Geodesic
Backward nonlinear smoothing diffusions
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 2, pp. 305-326

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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, stochastic semigroups, backward stochastic integration, backward Itô–Ventzell formula, time-reversed stochastic differential equations, Zakai and Kushner–Stratonovich equations.
Mots-clés : diffusion equations 
@article{TVP_2021_66_2_a4,
     author = {B. D. O. Anderson and A. N. Bishop and P. Del Moral and C. Palmier},
     title = {Backward nonlinear smoothing diffusions},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {305--326},
     publisher = {mathdoc},
     volume = {66},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2021_66_2_a4/}
}
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B. D. O. Anderson; A. N. Bishop; P. Del Moral; C. Palmier. Backward nonlinear smoothing diffusions. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 2, pp. 305-326. http://geodesic.mathdoc.fr/item/TVP_2021_66_2_a4/