Optimal switching problems with an infinite set of modes: an approach by randomization and constrained backward SDEs

International audienceWe address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingale part, that allows to give a probabilistic representation of the value function of the given problem. This is achieved by randomization of control, i.e. by introducing an auxiliary optimization problem which has the same value as the starting optimal switching problem and for which the desired BSDE representation is obtained. In comparison with the existing literature we do not rely on a system of reflected BSDE nor can we use the associated Hamilton-Jacobi-Bellman equation in our non-Markovian framework

Quadratic BSDEs driven by a continuous martingale and application to utility maximization problem

In this paper, we study a class of quadratic Backward Stochastic Differential Equations (BSDEs) which arises naturally when studying the problem of utility maximization with portfolio constraints. We first establish existence and uniqueness results for such BSDEs and then, we give an application to the utility maximization problem. Three cases of utility functions will be discussed: the exponential, power and logarithmic ones

Reflected backward stochastic differential equations and a class of non linear dynamic pricing rule

20 pages, partial modification of the contentIn that paper, we provide a new characterization of the solutions of specific reflected backward stochastic differential equations (or RBSDEs) whose driver $g$ is convex and has quadratic growth in its second variable: this is done by introducing the extended notion of $g$-Snell enveloppe. Then, in a second step, we relate this representation to a specific class of dynamic monetary concave functionals already introduced in a discrete time setting. This connection implies that the solution, characterized by means of non linear expectations, has again the time consistency property

Utility Maximization in a jump market model

In this paper, we consider the classical problem of utility maximization in a financial market allowing jumps. Assuming that the constraint set is a compact set, rather than a convex one, we use a dynamic method from which we derive a specific BSDE. We then aim at showing existence and uniqueness results for the introduced BSDE. This allows us to give an explicit expression of the value function and characterize optimal strategies for our problem

Equations différentielles stochastiques rétrogrades à croissance quadratique et applications

In my PhDthesis, I have been mainly interested in the theoretical study of Backward Stochastic Differential Equations with Quadratic Growth. The other major part of my study consists in focusing on applications to finance and especially in the classical utility maximization problem under portfolio constraints. To this end, I have extended results for non linear BSDEs by using martingale methods already known in the brownian setting to solve this problem in more general filtrations.Dans cette thèse, l'étude menée consiste à établir de nouveaux résultats théoriques concernant des problèmes d'existence et d'unicité pour des Equations Différentielles Stochastiques Rétrogrades (EDSR) à croissance quadratique : ceci a pour but de permettre la résolution d'un problème de Mathématiques Financières, à savoir la maximisation de l'utilité (exponentielle) d'un portefeuille sous contraintes. Généralisant des résultats déjà connus en filtration brownienne pour les EDSR quadratiques, ce travail permet ainsi d'apporter des réponses au problème financier dans des contextes plus généraux