Differentiability of quadratic forward-backward SDEs with rough drift

Abstract

In this paper, we consider quadratic forward-backward SDEs (QFBSDEs), for which the drift in the forward equation does not satisfy the standard globally Lipschitz condition and the driver of the backward system possesses nonlinearity of type $\mathit{f}(|\mathit{y}|)|\mathit{z}{|}^2$, where f is any locally integrable function. We prove both the Malliavin and classical differentiability of solutions to this type of QFBSDEs and provide representations of these derivatives processes. As a by-product, we derive a representation formula of the control variable ${\mathit{Z}}_{\mathit{t}}$ as a conditional expectation of the terminal value, the driver and the Malliavin weights, when the drift term is only bounded and Hölder continuous. We study a numerical approximation of this system in the sense of Imkeller and Dos Reis (Stochastic Process. Appl. 120 (2010) 2286–2288) in which the authors assume that the drift is Lipschitz and the driver of the BSDE is globally Lipschitz in y and quadratic in the traditional sense in z (i.e., f is a positive constant). We show that the rate of convergence is the same as in (Stochastic Process. Appl. 120 (2010) 2286–2288).