Communications in Mathematical Sciences

Volume 18 (2020)

Number 3

Forward backward doubly stochastic differential equations and the optimal filtering of diffusion processes

Pages: 635 – 661

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n3.a3

Authors

Feng Bao (Department of Mathematics, Florida State University, Tallahassee, Florida, U.S.A.)

Yanzhao Cao (Department of Mathematics and Statistics, Auburn University, Auburn, Alabama, U.S.A.)

Xiaoying Han (Department of Mathematics and Statistics, Auburn University, Auburn, Alabama, U.S.A.)

Abstract

The connection between forward backward doubly stochastic differential equations and the optimal filtering problem is established without using the Zakai equation. The solutions of forward backward doubly stochastic differential equations are expressed in terms of a conditional law of a partially observed Markov diffusion process. It then follows that the adjoint time-inverse forward backward doubly stochastic differential equations govern the evolution of the unnormalized filtering density in the optimal filtering problem.

Keywords

forward backward doubly stochastic differential equations, optimal filtering problem, Feynman–Kac formula, Itô’s formula, adjoint stochastic processes.

2010 Mathematics Subject Classification

60H10, 60H30

Full Text (PDF format)

The first author acknowledges the support by the Scientific Discovery through Advanced Computing (SciDAC) program funded by U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research through FASTMath Institute and CompFUSE project. The first author is also partially supported by National Science Foundation under grant number DMS1720222. The second author is partially supported by National Science Foundation under grant number DMS1620150.

Received 11 May 2017

Accepted 18 November 2019

Published 30 June 2020