SULEM Agnes

The treatment of uncertainties is a fundamental problem in the financial context. The variables studied are often time-dependent, with thick distribution tails. In this thesis, we are interested in tools to take into account uncertainties in its main forms: statistical uncertainties, parametric uncertainties and model error, keeping in mind that we wish to apply them to this context. The first part is devoted to the establishment of concentration inequalities in the context of variables with thick tails. The objective of these inequalities is to quantify the confidence that can be given to an estimator based on a finite size of observations. In this thesis, we establish new concentration inequalities, which cover in particular the case of estimators with lognormal distribution.In the second part, we deal with the impact of the model error for the estimation of the covariance matrix on stock returns, under the assumption that there is an instantaneous covariance process between the returns whose present value depends on its past value. We can then explicitly construct the best estimate of the covariance matrix for a given time and investment horizon, and show that it provides the smallest realized variance with high probability in the minimum variance portfolio framework.In the third part, we propose an approach to estimate the Sharpe ratio and the portfolio allocation when they depend on parameters considered uncertain. Our approach involves the adaptation of a stochastic approximation technique for the computation of the polynomial decomposition of the quantity of interest.Finally, in the last part of this thesis, we focus on portfolio optimization with target distribution. This technique can be formalized without any model assumptions on the returns. We propose to find these portfolios by minimizing divergence measures based on kernel functions and optimal transport theory.

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

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

We study the superhedging prices and the associated superhedging strategies for American options in a non-linear incomplete market model with default. The points of view of the seller and of the buyer are presented. The underlying market model consists of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. The portfolio processes follow non-linear dynamics with a non-linear driver f. We give a dual representation of the seller's (superhedging) price for the American option associated with a completely irregular payoff $(\xi_t)$ (not necessarily càdlàg) in terms of the value of a non-linear mixed control/stopping problem. The dual representation involves a suitable set of equivalent probability measures, which we call f-martingale probability measures. We also provide two infinitesimal characterizations of the seller's price process: in terms of the minimal supersolution of a constrained reflected BSDE and in terms of the minimal supersolution of an optional reflected BSDE. Under some regularity assumptions on $\xi$, we also show a duality result for the buyer's price in terms of the value of a non-linear control/stopping game problem.

This paper studies the superhedging prices and the associated superhedging strategies for European options in a non-linear incomplete market model with default. We present the seller's and the buyer's point of view. The underlying market model consists of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. The portfolio processes follow non-linear dynamics with a non-linear driver f. By using a dynamic programming approach, we first provide a dual formulation of the seller's (superhedging) price for the European option as the supremum, over a suitable set of equivalent probability measures Q ∈ Q, of the f-evaluation/expectation under Q of the payoff. We also provide a characterization of the seller's (superhedging) price process as the minimal supersolution of a constrained BSDE with default and a characterization in terms of the minimal weak supersolution of a BSDE with default. By a form of symmetry, we derive corresponding results for the buyer. Our results rely on first establishing a non-linear optional and a non-linear predictable decomposition for processes which are $\mathcal{E}^f$-strong supermartingales under Q, for all Q ∈ Q.

We introduce threshold growth in the classical threshold contagion model, or equivalently a network of Cramér-Lundberg processes in which nodes have downward jumps when there is a failure of a neighboring node. Choosing the configuration model as underlying graph, we prove fluid limits for the baseline model, as well as extensions to the directed case, state-dependent inter-arrival times and the case of growth driven by upward jumps. We obtain explicit ruin probabilities for the nodes according to their characteristics: initial threshold and in-(and out-) degree. We then allow nodes to choose their connectivity by trading off link benefits and contagion risk. We define a rational equilibrium concept in which nodes choose their connectivity according to an expected failure probability of any given link, and then impose condition that the expected failure probability coincides with the actual failure probability under the optimal connectivity. We show existence of an asymptotic equilibrium as well as convergence of the sequence of equilibria on the finite networks. In particular, our results show that systems with higher overall growth may have higher failure probability in equilibrium.

