BSDEs with terminal conditions that have bounded Malliavin derivative

TL;DR

This paper proves existence and uniqueness of solutions to certain backward stochastic differential equations (BSDEs) with terminal conditions having bounded Malliavin derivatives, even with non-Lipschitz drivers, and connects these to semilinear PDEs.

Contribution

It establishes new existence and uniqueness results for BSDEs with terminal conditions possessing bounded Malliavin derivatives, including cases with non-Lipschitz drivers and boundary conditions.

Findings

Existence and uniqueness of BSDE solutions with bounded Malliavin derivative terminal conditions.

Extension to non-Lipschitz drivers in BSDEs and related PDEs.

Results applicable to Markovian BSDEs with boundary conditions.

Abstract

We show existence and uniqueness of solutions to BSDEs of the form $$ Y_t = \xi + \int_t^T f(s,Y_s,Z_s)ds - \int_t^T Z_s dW_s$$ in the case where the terminal condition $\xi$ has bounded Malliavin derivative. The driver $f(s,y,z)$ is assumed to be Lipschitz continuous in $y$ but only locally Lipschitz continuous in $z$. In particular, it can grow arbitrarily fast in $z$. If in addition to having bounded Malliavin derivative, $\xi$ is bounded, the driver needs only be locally Lipschitz continuous in $y$. In the special case where the BSDE is Markovian, we obtain existence and uniqueness results for semilinear parabolic PDEs with non-Lipschitz nonlinearities. We discuss the case where there is no lateral boundary as well as lateral boundary conditions of Dirichlet and Neumann type.