Malliavin Regularity of Non-Markovian Quadratic BSDEs and Their Numerical Schemes

Abstract

We study both Malliavin regularity and numerical approximation schemes for a class of quadratic backward stochastic differential equations (QBSDEs for short) in cases where the terminal data need not be a function of a forward diffusion. By using the connection between the QBSDE under study and some backward stochastic differential equations (BSDEs) with global Lipschitz coefficients, we firstly prove Lq, (q≥2) existence and uniqueness results for QBSDE. Secondly, the Lp-Hölder continuity of the solutions is established for (q>4 and 2≤pq2). Then, we analyze some numerical schemes for our systems and establish their rates of convergence. Moreover, our results are illustrated with three examples.

Keywords

• quadratic backward stochastic differential equations
• Malliavin calculus
• explicit scheme
• implicit scheme
• rate of convergence
• Hölder continuity