Backward stochastic differential equations and stochastic control and applications to financial mathematics.

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