BSDEs with terminal conditions that have bounded Malliavin derivative

2012/11/06 by Patrick Cheridito, Cheridito, Patrick, Kihun Nam +1 · 2 citations
Computer Science · Economics, Econometrics and Finance · Mathematics · #35K58 #60H07 #60H10 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1211.1089

openalex publication_date 2012/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show existence and uniqueness of solutions to BSDEs of the form Y_t = ξ+ ∫_t^T f(s,Y_s,Z_s)ds - ∫_t^T Z_s dW_s in the case where the terminal condition ξ 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, ξ 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.

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