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

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We study Mean-field BSDEs with jumps and a generalized mean-field operator that can capture higher order interactions such as those occurring on an inhomogeneous random graph. We provide comparison and strict comparison results. Based on these, we interpret the BSDE solution as a global dynamic risk measure that can account for the intensity of system interactions and therefore incorporate systemic risk. Using Fenchel-Legendre transforms, we establish a dual representation for the risk measure, and in particular we exhibit its dependence on the mean-field operator.

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In this third edition, we have expanded and updated the second edition and includedmore recent developments within stochastic control and its applications.Specifically, we have replaced Section1.5on application tofinance by a morecomprehensive presentation offinancial markets modeled by jump diffusions (thenew Chap.2). We have added a new chapter on backward stochastic differentialequations, convex risk measures, and recursive utilities (Chap.4). Moreover, wehave expanded the optimal stopping chapter (was Chap. 2, now Chap.3) and thestochastic control chapter (was Chap. 3, now Chap.5) and added a new chapter onstochastic differential games (Chap.6). In addition, we have corrected errors andupdated and improved the presentation throughout the book.

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

This paper studies the superhedging prices and the associated superhedging strategies for European and American options in a non-linear incomplete market with default. We present the seller's and the buyer's point of view. The underlying market model consists of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. The portfolio process follows non-linear dynamics with a non-linear driver f. By using a dynamic programming approach, we first provide a dual formulation of the seller's (superhedging) price for the European option as the supremum over a suitable set of equivalent probability measures Q ∈ Q of the f-evaluation/expectation under Q of the payoff. We also provide an infinitesimal characterization of this price as the minimal supersolution of a constrained BSDE with default. By a form of symmetry, we derive corresponding results for the buyer. We also give a dual representation of the seller's (superhedging) price for the American option associated with an irregular payoff (ξ t) (not necessarily càdlàg) in terms of the value of a non-linear mixed control/stopping problem. We also provide an infinitesimal characterization of this price in terms of a constrained reflected BSDE. When ξ is càdlàg, we show a duality result for the buyer's price. These results rely on first establishing a non-linear optional decomposition for processes which are E f-strong supermartingales under Q, for all Q ∈ Q.

We study pricing and hedging for American options in an imperfect market model with default, where the imperfections are taken into account via the nonlinearity of the wealth dynamics. The payoff is given by an RCLL adapted process (ξt). We define the seller's price of the American option as the minimum of the initial capitals which allow the seller to build up a (super)hedging portfolio. We prove that this price coincides with the value function of an optimal stopping problem with a nonlinear expectation E g (induced by a BSDE), which corresponds to the solution of a nonlinear reflected BSDE with obstacle (ξt). Moreover, we show the existence of a (super)hedging portfolio strategy. We then consider the buyer's price of the American option, which is defined as the supremum of the initial prices which allow the buyer to select an exercise time τ and a portfolio strategy ϕ so that he/she is superhedged. We show that the buyer's price is equal to the value function of an optimal stopping problem with a nonlinear expectation, and that it can be characterized via the solution of a reflected BSDE with obstacle (ξt). Under the additional assumption of left upper semicontinuity along stopping times of (ξt), we show the existence of a super-hedge (τ, ϕ) for the buyer.

We study the problem of optimal control for mean-field stochastic partial differential equations (stochastic evolution equations) driven by a Brownian motion and an independent Poisson random measure, in the case of partial information control. One important novelty of our problem is represented by the introduction of general mean-field operators, acting on both the controlled state process and the control process. We first formulate a sufficient and a necessary maximum principle for this type of control. We then prove existence and uniqueness of the solution of such general forward and backward mean-field stochastic partial differential equations. We finally apply our results to find the explicit optimal control for an optimal harvesting problem.

