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Teoriya Veroyatnostei i ee Primeneniya, 2021, Volume 66, Issue 2, Pages 305–326
DOI: https://doi.org/10.4213/tvp5383 (Mi tvp5383) 		 
This article is cited in 1 scientific paper (total in 1 paper)

Backward nonlinear smoothing diffusions

B. D. O. Andersonabc, A. N. Bishopde, P. Del Moralfg, C. Palmierhi

a Research School of Electrical, Energy and Material Engineering, Australian National University, Canberra, Australia
b Hangzhou Dianzi University, China
c Data61-CSIRO in Canberra, Australia
d CSIRO, Australia
e University of Technology Sydney (UTS), Australia
f INRIA, Bordeaux Research Center, Talence, France
g CMAP, Polytechnique Palaiseau, France
h Institut de Mathématiques de Bordeaux (IMB), Bordeaux University, France
i ONERA Palaiseau, France
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DOI: https://doi.org/10.4213/tvp5383
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.
Keywords: 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, time-reversed stochastic differential equations, Zakai and Kushner–Stratonovich equations.
Funding agency Grant number
Australian Research Council DP160104500
DP190100887
Data61-CSIRO
BNP Paribas
The first author was supported by the Australian Research Council (ARC) via grants DP160104500 and DP190100887, and by Data61-CSIRO. The third author was supported in part by the Chair Stress Test, RISK Management and Financial Steering, led by the French Ecole Polytechnique and its Foundation and sponsored by BNP Paribas.
Received: 27.11.2019
Revised: 10.12.2020
Accepted: 01.12.2020
English version:
Theory of Probability and its Applications, 2021, Volume 66, Issue 2, Pages 245–262
DOI: https://doi.org/10.1137/S0040585X97T99037X
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: B. D. O. Anderson, A. N. Bishop, P. Del Moral, C. Palmier, “Backward nonlinear smoothing diffusions”, Teor. Veroyatnost. i Primenen., 66:2 (2021), 305–326; Theory Probab. Appl., 66:2 (2021), 245–262
Citation in format AMSBIB
\Bibitem{AndBisDel21}
\by B.~D.~O.~Anderson, A.~N.~Bishop, P.~Del Moral, C.~Palmier
\paper Backward nonlinear smoothing diffusions
\jour Teor. Veroyatnost. i Primenen.
\yr 2021
\vol 66
\issue 2
\pages 305--326
\mathnet{http://mi.mathnet.ru/tvp5383}
\crossref{https://doi.org/10.4213/tvp5383}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4252927}
\zmath{https://zbmath.org/?q=an:1470.60220}
\transl
\jour Theory Probab. Appl.
\yr 2021
\vol 66
\issue 2
\pages 245--262
\crossref{https://doi.org/10.1137/S0040585X97T99037X}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85129667840}
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    • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    		Теория вероятностей и ее применения Theory of Probability and its Applications
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