TY - JOUR
T1 - BACKWARD NONLINEAR SMOOTHING DIFFUSIONS
AU - Anderson, B. D.O.
AU - Bishop, A. N.
AU - Del Moral, P.
AU - Palmier, C.
N1 - Publisher Copyright:
© SIAM. Unauthorized reproduction of this article is prohibited.
PY - 2022
Y1 - 2022
N2 - 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.
AB - 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.
KW - Kalman–Bucy filter
KW - Rauch–Tung–Striebsmoother
KW - backward Itô–Ventzell formula
KW - backward stochastic integration
KW - diffusion equations
KW - nonlinear filtering and smoothing
KW - particle filtering and smoothing
KW - stochastic semigroups
KW - time-reversed stochastic differential equationZakai and Kushner–Stratonovich equations
UR - https://www.scopus.com/pages/publications/85129667840
U2 - 10.1137/S0040585X97T99037X
DO - 10.1137/S0040585X97T99037X
M3 - Article
SN - 0040-585X
VL - 66
SP - 245
EP - 262
JO - Theory of Probability and its Applications
JF - Theory of Probability and its Applications
IS - 2
ER -