# American Institute of Mathematical Sciences

September  2018, 8(3&4): 1021-1049. doi: 10.3934/mcrf.2018044

## Forward backward SDEs in weak formulation

 1 School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China 2 Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA

* Corresponding author: J. Zhang

Received  December 2017 Revised  February 2018 Published  September 2018

Fund Project: Wang is supported by Distinguished Middle-Aged and Young Scientist Encourage and Reward Foundation of Shandong Province (ZR2017BA033); Zhang is supported by NSF grant #1413717, and Zhang would like to thank Daniel Lacker for very helpful discussion on Section 2.3.2.

Although having been developed for more than two decades, the theory of forward backward stochastic differential equations is still far from complete. In this paper, we take one step back and investigate the formulation of FBSDEs. Motivated from several considerations, both in theory and in applications, we propose to study FBSDEs in weak formulation, rather than the strong formulation in the standard literature. That is, the backward SDE is driven by the forward component, instead of by the Brownian motion. We establish the Feyman-Kac formula for FBSDEs in weak formulation, both in classical and in viscosity sense. Our new framework is efficient especially when the diffusion part of the forward equation involves the $Z$-component of the backward equation.

Citation: Haiyang Wang, Jianfeng Zhang. Forward backward SDEs in weak formulation. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1021-1049. doi: 10.3934/mcrf.2018044
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