The Steepest Descent Method for Forward-Backward SDEs
Jaksa Cvitanic, California Institute of Technology, USA
Jianfeng Zhang, University of Southern California, USA
Abstract
This paper aims to open a door to Monte-Carlo methods for numerically solving
Forward-Backward SDEs, without computing over all Cartesian grids as usually done
in the literature. We transform the FBSDE to a control problem and
propose the steepest descent method to solve the latter one. We show
that the original (coupled) FBSDE can be approximated by {it decoupled}
FBSDEs, which further comes down to computing a sequence of conditional expectations.
The rate of convergence is obtained, and the key to its proof is a new well-posedness
result for FBSDEs. However, the approximating decoupled FBSDEs are
non-Markovian. Some Markovian type of modification is needed in
order to make the algorithm efficiently implementable.
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