets. This is done in six steps. In the first part, we explain this property from the typical behaviors of agents in the market. More precisely, we build a microscopic price model based on Hawkes processes reproducing the important stylized facts of the market microstructure. By studying the long-run price behavior, we show the emergence of a rough version of the Heston model (called rough Heston model) with leverage. Using this original link between Hawkes processes and Heston models, we compute in the second part of this thesis the characteristic function of the log-price of the rough Heston model. This characteristic function is given in terms of a solution of a Riccati equation in the case of the classical Heston model. We show the validity of a similar formula in the case of the rough Heston model, where the Riccati equation is replaced by its fractional version. This formula allows us to overcome the technical difficulties due to the non-Markovian character of the model in order to value derivatives. In the third part, we address the issue of risk management of derivatives in the rough Heston model. We present hedging strategies using the underlying asset and the forward variance curve as instruments. This is done by specifying the infinite-dimensional Markovian structure of the model. Being able to value and hedge derivatives in the rough Heston model, we confront this model with the reality of financial markets in the fourth part. More precisely, we show that it reproduces the behavior of implied and historical volatility. We also show that it generates the Zumbach effect, which is a time-reversal asymmetry observed empirically on financial data. In the fifth part, we study the limiting behavior of the implied volatility at low maturity in the framework of a general stochastic volatility model (including the rough Bergomi model), by applying a density development of the asset price. While the approximation based on Hawkes processes has addressed several questions related to the rough Heston model, in Part 6 we consider a Markovian approximation applying to a more general class of rough volatility models. Using this approximation in the particular case of the rough Heston model, we obtain a numerical method for solving the fractional Riccati equations. Finally, we conclude this thesis by studying a problem not related to the rough volatility literature. We consider the case of a platform seeking the best make-take fee scheme to attract liquidity. Using the principal-agent framework, we describe the best contract to offer to the market maker as well as the optimal quotes displayed by the latter. We also show that this policy leads to better liquidity and lower transaction costs for investors.

No summary available.

No summary available.

We address the mechanism design problem of an exchange setting suitable make-take fees to attract liquidity on its platform. Using a principal-agent approach, we provide the optimal compensation scheme of a market maker in quasi-explicit form. This contract depends essentially on the market maker inventory trajectory and on the volatility of the asset. We also provide the optimal quotes that should be displayed by the market maker. The simplicity of our formulas allows us to analyze in details the effects of optimal contracting with an exchange, compared to a situation without contract. We show in particular that it improves liquidity and reduces trading costs for investors. We extend our study to an oligopoly of symmetric exchanges and we study the impact of such common agency policy on the system.

No summary available.

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.

We provide a probabilistic representations of the solution of some semilinear hyperbolic and high-order PDEs based on branching diffusions. These representations pave the way for a Monte-Carlo approximation of the solution, thus bypassing the curse of dimensionality. We illustrate the numerical implications in the context of some popular PDEs in physics such as nonlinear Klein-Gordon equation, a simplied scalar version of the Yang-Mills equation, a fourth-order nonlinear beam equation and the Gross-Pitaevskii PDE as an example of nonlinear Schrodinger equations.

This paper examines the Root solution of the Skorohod embedding problem given full marginals on some compact time interval. Our results are obtained by limiting arguments based on finitely-many marginals Root solution of Cox, Oblój, and Touzi. Our main result provides a characterization of the corresponding potential function by means of a convenient parabolic PDE.

This thesis deals with the theory of stochastic equations in finite dimension. In the first part, we derive necessary and sufficient geometric conditions on the coefficients of a stochastic differential equation for the existence of a solution constrained to remain in a closed domain, under weak regularity conditions on the coefficients.In the second part, we address existence and uniqueness problems of stochastic Volterra equations of convolutional type. These equations are in general non-Markovian. We establish their correspondence with infinite-dimensional equations which allows us to approximate them by finite-dimensional Markovian stochastic differential equations. Finally, we illustrate our results by an application in mathematical finance, namely the modeling of rough volatility. In particular, we propose a stochastic volatility model that provides a good compromise between flexibility and tractability.

In this thesis we study various aspects of martingale optimal transport in dimension greater than one, from duality to local structure, and finally propose methods of numerical approximation.We first prove the existence of irreducible components intrinsic to martingale transports between two given measures, and the canonicity of these components. We then prove a duality result for the optimal martingale transport in any dimension, the point by point duality is no longer true but a form of quasi-safe duality is proved. This duality allows us to prove the possibility of decomposing the quasi-safe optimal transport into a series of subproblems of point by point optimal transports on each irreducible component. We finally use this duality to prove a martingale monotonicity principle, analogous to the famous monotonicity principle of classical optimal transport. We then study the local structure of optimal transports, deduced from differential considerations. We obtain a characterization of this structure using real algebraic geometry tools. We deduce the structure of martingale optimal transports in the case of Euclidean norm power costs, thus solving a conjecture dating back to 2015. Finally, we compare existing numerical methods and propose a new method that is shown to be more efficient and to deal with an intrinsic problem of the martingale constraint that is the convex order defect. We also give techniques to handle the numerical problems in practice.

