Backward Nonlinear Smoothing Diffusions

Date

2021

Authors

Anderson, Brian
Bishop, A N
Del Moral, P
Palmier, C

Journal Title

Journal ISSN

Volume Title

Publisher

Society for Industrial and Applied Mathematics

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 later 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.

Description

Keywords

nonlinear filtering and smoothing, Kalman–Bucy filter, Rauch–Tung–Striebelsmoother, particle filtering and smoothing, diffusion equations, stochastic semigroups, backward stochastic integration, backward Itô--Ventzell formula, time-reversed stochastic differential equations, Zakai and Kushner–Stratonovich equations

Citation

Source

Theory of Probability and its Applications

Type

Journal article

Book Title

Entity type

Access Statement

License Rights

DOI

10.1137/S0040585X97T99037X

Restricted until

2099-12-31