Forward Backward Doubly Stochastic Differential Equations and the Optimal Filtering of Diffusion Processes

TL;DR

This paper explores the relationship between forward backward doubly stochastic differential equations and optimal filtering of diffusion processes, providing a new approach that bypasses Zakai's equation and links solutions to conditional laws.

Contribution

It establishes a novel connection between doubly stochastic differential equations and optimal filtering without relying on Zakai's equation, offering new insights into filtering density evolution.

Findings

Solutions expressed in terms of conditional law

Doubly stochastic equations govern filtering density

Provides a new framework for optimal filtering

Abstract

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