Backward Nonlinear Smoothing Diffusions

TL;DR

This paper introduces a novel backward nonlinear diffusion flow that characterizes the smoothing distribution in stochastic processes, offering a new perspective and derivation methods for classical smoothing equations.

Contribution

It presents a new backward nonlinear diffusion flow interpretation of smoothing distributions, complementing existing deterministic approaches and deriving classical smoothing equations.

Findings

Derivation of a backward stochastic differential equation for smoothing distributions

Connection between nonlinear diffusion flows and classical Rauch-Tung-Striebel smoothing

New insights into time-reversal of stochastic differential equations

Abstract

We present a backward diffusion flow (i.e. a backward-in-time stochastic differential equation) whose marginal distribution at any (earlier) time is equal to the smoothing distribution when the terminal state (at a latter time) is distributed according to the filtering distribution. This is a novel interpretation of the smoothing solution in terms of a nonlinear diffusion (stochastic) flow. This solution contrasts with, and complements, the (backward) deterministic flow of probability distributions (viz. a type of Kushner smoothing equation) studied in a number of prior works. A number of corollaries of our main result are given including a derivation of the time-reversal of a stochastic differential equation, and an immediate derivation of the classical Rauch-Tung-Striebel smoothing equations in the linear setting.