Malliavin Smoothness of Solutions of BSDE and Applications

Abstract

his thesis studies two different topics in the stochastic systems fields: The solutions’ Malliavin regularity and control theory. The first is related to the Malliavin smoothness of the solutions of a specific type of quadratic backward stochastic differential equation (chapter 1) and the convergence of their numerical approximating schemes (chapter 2). The second topic refers to optimal control problems for stochastic systems with non-smooth coefficients (chapter 3). Chapter one focuses on the Lq(q ≥ 2)-existence and uniqueness of the solutions of the one-dimensional quadratic backward stochastic differential equation (Q-BSDEs for short) and their properties. The Lp-Hölder continuity of the solutions for any (q > 4 and 2 ≤ p < q2 ) are established and some important results concerning the smoothness of the solution of Q-BSDEs are presented. These findings are obtained based on the connection between the underlying Q-BSDEs and the related Lipschitz BSDE (L-BSDEs for short). The natural tools are the Malliavin calculus and the so-called Zvonkin’s transformation. Chapter two uses some existing results on L-BSDEs literature to construct and study the convergence rates of different types of numerical schemes for the solution of Q- BSDE in different cases: explicit and implicit. Those schemes are not completely discrete with respect to the z-variable. However, under some restrictive conditions, a completely discrete scheme” is introduced and studied. The last chapter investigates the necessary and sufficient optimality conditions for a class of controlled stochastic differential equations where the coefficients are merely Lipschitz continuous in the state variable but not necessarily differentiable everywhere. The Malliavin calculus and Radmecher’s theorem are the main tools in this analysis.