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We study (nonlinear) Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale attached to a default jump with intensity process λ = (λ t). The driver of the BSDEs can be of a generalized form involving a singular optional finite variation process. In particular, we provide a comparison theorem and a strict comparison theorem. In the special case of a generalized λ-linear driver, we show an explicit representation of the solution, involving conditional expectation and an adjoint exponential semimartingale. for this representation, we distinguish the case where the singular component of the driver is predictable and the case where it is only optional. We apply our results to the problem of (nonlinear) pricing of European contingent claims in an imperfect market with default. We also study the case of claims generating intermediate cashflows, in particular at the default time, which are modeled by a singular optional process. We give an illustrating example when the seller of the European option is a large investor whose portfolio strategy can influence the probability of default.

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

We investigate network formation for a set of financial institutions represented as nodes. Linkages are source of income, and at the same time they bear the risk of contagion. The optimal connectivity of the nodes results from a game, in which the risk of contagion depends on the choices of all nodes in the system. Our financial network model can be interpreted as a set of banks connected through funding relations, in which a node's threshold to contagion is represented by its external funding capacity. A second interpretation is that of a set of insurers connected through reinsurance contracts, in which the threshold to contagion is represented by their capital. Our results show that when the threshold distribution across the nodes has higher variance, then, in equilibrium, the average connectivity is in general increasing, but the link failure probability decreases. This suggests that financial stability is best described in terms of the mechanism of network formation than in terms of simple statistics of the network topology like the average connectivity.

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We study optimal equity infusions into a financial network prone to the risk of contagious failures, which may be due to insolvency or to bank runs by short term creditors. Bank runs can be triggered by failures of connected banks. Under complete information on interbank linkages, we show that the problem reduces to a combinatorial optimization problem. Subject to budget constraints, the government chooses the set of minimal cost whose survival induces the maximum network stability. Our results demonstrate that the optimal equity infusion might significantly mitigate failure contagion risk and stabilize the system. In the case of partial information on the network, the controllers' focus swiftly changes from preventing insolvencies to preventing runs by short term creditors.

We study pricing and superhedging strategies for game options in an imperfect market with default. We extend the results obtained by Kifer in [Game Options, Finance. Stoch., 4 (2000), pp. 443–463] in the case of a perfect market model to the case of an imperfect market with default, when the imperfections are taken into account via the nonlinearity of the wealth dynamics. We introduce the seller's price of the game option as the infimum of the initial wealths which allow the seller to be superhedged. We prove that this price coincides with the value function of an associated generalized Dynkin game, recently introduced in [R. Dumitrescu, M.-C. Quenez, and A. Sulem, Elect. J. Probab., 21 (2016), 64], expressed with a nonlinear expectation induced by a nonlinear backward SDE with default jump. We, moreover, study the existence of superhedging strategies. We then address the case of ambiguity on the model—for example ambiguity on the default probability—and characterize the robust seller's price of a game option as the value function of a mixed generalized Dynkin game. We study the existence of a cancellation time and a trading strategy which allow the seller to be superhedged, whatever the model is.

This article studies singular mean field control problems and singular mean field two-players stochastic differential games. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions the optimality conditions for singular mean-field control are reduced to a reflected Skorohod problem, whose solution is proved to exist uniquely. Motivations are given as optimal harvesting of stochastic mean-field systems, optimal irreversible investments under uncertainty and mean-field singular investment games. In particular, a simple singular mean-field investment game is studied, where the Nash equilibrium exists but is not unique.

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In this thesis, we consider the joint law modeling of order statistics, i.e. random vectors with almost surely ordered components. The first part is dedicated to the probabilistic modeling of order statistics of maximum entropy with fixed marginals. Since the marginals are fixed, the characterization of the joint distribution amounts to considering the associated copula. In Chapter 2, we present an auxiliary result on maximum entropy copulas with fixed diagonal. A necessary and sufficient condition is given for the existence of such a copula, as well as an explicit formula for its density and entropy. The solution of the entropy maximization problem for order statistics with fixed marginals is presented in Chapter 3. Explicit formulas for its copula and joint density are given. In the second part of the thesis, we study the problem of non-parametric estimation of maximum entropy densities of order statistics in Kullback-Leibler distance. Chapter 5 describes an aggregation method for probability and spectral densities, based on a convex combination of its logarithms, and shows non-asymptotic optimal bounds in deviation. In Chapter 6, we propose an adaptive method based on an exponential log-additive model to estimate the considered densities, and we show that it reaches the known minimax speeds. The application of this method to estimate the dimensions of defects is presented in Chapter 7.

