On path-dependent multidimensional forward-backward SDEs

Abstract

<p style='text-indent:20px;'>This paper extends the results of Ma, Wu, Zhang, Zhang [<xref ref-type="bibr" rid="b11">11</xref>] to the context of path-dependent multidimensional forward-backward stochastic differential equations (FBSDE). By path-dependent we mean that the coefficients of the forward-backward SDE at time <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula> can depend on the whole path of the forward process up to time <inline-formula><tex-math id="M2">\begin{document}$ t $\end{document}</tex-math></inline-formula>. Such a situation appears when solving path-dependent stochastic control problems by means of variational calculus. At the heart of our analysis is the construction of a decoupling random field on the path space. We first prove the existence and the uniqueness of decoupling field on small time interval. Then by introducing the characteristic BSDE, we show that a global decoupling field can be constructed by patching local solutions together as long as the solution of the characteristic BSDE remains bounded. Finally, we provide a stability result for path-dependent forward-backward SDEs.</p>