We provide a representation result of parabolic semi-linear PD-Es, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod [23], Watanabe [27] and McKean [18], by allowing for polynomial nonlinearity in the pair (u, Du), where u is the solution of the PDE with space gradient Du. Similar to the previous literature, our result requires a non-explosion condition which restrict to " small maturity " or " small nonlinearity " of the PDE. Our main ingredient is the automatic differentiation technique as in [15], based on the Malliavin integration by parts, which allows to account for the nonlin-earities in the gradient. As a consequence, the particles of our branching diffusion are marked by the nature of the nonlinearity. This new representation has very important numerical implications as it is suitable for Monte Carlo simulation. Indeed, this provides the first numerical method for high dimensional nonlinear PDEs with error estimate induced by the dimension-free Central limit theorem. The complexity is also easily seen to be of the order of the squared dimension. The final section of this paper illustrates the efficiency of the algorithm by some high dimensional numerical experiments.

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.

The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuous-time martingale transport on the Skorokhod space of c`adì ag paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S−topology and the dynamic programming principle 1 .

The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuous-time martingale transport on the Skorokhod space of c`adì ag paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S−topology and the dynamic programming principle 1 .

We consider a contracting problem in which a principal hires an agent to manage a risky project. When the agent chooses volatility components of the output process and the principal observes the output continuously, the principal can compute the quadratic variation of the output, but not the individual components. This leads to moral hazard with respect to the risk choices of the agent. We identify a family of admissible contracts for which the optimal agent's action is explicitly characterized, and, using the recent theory of singular changes of measures for Itô processes, we study how restrictive this family is. In particular, in the special case of the standard Homlstrom-Milgrom model with fixed volatility, the family includes all possible contracts. We solve the principal-agent problem in the case of CARA preferences, and show that the optimal contract is linear in these factors: the contractible sources of risk, including the output, the quadratic variation of the output and the cross-variations between the output and the contractible risk sources. Thus, like sample Sharpe ratios used in practice, path-dependent contracts naturally arise when there is moral hazard with respect to risk management. In a numerical example, we show that the loss of efficiency can be significant if the principal does not use the quadratic variation component of the optimal contract.

No summary available.

We propose an unbiased Monte-Carlo estimator for E[g(X t 1 , · · · , X tn)], where X is a diffusion process defined by a multi-dimensional stochastic differential equation (SDE). The main idea is to start instead from a well-chosen simulatable SDE whose coefficients are updated at independent exponential times. Such a simulatable process can be viewed as a regime-switching SDE, or as a branching diffusion process with one single living particle at all times. In order to compensate for the change of the coefficients of the SDE, our main representation result relies on the automatic differentiation technique induced by Bismu-Elworthy-Li formula from Malliavin calculus, as exploited by Fournié et al. [14] for the simulation of the Greeks in financial applications. In particular, this algorithm can be considered as a variation of the (infinite variance) estimator obtained in Bally and Kohatsu-Higa [3, Section 6.1] as an application of the parametrix method. MSC2010. Primary 65C05, 60J60. secondary 60J85, 35K10.

We consider a general formulation of the Principal-Agent problem with a lump-sum payment on a finite horizon, providing a systematic method for solving such problems. Our approach is the following: we first find the contract that is optimal among those for which the agent's value process allows a dynamic programming representation, for which the agent's optimal effort is straightforward to find. We then show that the optimization over the restricted family of contracts represents no loss of generality. As a consequence, we have reduced this non-zero sum stochastic differential game to a stochastic control problem which may be addressed by the standard tools of control theory. Our proofs rely on the backward stochastic differential equations approach to non-Markovian stochastic control, and more specifically, on the recent extensions to the second order case.

No summary available.

This thesis studies risk management and hedging using Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) as risk measures. The first part proposes a price evolution model that we confront with real data from the Paris stock exchange (Euronext PARIS). Our model takes into account the probabilities of occurrence of extreme losses and the regime changes observed in the data. Our approach consists in detecting the different periods of each regime by constructing a hidden Markov chain and estimating the tail of each regime distribution by power laws. We show empirically that the latter are more suitable than normal and stable distributions. The VaR estimation is validated by several backtests and compared to the results of other classical models on a base of 56 stock assets. In the second part, we assume that stock prices are modeled by exponential Lévy processes. First, we develop a numerical method for computing the cumulative VaR and CVaR. This problem is solved using the formalization of Rockafellar and Uryasev, which we evaluate numerically by Fourier inversion. In a second step, we focus on minimizing the hedging risk of European options, under a budget constraint on the initial capital. By measuring this risk by the CVaR, we establish an equivalence between this problem and a Neyman-Pearson type problem, for which we propose a numerical approximation based on the relaxation of the constraint.

We provide an extension of the martingale version of the Fréchet-Hoeffding coupling to the infinitely-many marginals constraints setting. In the two-marginal context, this extension was obtained by Beiglböck & Juillet [7], and further developed by Henry-Labordère & Touzi [40], see also [6]. Our main result applies to a special class of reward functions and requires some restrictions on the marginal distributions. We show that the optimal martingale transference plan is induced by a pure downward jump local Lévy model. In particular, this provides a new martingale peacock process (PCOC " Processus Croissant pour l'Ordre Convexe, " see Hirsch, Profeta, Roynette & Yor [43]), and a new remarkable example of discontinuous fake Brownian motions. Further, as in [40], we also provide a duality result together with the corresponding dual optimizer in explicit form. As an application to financial mathematics, our results give the model-independent optimal lower and upper bounds for variance swaps.

No summary available.