A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: (i) The optimal terminal wealth X^*(T) : = X_{\varphi ^*}(T) of the problem to maximize the expected U-utility of the terminal wealth X_{\varphi }(T) generated by admissible portfolios \varphi (t). 0 \le t \le T in a market with the risky asset price process modeled as a semimartingale. (ii) The optimal scenario \frac{dQ^*}{dP} of the dual problem to minimize the expected V-value of \frac{dQ}{dP} over a family of equivalent local martingale measures Q, where V is the convex conjugate function of the concave function U. In this paper we consider markets modeled by Itô-Lévy processes. In the first part we use the maximum principle in stochastic control theory to extend the above relation to a dynamic relation, valid for all t \in [0,T]. We prove in particular that the optimal adjoint process for the primal problem coincides with the optimal density process, and that the optimal adjoint process for the dual problem coincides with the optimal wealth process.

We study a combined optimal control/stopping problem under a nonlinear expectation E f induced by a BSDE with jumps, in a Markovian framework. The terminal reward function is only supposed to be Borelian. The value function u associated with this problem is generally irregular. We first establish a sub-(resp., super-) optimality principle of dynamic programming involving its upper-(resp., lower-) semicontinuous envelope u * (resp., u *). This result, called the weak dynamic programming principle (DPP), extends that obtained in [Bouchard and Touzi, SIAM J. Control Optim., 49 (2011), pp. 948–962] in the case of a classical expectation to the case of an E f-expectation and Borelian terminal reward function. Using this weak DPP, we then prove that u * (resp., u *) is a viscosity sub-(resp., super-) solution of a nonlinear Hamilton–Jacobi–Bellman variational inequality.

These Proceedings offer a selection of peer-reviewed research and survey papers by some of the foremost international researchers in the fields of finance, energy, stochastics and risk, who present their latest findings on topical problems. The papers\ua0cover the areas\ua0of\ua0stochastic modeling\ua0in energy and financial markets.\ua0risk management with environmental factors from a stochastic control perspective. and\ua0valuation and hedging of derivatives\ua0in markets dominated by renewables, all of which further develop the theory of stochastic analysis and mathematical finance. The papers were presented at the first conference on "Stochastics of Environmental and Financial Economics (SEFE)", being\ua0part of the activity in the SEFE research group of\ua0the Centre of Advanced Study (CAS)\ua0at the Academy of Sciences in Oslo, Norway during the 2014/2015 academic year.

We introduce a game problem which can be seen as a generalization of the classical Dynkin game problem to the case of a nonlinear expectation ${\cal E}^g$, induced by a Backward Stochastic Differential Equation (BSDE) with jumps with nonlinear driver $g$. Let $\xi, \zeta$ be two RCLL adapted processes with $\xi \leq \zeta$. The criterium is given by $ {\cal J}_{\tau, \sigma}= {\cal E}^g_{0, \tau \wedge \sigma } \left(\xi_{\tau}\textbf{1}_{\{ \tau \leq \sigma\}}+\zeta_{\sigma}\textbf{1}_{\{\sigma<\tau\}}\right)$ where $\tau$ and $ \sigma$ are stopping times valued in $[0,T]$. Under Mokobodzki's condition, we establish the existence of a value function for this game, i.e. $\inf_{\sigma}\sup_{\tau} {\cal J}_{\tau, \sigma} = \sup_{\tau} \inf_{\sigma} {\cal J}_{\tau, \sigma}$. This value can be characterized via a doubly reflected BSDE. Using this characterization, we provide some new results on these equations, such as comparison theorems and a priori estimates. When $\xi$ and $\zeta$ are left upper semicontinuous along stopping times, we prove the existence of a saddle point. We also study a generalized mixed game problem when the players have two actions: continuous control and stopping. We then study the generalized Dynkin game in a Markovian framework and its links with parabolic partial integro-differential variational inequalities with two obstacles.

