Backward Nonlinear Smoothing Diffusions

Brian D O Anderson (1, 2) , Adrian N Bishop (3) , Pierre del Moral (4) , Camille Palmier (5)

1 ANU - Australian National University
2 CSIRO - Commonwealth Scientific and Industrial Research Organisation [Canberra]
3 UTS - University of Technology Sydney
4 CQFD - Quality control and dynamic reliability
5 IMB - Institut de Mathématiques de Bordeaux

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.

Mots clés: Nonlinear filtering and smoothing; Kalman-Bucy filter; Rauch-Tung-Striebel smoother; Particle filtering and smoothing; Diffusion equations; Stochastic semigroups; Backward stochastic integration; Backward Itô-Ventzell formula; Tme-reversed stochastic differential equations; Zakai and Kushner-Stratonovich equations

Domaines: Probabilités [math.PR]; Optimisation et contrôle [math.OC]

Brian D O Anderson, Adrian N Bishop, Pierre del Moral, Camille Palmier. Backward Nonlinear Smoothing Diffusions. [Research Report] ANU; UTS; INRIA; IMB. 2019. ⟨hal-02342600⟩