The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this paper, we consider an extension of the optimal Skorokhod embedding problem in Beiglböck , Cox & Huesmann [1] to the case of finitely-many marginal constraints 1. Using the classical convex duality approach together with the optimal stopping theory, we obtain the duality results which are formulated by means of probability measures on an enlarged space. We also relate these results to the problem of martingale optimal transport under multiple marginal constraints.

We consider the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval [0, 1]. The problem is related to the study of extremal martingales associated with a peacock (" process increasing in convex order " , by Hirsch, Profeta, Roynette and Yor [16]). A general duality result is obtained by convergence techniques. We then study the case where the reward function depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labord ere, Ob lój , Spoida and Touzi [13]. Under technical conditions, some explicit characteristics of the solutions to the optimal SEP as well as to its dual problem are obtained. We also discuss the associated martingale inequality.

We provide an extension of the martingale version of the Fréchet-Hoeffding coupling to the infinitely-many marginals constraints setting. In the two-marginal context, this extension was obtained by Beiglböck & Juillet [7], and further developed by Henry-Labordère & Touzi [40], see also [6]. Our main result applies to a special class of reward functions and requires some restrictions on the marginal distributions. We show that the optimal martingale transference plan is induced by a pure downward jump local Lévy model. In particular, this provides a new martingale peacock process (PCOC " Processus Croissant pour l'Ordre Convexe, " see Hirsch, Profeta, Roynette & Yor [43]), and a new remarkable example of discontinuous fake Brownian motions. Further, as in [40], we also provide a duality result together with the corresponding dual optimizer in explicit form. As an application to financial mathematics, our results give the model-independent optimal lower and upper bounds for variance swaps.

This is a continuation of our accompanying paper [18]. We provide an alternative proof of the monotonicity principle for the optimal Skorokhod embedding problem established in Beiglböck , Cox and Huesmann [2]. Our proof is based on the adaptation of the Monge-Kantorovich duality in our context, a delicate application of the optional cross-section theorem, and a clever conditioning argument introduced in [2].

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.

This volume is dedicated to the memory of Marc Yor, who passed away in 2014. The invited contributions by his collaborators and former students bear testament to the value and diversity of his work and of his research focus, which covered broad areas of probability theory. The volume also provides personal recollections about him, and an article on his essential role concerning the Doeblin documents. With contributions by P. Salminen, J-Y. Yen & M. Yor. J. Warren. T. Funaki. J. Pitman& W. Tang. J-F.

No summary available.

We study a class of martingale inequalities involving the running maximum process. They are derived from pathwise inequalities introduced by Henry-Labordere et al. (Ann. Appl. Probab., 2015 [arxiv:1203.6877v3]) and provide an upper bound on the expectation of a function of the running maximum in terms of marginal distributions at n intermediate time points. The class of inequalities is rich and we show that in general no inequality is uniformly sharp—for any two inequalities we specify martingales such that one or the other inequality is sharper. We use our inequalities to recover Doob’s L p inequalities. Further, for p = 1 we refine the known inequality and for p < 1 we obtain new inequalities.

In the context of the multi-dimensional infinite horizon optimal consumption-investment problem with proportional transaction costs, we provide the first order expansion in small transact costs. Similar to the one-dimensional derivation in our accompanying paper [42], the asymptotic expansion is expressed in terms of a singular ergodic control problem, and our arguments are based on the theory of viscosity solutions, and the techniques of homogenization which leads to a system of corrector equations. In contrast with the one-dimensional case, no explicit solution of the first corrector equation is available anymore. Finally, we provide some numerical results which illustrate the structure of the first order optimal controls.

We develop a weak exact simulation technique for a process X defined by a multi-dimensional stochastic differential equation (SDE). Namely, for a Lipschitz function g, we propose a simulation based approximation of the expectation E[g(X_{t_1}, \cdots, X_{t_n})], which by-passes the discretization error. The main idea is to start instead from a well-chosen simulatable SDE whose coefficients are up-dated at independent exponential times. Such a simulatable process can be viewed as a regime-switching SDE, or as a branching diffusion process with one single living particle at all times. In order to compensate for the change of the coefficients of the SDE, our main representation result relies on the automatic differentiation technique induced by Elworthy's formula from Malliavin calculus, as exploited by Fournie et al. for the simulation of the Greeks in financial applications.Unlike the exact simulation algorithm of Beskos and Roberts, our algorithm is suitable for the multi-dimensional case. Moreover, its implementation is a straightforward combination of the standard discretization techniques and the above mentioned automatic differentiation method.

Path Dependent PDE's (PPDE's) are natural objects to study when one deals with non Markovian models. Recently, after the introduction (see [12]) of the so-called pathwise (or functional or Dupire) calculus, various papers have been devoted to study the well-posedness of such kind of equations, both from the point of view of regular solutions (see e.g. [18]) and viscosity solutions (see e.g. [13]), in the case of finite dimensional underlying space. In this paper, motivated by the study of models driven by path dependent stochastic PDE's, we give a first well-posedness result for viscosity solutions of PPDE's when the underlying space is an infinite dimensional Hilbert space. The proof requires a substantial modification of the approach followed in the finite dimensional case. We also observe that, differently from the finite dimensional case, our well-posedness result, even in the Markovian case, apply to equations which cannot be treated, up to now, with the known theory of viscosity solutions.