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In this paper, three different topics related to stochastic control and computation are discussed, based on the notion of a random measure directed stochastic backward differential equation (SRDE). The first three chapters of the thesis deal with optimal control problems for different classes of non-diffusive Markovian processes, with finite or infinite horizon. In each case, the value function, which is the unique solution of a Hamilton-Jacobi-Bellman (HJB) integro-differential equation, is represented as the unique solution of an appropriate RLS. In the first chapter, we control a class of finite-horizon semi-Markovian processes. The second chapter is devoted to the optimal control of pure jump Markovian processes, while in the third chapter we consider the case of infinite-horizon piecewise deterministic Markovian processes (PDMPs). In the second and third chapters the HJB equations associated with optimal control are completely nonlinear. This situation arises when the laws of the controlled processes are not absolutely continuous with respect to the law of a given process. Given this completely nonlinear character, these equations cannot be represented by classical EDSRs. In this framework, we obtained nonlinear Feynman-Kac formulas by generalizing the control randomization method introduced by Kharroubi and Pham (2015) for diffusions. These techniques allow us to relate the value function of the control problem to a random measure directed SDE, one component of whose solution is sign constrained. Furthermore, we show that the value function of the original non-dominated control problem coincides with the value function of an auxiliary dominated control problem, expressed in terms of changes in equivalent probability measures. In the fourth chapter, we study a finite horizon backward stochastic differential equation directed by an integer-valued random measure on $R_+ times E$, where $E$ is a Lusinian space, with compensator of the form $nu(dt, dx) = dA_t phi_t(dx)$. The generator of this equation satisfies a uniform Lipschitz condition with respect to the unknowns. In the literature, the existence and uniqueness for EDSRs in this framework have been established only when $A$ is continuous or deterministic. We provide an existence and uniqueness theorem even when $A$ is a predictable, nondecreasing, right-hand continuous process. This result applies, for example, to the case of control related to PDMPs. Indeed, when $mu$ is the jump measure of a PDMP on a bounded domain, $A$ is predictable and discontinuous. Finally, in the last two chapters of the thesis we deal with stochastic computation for general discontinuous processes. In the fifth chapter, we develop the stochastic calculus via regularizations of jump processes which are not necessarily semimartingales. In particular we continue the study of so-called weak Dirichlet processes in the discontinuous framework. Such a process $X$ is the sum of a local martingale and an adapted process $A$ such that $[N, A] = 0$, for any continuous local martingale $N$. For a function $u: [0, T] times R rightarrow R$ of class $C^{0,1}$ (or sometimes less), we express a development of $u(t, X_t)$, in the spirit of a generalization of the Itô lemma, which holds when $u$ is of class $C^{1,2}$. The computation is applied in the sixth chapter to the theory of EDSRs directed by random measures. In many situations, when the underlying process $X$ is a special semimartingale, or more generally, a special weak Dirichlet process, we identify the solutions of the considered EDSRs via the process $X$ and the solution $u$ of an associated integro-differential PDE.

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We study optimal stochastic control problems of general coupled systems of forward-backward stochastic differential equations with jumps. By means of the Ito-Ventzell formula, the system is transformed into a controlled backward stochastic partial differential equation. Using a comparison principle for such equations we obtain a general stochastic Hamilton-Jacobi-Bellman (HJB) equation for the value function of the control problem. In the case of Markovian optimal control of jump diffusions, this equation reduces to the classical HJB equation. The results are applied to study risk minimization in financial markets.

We consider a stylized core-periphery financial network in which links lead to the creation of projects in the outside economy but make banks prone to contagion risk. The controller seeks to maximize, under budget constraints, the value of the financial system defined as the total amount of external projects. Under partial information on interbank links, revealed in conjunction with the spread of contagion, the optimal control problem is shown to become a Markov decision problem. We find the optimal intervention policy using dynamic programming. Our numerical results show that the value of the system depends on the connectivity in a non- monotonous way: it first increases with connectivity and then decreases with connectivity. The maximum value attained depends critically on the budget of the controller and the availability of an adapted intervention strategy. Moreover, we show that for highly connected systems, it is optimal to increase the rate of intervention in the peripheral banks rather than in core banks. Keywords: Systemic risk, Optimal control, Financial networks.

