Model risk analysis in finance: reflected backward stochastic differential equations with random terminal time.

Summary This thesis is divided into three parts. The first two parts are devoted to model risk in finance (valuation, management). The third part is devoted to backward stochastic differential equations with random terminal time and some of their applications. In the first part, we study the speed of convergence of the numerical approximation of quantiles of the law of a component of (X_t), when (X_t) is a diffusion process and when we use a Monte-Carlo method combined with the Euler time discretization scheme of the process. The speed of convergence is obtained under two different assumptions: either (X_t) has a uniformly hypoelliptic generator, or the inverse of the Malliavin covariance matrix of the component of X_t considered satisfies a certain condition (M). We then show that this condition (M) is satisfied in various contexts in finance. In the second part, we focus on model risk control. We study a strategy that, in a sense, guarantees good performance regardless of the (unknown) model of the underlying assets used in the hedge portfolio. We consider the model risk control problem as a two-player (trader vs. market) zero-sum stochastic game problem corresponding to a 'worst case' protection. We prove that the corresponding value function is the unique viscosity solution of a Hamilton-Jacobi-Bellman-Isaacs equation. The third part of the thesis deals with various issues related to backward stochastic differential equations with random terminal time, their relations with Dynkin sets and viscosity solutions of various elliptic problems.