This thesis studies some stochastic control problems in a context of liquidity risk and impact on asset prices. The thesis is composed of four chapters.In the second chapter, we propose a modeling of a market making problem in a liquidity risk context in the presence of inventory constraints and regime shifts. This formulation can be considered as an extension of previous studies on this subject. The main result of this part is the characterization of the value function as a unique solution, in the sense of viscosity, of a system of Hamilton-Jacobi-Bellman equations . In the third chapter, we propose a numerical approximation scheme to solve a portfolio optimization problem in a context of liquidity risk and impact on asset prices. We show that the value function can be obtained as a limit of an iterative procedure where each iteration represents an optimal stopping problem and we use a numerical algorithm, based on optimal quantization, to compute the value function and the control policy. The convergence of the numerical scheme is obtained via monotonicity, stability and consistency criteria.In the fourth chapter, we focus on a coupled problem of singular control and impulse control in an illiquidity context. We propose a mathematical formulation to model the dividend distribution and the investment policy of a firm subject to liquidity constraints. We show that, under transaction costs and an impact on the price of illiquid assets, the firm's value function is the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. An iterative numerical method is also proposed to compute the optimal buy, sell and dividend strategy.

We provide an extension of the martingale version of the Fréchet-Hoeffding coupling to the infinitely-many marginals constraints setting. In the two-marginal context, this extension was obtained by Beiglböck & Juillet [7], and further developed by Henry-Labordère & Touzi [40], see also [6]. Our main result applies to a special class of reward functions and requires some restrictions on the marginal distributions. We show that the optimal martingale transference plan is induced by a pure downward jump local Lévy model. In particular, this provides a new martingale peacock process (PCOC " Processus Croissant pour l'Ordre Convexe, " see Hirsch, Profeta, Roynette & Yor [43]), and a new remarkable example of discontinuous fake Brownian motions. Further, as in [40], we also provide a duality result together with the corresponding dual optimizer in explicit form. As an application to financial mathematics, our results give the model-independent optimal lower and upper bounds for variance swaps.

This paper provides a large deviation principle for Non-Markovian, Brownian motion driven stochastic differential equations with random coefficients. Similar to Gao \& Liu \cite{GL}, this extends the corresponding results collected in Freidlin \& Wentzell \cite{FreidlinWentzell}. However, we use a different line of argument, adapting the PDE method of Fleming \cite{Fleming} and Evans \& Ishii \cite{EvansIshii} to the path-dependent case, by using backward stochastic differential techniques. Similar to the Markovian case, we obtain a characterization of the action function as the unique bounded solution of a path-dependent version of the Eikonal equation. Finally, we provide an application to the short maturity asymptotics of the implied volatility surface in financial mathematics.

This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial di erential equations. We start by a quick review of the Crandall- Ishii notion of viscosity solutions, so as to motivate the relevance of our de nition in the path-dependent case. We focus on the wellposedness theory of such equations. In partic- ular, we provide a simple presentation of the current existence and uniqueness arguments in the semilinear case. We also review the stability property of this notion of solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme approximation method. Our results rely crucially on the theory of optimal stopping under nonlinear expectation. In the dominated case, we provide a self-contained presentation of all required results. The fully nonlinear case is more involved and is addressed in [12].

Let X : [0, T ] × Ω −→ R be a bounded c` adl` ag process with positive jumps defined on the canonical space of continuous paths Ω . We consider the problem of optimal stopping the process X under a nonlinear expectation operator E defined as the supremum of expectations over a weakly compact but nondominated family of probability measures. We introduce the corresponding nonlinear Snell envelope. Our main objective is to extend the Snell envelope characterization to the present context. Namely, we prove that the nonlinear Snell envelope is an E-supermartingale, and an E-martingale up to its first hitting time of the obstacle X . This result is obtained under an additional uniform continuity property of X . We also extend the result in the context of a random horizon optimal stopping problem. This result is crucial for the newly developed theory of viscosity solutions of path-dependent PDEs as introduced in Ekren et al. (2014), in the semilinear case, and extended to the fully nonlinear case in the accompanying papers (Ekren et al. [6,7]). c.

We consider an extension of the Monge-Kantorovitch optimal transportation problem. The mass is transported along a continuous semimartingale, and the cost of transportation depends on the drift and the diffusion coefficients of the continuous semimartingale. The optimal transportation problem minimizes the cost among all continuous semimartingales with given initial and terminal distributions. Our first main result is an extension of the Kantorovitch duality to this context. We also suggest a finite-difference scheme combined with the gradient projection algorithm to approximate the dual value. We prove the convergence of the scheme, and we derive a rate of convergence. We finally provide an application in the context of financial mathematics, which originally motivated our extension of the Monge-Kantorovitch problem. Namely, we implement our scheme to approximate no-arbitrage bounds on the prices of exotic options given the implied volatility curve of some maturity.

In the recent literature, martingale inequalities have been emphasized to be induced by pathwise inequalities independently of any reference probability measure on the paths space. This feature is closely related to the problem of robust hedging in nancial mathematics, which was originally addressed in some specic cases by means of the Skorohod embedding problem. The martingale optimal transport problem provides a systematic framework for the robust hedging problem and, therefore, allows to derive sharp martingale inequalities. We illustrate this methodology by deriving the sharpest possible control of the running maximum of a martingale by means of a nite number of marginals.

No summary available.

This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial di erential equations. We start by a quick review of the Crandall- Ishii notion of viscosity solutions, so as to motivate the relevance of our de nition in the path-dependent case. We focus on the wellposedness theory of such equations. In partic- ular, we provide a simple presentation of the current existence and uniqueness arguments in the semilinear case. We also review the stability property of this notion of solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme approximation method. Our results rely crucially on the theory of optimal stopping under nonlinear expectation. In the dominated case, we provide a self-contained presentation of all required results. The fully nonlinear case is more involved and is addressed in [12].