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We study a coupled system of controlled stochastic differential equations (SDEs) driven by a Brownian motion and a compensated Poisson random measure, consisting of a forward SDE in the unknown process X(t) and a predictive mean-field backward SDE (BSDE) in the unknowns Y(t),Z(t),K(t,⋅). The driver of the BSDE at time t may depend not just upon the unknown processes Y(t),Z(t),K(t,⋅), but also on the predicted future value Y(t+δ), defined by the conditional expectation A(t):=E[Y(t+δ)|Ft]. We give a sufficient and a necessary maximum principle for the optimal control of such systems, and then we apply these results to the following two problems: (i) Optimal portfolio in a financial market with an insider influenced asset price process. (ii) Optimal consumption rate from a cash flow modeled as a geometric Itô-Lévy SDE, with respect to predictive recursive utility.acceptedVersio.

We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). We first provide general existence, uniqueness and comparison theorems for RBSDEs with jumps in the case of a RCLL adapted obstacle. We then show that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of an optimal stopping time is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, robust optimal stopping problems related to the case with model ambiguity are investigated.

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We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). The financial position is given by an RCLL adapted process. We first state some properties of RBSDEs with jumps when the obstacle process is RCLL only. We then prove that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of optimal stopping times is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, we investigate robust optimal stopping problems related to the case with model ambiguity and their links with mixed control/optimal stopping game problems. We prove that, under some hypothesis, the value function is equal to the solution of an RBSDE. We then study the existence of saddle points when the obstacle is left-upper semi-continuous along stopping times.

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We study the optimal stopping problem for a monotonous dynamic risk measure induced by a Backward Stochastic Differential Equation with jumps in the Markovian case.We show that the value function is a viscosity solution of an obstacle problem for a partial integro-differential variational inequality and we provide an uniqueness result for this obstacle problem.

We study the optimal control of interbank contagion, when the government has complete information on interbank exposures. Financial institutions are prone to insolvency risk channeled through the network of exposures and to liquidity risk through bank runs. The government seeks to maximize, under budget constraints the total value of the financial system or, equivalently, to minimize the dead-weight loss induced by bank runs. The problem can be expressed as a convex optimization problem with a combinatorial aspect, tractable when the set of banks eligible for intervention is sufficiently, yet realistically, small.

This paper is mainly a survey of recent research developments regarding methods for risk minimization in financial markets modeled by Itô-Lévy processes, but it also contains some new results on the underlying stochastic maximum principle. The concept of a convex risk measure is introduced, and two representations of such measures are given, namely: (i) the dual representation and (ii) the representation by means of backward stochastic differential equations (BSDEs) with jumps. Depending on the representation, the corresponding risk minimal portfolio problem is studied, either in the context of stochastic differential games or optimal control of forward-backward SDEs. The related concept of recursive utility is also introduced, and corresponding recursive utility maximization problems are studied. In either case the maximum principle for optimal stochastic control plays a crucial role, and in the paper we prove a version of this principle which is stronger than what was previously known. The theory is illustrated by examples, showing explicitly the risk minimizing portfolio in some cases. Afrika Matematika May 2014, Available online on 18 May 2014 The final publication is available at Springe.

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We consider a financial market model with a single risky asset whose price process evolves according to a general jump-diffusion with locally bounded coefficients and where market participants have only access to a partial information flow $(\E_t)_{t\geq0}\subseteq(\F_t)_{t\geq0}$. For any utility function, we prove that the partial information financial market is locally viable, in the sense that the problem of maximizing the expected utility of terminal wealth has a solution up to a stopping time, if and only if the marginal utility of the terminal wealth is the density of a partial information equivalent martingale measure (PIEMM). This equivalence result is proved in a constructive way by relying on maximum principles for stochastic control under partial information. We then show that the financial market is globally viable if and only if there exists a partial information local martingale deflator (PILMD), which can be explicitly constructed. In the case of bounded coefficients, the latter turns out to be the density process of a global PIEMM. We illustrate our results by means of an explicit example.

