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 Electronic Journal of Probability > Vol. 10 (2005) > Paper 45 open journal systems 


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.


Full text: PDF

Pages: 1468-1495

Published on: December 19, 2005
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Electronic Journal of Probability. ISSN: 1083-6489