This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in [8] which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions, and reduces to the notion of stochastic viscosity solutions analyzed in [1, 2]. Our main result takes advantage of this enlargement of the test functions, and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions [3], and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in [8]. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution.

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.

This thesis presents two main independent research topics, the last one grouping two distinct problems. In the first part we focus on the martingale optimal transport problem, whose primary goal is to find no-arbitrage bounds for any option. We are first interested in the discrete time question of the existence of a probability law under which the canonical process is martingale, having two fixed marginal laws. This result due to Strassen (1965) is the starting point for the primal martingale optimal transport problem. We give a proof based on financial techniques of utility maximization, adapting a method developed by Rogers to prove the fundamental asset pricing theorem. These techniques correspond to a discretized time version of the martingale optimal transport. We then consider the continuous-time martingale optimal transport problem introduced in the lookback options framework by Galichon, Henry-Labordère and Touzi. We start by establishing a partial duality result concerning the robust over-coverage of any option. We adapt recent work by Neufeld and Nutz to the optimal martingale transport. We then study the robust utility maximization problem of any option with an exponential utility function in the framework of martingale optimal transport, and derive the robust utility indifference price, under a dynamic where the sharpe ratio is constant and known. In particular, we prove that this robust utility indifference price is equal to the robust over-coverage price. The second part of this thesis deals first with an optimal liquidation problem of an indivisible asset. We study the profitability of adding a strategy of buying and selling an asset orthogonal to the first one on the optimal liquidation strategy of the indivisible asset. We then provide some illustrative examples. The last chapter of this thesis concerns the indifference utility pricing problem of a European option in the presence of small transaction costs. We draw on recent work by Soner and Touzi to obtain asymptotic developments of the value functions of the Merton problems with and without the option. These developments are obtained using homogenization techniques. We formally obtain a system of equations verified by the components of the problem and we check that they are indeed solutions. Finally, we deduce an asymptotic development of the desired indifference utility price.

This book collects some recent developments in stochastic control theory with applications to financial mathematics. In the first part of the volume, standard stochastic control problems are addressed from the viewpoint of the recently developed weak dynamic programming principle. A special emphasis is put on regularity issues and, in particular, on the behavior of the value function near the boundary. Then a quick review of the main tools from viscosity solutions allowing one to overcome all regularity problems is provided.  The second part is devoted to the class of stochastic target problems, which extends in a nontrivial way the standard stochastic control problems. Here the theory of viscosity solutions plays a crucial role in the derivation of the dynamic programming equation as the infinitesimal counterpart of the corresponding geometric dynamic programming equation. The various developments of this theory have been stimulated by applications in finance and by relevant connections with geometric flows. namely, the second order extension was motivated by illiquidity modeling, and the controlled loss version was introduced following the problem of quantile hedging. The third part presents an overview of backward stochastic differential equations and their extensions to the quadratic case. Backward stochastic differential equations are intimately related to the stochastic version of Pontryagin’s maximum principle and can be viewed as a strong version of stochastic target problems in the non-Markov context. The main applications to the hedging problem under market imperfections, the optimal investment problem in the exponential or power expected utility framework, and some recent developments in the context of a Nash equilibrium model for interacting investors, are presented. The book concludes with a review of the numerical approximation techniques for nonlinear partial differential equations based on monotonic schemes methods in the theory of viscosity solutions.

No summary available.

We give a study to the algorithm for semi-linear parabolic PDEs in Henry-Labordere (2012) and then generalize it to the non-Markovian case for a class of Backward SDEs (BSDEs). By simulating the branching process, the algorithm does not need any backward regression. To prove that the numerical algorithm converges to the solution of BSDEs, we use the notion of viscosity solution of path dependent PDEs introduced by Ekren et al. (to appear) [5] and extended in Ekren et al. (2012) [6,7].

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present martingale version. We provide the explicit martingale optimal transference plans for a remarkable class of coupling functions corresponding to the lower and upper bounds. These explicit extremal probability measures coincide with the unique left and right monotone martingale transference plans, which were introduced in \cite{BeiglbockJuillet} by suitable adaptation of the notion of cyclic monotonicity. Instead, our approach relies heavily on the (weak) duality result stated in \cite{BeiglbockHenry-LaborderePenkner}, and provides, as a by-product, an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.

We consider the problem of optimal investment when agents take into account their relative performance by comparison to their peers. Given N interacting agents, we consider the following optimization problem for agent i,: where is the utility function of agent i, his portfolio, his wealth, the average wealth of his peers, and is the parameter of relative interest for agent i. Together with some mild technical conditions, we assume that the portfolio of each agent i is restricted in some subset. We show existence and uniqueness of a Nash equilibrium in the following situations: We also investigate the limit when the number of agents N goes to infinity. Finally, when the constraints sets are vector spaces, we study the impact of the s on the risk of the market.