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We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model certainty, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index. Published online: 01 Sep 2012 The final publication is available at Springe.

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We consider general singular control problems for random fields given by a stochastic partial differential equation (SPDE). We show that under some conditions the optimal singular control can be identified with the solution of a coupled system of SPDE and a reflected backward SPDE (RBSPDE). As an illustration we apply the result to a singular optimal harvesting problem from a population whose density is modeled as a stochastic reaction-diffusion equation. Existence and uniqueness of solutions of RBSPDEs are established, as well as comparison theorems. We then establish a relation between RBSPDEs and optimal stopping of SPDEs, and apply the result to a risk-minimizing stopping problem.

We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). We first provide general existence, uniqueness and comparison theorems for RBSDEs with jumps in the case of a RCLL adapted obstacle. We then show that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of an optimal stopping time is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, robust optimal stopping problems related to the case with model ambiguity are investigated.

We study double barrier reflected BSDEs (DBBSDEs) with jumps and RCLL barriers, and their links with generalized Dynkin games. We provide existence and uniqueness results and prove that for any Lipschitz driver, the solution of the DBBSDE coincides with the value function of a game problem, which can be seen as a generalization of the classical Dynkin problem to the case of $g$-conditional expectations. Using this characterization, we prove some new results on DBBSDEs with jumps, such as comparison theorems and a priori estimates.

We study the links between reflected backward stochastic differential equations (reflected BSDEs) with jumps and partial integro-differential variational inequalities (PIDVIs). In a Markovian framework, we show that the solution of the reflected BSDE corresponds to the unique viscosity solution of the PIDVI. We apply these results to an optimal stopping problem for dynamic risk measures induced by BSDEs with jumps.

We study optimal stochastic control problems of general coupled systems of forward- backward stochastic di erential equations with jumps. By means of the It^o-Ventzell formula the system is transformed to a controlled backward stochastic partial di eren- tial equation (BSPDE) with jumps. Using a comparison principle for such BSPDEs we obtain a general stochastic Hamilton-Jacobi- Bellman (HJB) equation for such control problems. In the classical Markovian case with optimal control of jump di usions, the equation reduces to the classical HJB equation. The results are applied to study risk minimization in nancial markets.

A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: $$\text{Unknown environment 'myenumerate'}$$ In this paper we consider markets modeled by Itô-Lévy processes, and in the first part we give a new proof of the above result in this setting, based on the maximum principle in stochastic control theory. An advantage with our approach is that it also gives an explicit relation between the optimal portfolio $\varphi^*$ and the optimal measure $Q^*$, in terms of backward stochastic differential equations. In the second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a relation between them. In particular, we show explicitly how to get from the solution of one of the problems to the solution of the other. We illustrate the results with explicit examples.

In the Brownian case, the links between dynamic risk measures and BSDEs have been widely studied. In this paper, we study the case with jumps. We first study the properties of BSDEs driven by a Brownian motion and a Poisson random measure. In particular, we provide a comparison theorem, under quite weak assumptions, extending that of Royer \cite{R}. We then give some properties of dynamic risk measures induced by BSDEs with jumps.

We study a preferred equity infusion government program set to mitigate interbank contagion. Financial institutions are prone to insolvency risk channeled through the network of interbank debt and to funding liquidity risk. The government seeks to maximize, under budget constraints, the total net worth of the financial system or, equivalently, to minimize the dead-weight losses induced by bank runs. The government is assumed to have complete information on interbank debt. The problem of quantifying the optimal amount of infusions can be expressed as a convex combinatorial optimization problem, tractable when the set of banks eligible for intervention (core banks) is sufficiently, yet realistically, small. We find that no bank has an incentive to withdraw from the program, when the preferred dividend rate paid to the government is equal to the government's outside return on the intervention budget. On the other hand, it may be optimal for the government to make infusions in a strict subset of core banks.

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

This thesis gives two applications of Malliavin's calculus for jump processes. In the first part, we deal with the density minimization of hopping diffusions whose continuous part is directed by a Brownian motion. For this purpose, we use a conditional integration by parts formula based on the Brownian motion only. We then treat the computation of financial options whose underlying price is a pure jump process. In the second part, we develop an abstract Malliavin-type calculus based on non-independent random variables of discontinuous conditional density. We establish an integration by parts formula that we apply to the amplitudes and jump times of the considered jump processes. In the third part, we use this integration by parts to compute the Delta of European and Asian options and the price and Delta of American options.

We study issues related to the valuation and hedging of basket credit derivatives. In particular, we are interested, on the one hand, in the modeling of default correlation and, on the other hand, in the market incompleteness introduced by the correlation risk. This thesis also contributes to the literature on numerical methods for valuing basket products. In the first chapter, we study the conditional-hop diffusion approach and the impact of filter magnification on the dynamics of intensity processes. The second chapter is devoted to the Marshall-Olkin copula. A detailed study of its properties is made, as well as the parameterization of this correlation structure. In the third chapter, we develop semi-analytical methods to evaluate basket credit derivatives in a Marshall-Olkin model. In the fourth chapter, we solve the hedging problem of basket products. Finally, in the fifth chapter, we analyze the correlation risk found in a new generation of products known as CDO-squared.

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The first part is devoted to stochastic and impulse optimal control. We propose two algorithms to numerically solve Quasi Variational inequalities that arise in a portfolio management problem with fixed and proportional transaction costs. In the second part we apply Malliavin's calculus to the calculation of sensitivities. We study pure jump processes and establish part-integration formulas using the densities of the jump amplitudes that we assume to be differentiable. Then we weaken the assumption on the densities by assuming them piecewise differentiable. Thus we use the density of jump times to establish PPI formulas. We also study models of continuous multi-factor scattering. The ellipticity of the diffusion is necessary for the classical Malliavin calculus approach. For European options we establish several PPIs independently of the ellipticity of the diffusion, using other variables that aggregate the multidimensional diffusion and reduce the dimension of the Malliavin covariance matrix. In the last chapter we study the calibration of the local volatility by minimizing the relative entropy. This involves solving a stochastic control problem. We propose improvements to existing algorithms.

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This thesis studies the discrete-time hedging of European options. In the first part, we introduce hedging restrictions in the black-scholes model: we assume that the market-maker can only hedge a fixed maximum number of times at random times of his choice. We identify the strategy that minimizes the variance of the hedging error. We show that the minimum variance is a solution of a sequence of optimal stopping problems that lead to variational inequalities (i. V. ). Using the viscosity solution technique, we study the existence and uniqueness of solutions of these i. V. And we show the convergence of the solution of the discretized problem by the finite difference method to the solution of the continuous problem. Finally, we extend these results to other criteria. In the second part, we determine the smallest initial wealth needed to over-cover the option in the black-scholes model in the following real-world setting: the market-maker can only hedge at random times of his choice. When the number of covers is fixed, we show that this price corresponds to the buy-and-hold strategy for a call, or the corresponding strategy for any option with a continuous payoff. In the case where the number can depend on the trajectory of the spot and the delta of the contingent asset black-scholes option is a finite variation process (which excludes all standard options in general), we show that the smallest price is the black-scholes price of the option.

The author proposes a general alternative model to the classical Gaussian models. This model is based on alpha-stable processes or Levy processes. These processes have many important characteristics: much larger occurrence of large events (thick-tailed laws), asymmetry of ups and downs, self-similarity in agreement with statistical measures, taking into account the volatility smile. The thesis presents a complete set of analytical and numerical solutions allowing an effective implementation and computation associated with the use of these processes. It also proposes a closed-form pricing formula for foreign exchange options, extensible to equities and interest rates, and studies different strategies of use: arbitrage on the volatility smile, improvement of hedging management costs.