We consider the problem of superhedging under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We present a general duality result which converts this problem into a min-max calculus of variations problem where the Lagrange multipliers correspond to the static part of the hedge. Following Galichon, Henry-Labordére and Touzi \cite{ght}, we apply stochastic control methods to solve it explicitly for Lookback options with a non-decreasing payoff function. The first step of our solution recovers the extended optimal properties of the Azéma-Yor solution of the Skorokhod embedding problem obtained by Hobson and Klimmek \cite{hobson-klimmek} (under slightly different conditions). The two marginal case corresponds to the work of Brown, Hobson and Rogers \cite{brownhobsonrogers}. The robust superhedging cost is complemented by (simple) dynamic trading and leads to a class of semi-static trading strategies. The superhedging property then reduces to a functional inequality which we verify independently. The optimality follows from existence of a model which achieves equality which is obtained in Ob\lój and Spoida \cite{OblSp}.

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present martingale version. We provide the explicit martingale optimal transference plans for a remarkable class of coupling functions corresponding to the lower and upper bounds. These explicit extremal probability measures coincide with the unique left and right monotone martingale transference plans, which were introduced in \cite{BeiglbockJuillet} by suitable adaptation of the notion of cyclic monotonicity. Instead, our approach relies heavily on the (weak) duality result stated in \cite{BeiglbockHenry-LaborderePenkner}, and provides, as a by-product, an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry-LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present martingale version. We provide the explicit martingale optimal transference plans for a remarkable class of coupling functions corresponding to the lower and upper bounds. These explicit extremal probability measures coincide with the unique left and right monotone martingale transference plans, which were introduced in \cite{BeiglbockJuillet} by suitable adaptation of the notion of cyclic monotonicity. Instead, our approach relies heavily on the (weak) duality result stated in \cite{BeiglbockHenry-LaborderePenkner}, and provides, as a by-product, an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.

We generalize the algorithm for semi-linear parabolic PDEs in Henry-Labordére \cite{Henry-Labordere_branching} to the non-Markovian case for a class of Backward SDEs (BSDEs). By simulating the branching process, the algorithm does not need any backward regression. To prove that the numerical algorithm converges to the solution of BSDEs, we use the notion of viscosity solution of path dependent PDEs introduced by Ekren, Keller, Touzi and Zhang \cite{EkrenKellerTouziZhang} and extended in Ekren, Touzi and Zhang \cite{EkrenTouziZhang1, EkrenTouziZhang2}.

The problem of robust hedging requires to solve the problem of superhedging under a nondominated family of singular measures. Recent progress was achieved by van Handel, Neufeld, and Nutz. We show that the dual formulation of this problem is valid in a context suitable for martingale optimal transportation or, more generally, for optimal transportation under controlled stochastic dynamics.

The problem of robust hedging requires to solve the problem of superhedging under a nondominated family of singular measures. Recent progress was achieved by van Handel, Neufeld, and Nutz. We show that the dual formulation of this problem is valid in a context suitable for martingale optimal transportation or, more generally, for optimal transportation under controlled stochastic dynamics.

I was interested in solving some financial problems by stochastic control. We first considered a mixed problem of optimal investment and optimal sale. We studied the behavior of an investor who owns an indivisible asset that he wants to sell while continuously managing a portfolio of risky assets. Then, the study of first and second order backward stochastic equations with convex constraints was performed. In each case, we proved the existence of a minimal solution and a stochastic representation for this problem. Finally, we have studied a stochastic volatility model where the instantaneous volatility depends on the forward volatility curve. We propose an asymptotic development of the option price for small variations of the volatility.

Vanilla calibration is a major problem in finance. We try to solve it for three classes of models: local and stochastic volatility models, the so-called "local correlation" model and a hybrid model of local volatility with stochastic rates. From a mathematical point of view, the calibration equation is a particularly complex nonlinear and integro-differential equation. In a first part, we prove the existence of solutions for this equation, as well as for its adjoint (simpler to solve). These results are based on fixed point methods in Hölder spaces and require classical theorems related to parabolic partial differential equations, as well as some a priori estimates in short time. The second part deals with the application of these existence results to the three financial models mentioned above. We also present the numerical results obtained by solving the edp. The calibration by this method is quite satisfactory. Finally, we focus on the algorithm used for the numerical solution: a predictor-corrector ADI scheme, which is modified to take into account the nonlinear character of the equation. We also describe an instability phenomenon of the edp solution that we try to explain from a theoretical point of view thanks to the so-called "Hadamard instability".

This thesis deals with numerical methods for degenerate nonlinear partial differential equations (PDEs), as well as for control problems of nonlinear PDEs resulting from a new optimal transport problem. All these questions are motivated by applications in financial mathematics. The thesis is divided into four parts. In the first part, we focus on the necessary and sufficient condition of monotonicity of the finite difference theta-schema for the diffusion equation in dimension one. We give the explicit formula in the case of the heat equation, which is weaker than the classical Courant-Friedrichs-Lewy (CFL) condition. In a second part, we consider a degenerate nonlinear parabolic PDE and propose a splitting scheme to solve it. This scheme combines a probabilistic scheme and a semi-Lagrangian scheme. Finally, it can be considered as a Monte-Carlo scheme. We give a convergence result and also a convergence rate of the scheme. In a third part, we study an optimal transport problem, where the mass is transported by a controlled drift-diffusion state process. The associated cost depends on the trajectories of the state process, its drift and its diffusion coefficient. The transport problem consists in minimizing the cost among all dynamics verifying the initial and terminal constraints on the marginal distributions. We prove a duality formulation for this transport problem, thus extending Kantorovich's duality to our context. The dual formulation maximizes a value function on the space of bounded continuous functions, and the corresponding value function for each bounded continuous function is the solution of an optimal stochastic control problem. In the Markovian case, we prove a dynamic programming principle for these optimal control problems, propose a projected gradient algorithm for the numerical solution of the dual problem, and prove its convergence. Finally, in a fourth part, we further develop the dual approach for the optimal transportation problem with an application to the search for arbitrage-free price bounds of variance options given European option prices. After a first analytical approximation, we propose a projected gradient algorithm to approximate the bound and the corresponding static strategy in vanilla options.

This thesis presents two main independent research topics, the last one being declined as two distinct problems. In the first part of the thesis, we focus on the notion of second order stochastic backward differential equations (in the following 2EDSR), first introduced by Cheredito, Soner, Touzi and Victoir and recently reformulated by Soner, Touzi and Zhang. We first prove an extension of their existence and uniqueness results when the considered generator is only continuous and linearly growing. Then, we continue our study by a new extension to the case of a quadratic generator. These theoretical results then allow us to solve a utility maximization problem for an investor in an incomplete market, both because constraints are imposed on his investment strategies, and because the volatility of the market is assumed to be unknown. We prove in our framework the existence of optimal strategies, characterize the value function of the problem thanks to a second-order SRGE and solve explicitly some examples that allow us to highlight the modifications induced by the addition of volatility uncertainty compared to the usual framework. We end this first part by introducing the notion of second order EDSR with reflection on an obstacle. We prove the existence and uniqueness of the solutions of such equations, and provide a possible application to the problem of shorting American options in a market with uncertain volatility. The first chapter of the second part of this thesis deals with an option pricing problem in a model where market liquidity is taken into account. We provide asymptotic developments of these prices in the neighborhood of infinite liquidity and highlight a phase transition phenomenon depending on the regularity of the payoff of the considered options. Some numerical results are also proposed. Finally, we conclude this thesis by studying a Principal/Agent problem in a moral hazard framework. A bank (which plays the role of the agent) has a number of loans for which it is willing to exchange the interest for capital flows. The bank can influence the default probabilities of these loans by performing or not performing costly monitoring activity. These choices of the bank are known only to the bank. Investors (who play the role of principal) want to set up contracts that maximize their utility while implicitly incentivizing the bank to perform constant monitoring activity. We solve this optimal control problem explicitly, describe the associated optimal contract and its economic implications, and provide some numerical simulations.

This thesis presents two main independent research topics, the last one being declined as two distinct problems. In the first part of the thesis, we focus on the notion of second order stochastic backward differential equations (in the following 2EDSR), first introduced by Cheredito, Soner, Touzi and Victoir and recently reformulated by Soner, Touzi and Zhang. We first prove an extension of their existence and uniqueness results when the considered generator is only continuous and linearly growing. Then, we continue our study by a new extension to the case of a quadratic generator. These theoretical results then allow us to solve a utility maximization problem for an investor in an incomplete market, both because constraints are imposed on his investment strategies, and because the volatility of the market is assumed to be unknown. We prove in our framework the existence of optimal strategies, characterize the value function of the problem thanks to a second-order SRGE and solve explicitly some examples that allow us to highlight the modifications induced by the addition of volatility uncertainty compared to the usual framework. We end this first part by introducing the notion of second order EDSR with reflection on an obstacle. We prove the existence and uniqueness of the solutions of such equations, and provide a possible application to the problem of shorting American options in a market with uncertain volatility. The first chapter of the second part of this thesis deals with an option pricing problem in a model where market liquidity is taken into account. We provide asymptotic developments of these prices in the vicinity of infinite liquidity and highlight a phase transition phenomenon depending on the regularity of the payoff of the considered options. Some numerical results are also proposed. Finally, we conclude this thesis by studying a Principal/Agent problem in a moral hazard framework. A bank (which plays the role of the agent) has a number of loans for which it is willing to exchange the interest for capital flows. The bank can influence the default probabilities of these loans by performing or not performing costly monitoring activity. These choices of the bank are known only to the bank. Investors (who play the role of principal) wish to set up contracts that maximize their utility while implicitly incentivizing the bank to perform constant monitoring activity. We solve this optimal control problem explicitly, describe the associated optimal contract and its economic implications, and provide some numerical simulations.

This thesis presents three independent research topics, the last one being declined as two distinct problems. These different topics have in common that they apply stochastic control methods to optimal portfolio management problems. In the first part, we study an asset management model that takes into account capital gains taxes. In a second part, we study a problem of detecting the maximum of a mean-reverting process. In the third and fourth parts, we look at an optimal investment problem when agents look at each other. Finally, in a fifth part, we study a variant of this problem including a penalty term instead of constraints on the admissible portfolios.

Over the last thirty years, the electricity industry has undergone considerable changes in its organization, all over the world. Traditionally a centralized industry, organized as a monopoly, at least locally, and often publicly owned, the production and supply of electricity has gradually opened up to the rules of the market and competition. Today, a large number of countries have established wholesale markets where producers and suppliers exchange electricity to meet the needs of end consumers. These markets are no longer only national and cross-border exchanges are becoming more and more frequent. The question of the organization of these markets is crucial, given the magnitude of the consequences in the event of a failure (we remember the California crisis or the Enron scandal in 2001). More generally, it is the energy markets as a whole that have been transformed in a context of growing demand, the threat of depletion of fossil fuels, environmental awareness and political tensions over access to resources. Around an oil market under pressure, the gas and coal markets have seen their volumes increase. The signing of the Kyoto Protocol and the implementation of a proactive policy to reduce greenhouse gas emissions have led to the creation of emission permit markets and stimulated the use of renewable energies, biofuels and nuclear energy. The increasing exposure of the global economy to energy prices has led financial markets to develop risk management products adapted to the commodities world (energy, agriculture, metals). These products manage price risk, but also volume risk, in order to hedge the risk of fluctuating demand. They are now also derivatives or insurance products on climate risk. This doctoral thesis is part of this context. It is composed of four chapters that can be read independently. The first chapter deals with the valuation of production assets such as thermal power plants. This Real Option valuation method is based on a utility indifference calculation and allows for the consideration of production constraints and market frictions. The asset value is characterized using backward-looking stochastic differential equations and gives access to Monte Carlo and PDE-type resolution methods. The second chapter focuses on the valuation of storage assets such as hydraulic dams. The average yield of the asset is characterized as the unique viscosity solution of a nonlinear partial differential equation. The numerical implementation by a finite difference scheme is discussed. The third chapter is dedicated to a modeling of the CO2 emission permits market. A competitive equilibrium model with two markets (electricity spot and emission rights) is introduced in a random environment. The equilibrium prices and the strategies of the actors are characterized thanks to a representative agent property. We then discuss the impact of CO2 regulation on electricity prices and the introduction of new market rules to reduce the increase in electricity prices. The last chapter is a study of the relationship between risk management and the industrial structure of the players. We introduce and characterize the competitive equilibrium of an economy with three markets, retail, forward and spot, in the presence of uncertainty. The players can have different industrial structures, supplier, producer, trader or vertically integrated. We derive the relationship between prices on the different markets and compare the impact on prices and utilities of the presence of a forward market or integrated players.

In the first part, we present a non-parametric method for estimating option price sensitivities using random perturbation of the parameter of interest, Monte Carlo simulations and kernel regression. For an irregular function, the estimator converges faster than finite difference estimators, which is numerically verified. The 2nd part proposes an algorithm for solving decoupled EDSPR systems with jumps. The discretization error in time is parametric. And the statistical error is controlled. We present numerical examples on coupled systems of semi-linear PDE. The 3rd part studies the behavior of a fund manager, maximizing the intertemporal utility of his consumption, under a drawdown constraint. We obtain in explicit form the optimal strategy in infinite horizon, and we characterize the value function in finite horizon as the unique viscosity solution of the corresponding HJB equation.

In the first part, we generalize results from the theory of arbitrage to the case of non-linear exchanges. We show the equivalence between two notions of no arbitrage and the existence of consistent price systems, and we consider the over-replication problem. The second part presents two original principles of conditional deviations, derived from financial problems. We give a large deviation approximation of the law of a pair of unobservable processes conditional on an observed function. We also show a conditional large deviations principle for the law of the solution of a couple-dependent SRDE. In the third part, we study the case of a hydroelectric power producer facing two sources of hazards: rainfall and price. We show that the value function of the corresponding optimal control problem is the unique viscosity constrained solution of a nonlinear PDE.

The first part of this thesis deals with vector risk measures. In Chapter 1, we generalize the notion of a coherent vector risk measure into a convex measure. We obtain a dual representation result that extends the characterization of coherent risk measures. In Chapter 2, we define a distribution-based vector risk measure and show that it coincides, under certain conditions, with a coherent risk measure. In the second part, we use stochastic control techniques to address two problems. Chapter 3 presents a characterization and a numerical solution of the value function for a consumption and investment optimization problem in a financial market with taxes on capital gains. Chapter 4 presents an explicit solution of the over-replication strategy of a contingent asset in the presence of partial transaction costs.

No summary available.

This thesis consists of 3 parts. The first part is devoted to over-replication problems in financial asset models with "uncertain volatility" (UV). The second part deals with probabilistic methods for solving some nonlinear multidimensional problems in finance, such as the pricing and hedging of American options on a basket of underlyings. In the third part, some of these methods are applied to the problems presented in the first part. The first part recalls the main results of the literature for stocks and extends them to interest rates by introducing an HJM framework with uncertain volatility. The second part is devoted to the acceleration of various probabilistic numerical methods for the solution of optimal stopping time problems. Three types of approaches are presented: regression methods (Longstaff-Schwartz, Tsitsiklis-Van Roy), Markov chain approximation methods (Broadie-Glasserman, Quantification) and Monte-Carlo Malliavin type methods. In the third part, we apply successively quantification methods and Monte-Carlo Malliavin type methods to the valuation and hedging of European options on a multidimensional underlying in a UV framework.

No summary available.

This work consists of six independent chapters. The first chapter exploits the characterization of the set of admissible prices by the no-arbitrage principle to examine the bias on hedging by the classical model of Black and Sholes (1973). The second chapter generalizes a result of A Rochet and Bajeux (1992) and shows that any option, European or American, completes the market in the presence of a correlation between the asset price and its volatility. The third chapter presents recent developments in portfolio choice and consumption theory. The next chapter gives a necessary and sufficient condition on a given price system (or risk premium pair) to be consistent with an additive multi-agent intertemporal equilibrium model. These statistical inference problems are addressed in the last part of the thesis. The fifth chapter introduces a method for estimating the parameters of the volatility process of the e. The convergence of the proposed estimators is proved and their asymptotic distribution is characterized. The last chapter compares, by Monte Carlo methods, this estimation method to indirect inference.