TY  - JOUR
A1  - Pathiraja, Sahani Darschika
A1  - Reich, Sebastian
A1  - Stannat, Wilhelm
T1  - McKean-Vlasov SDEs in nonlinear filtering
JF  - SIAM journal on control and optimization : a publication of the Society for Industrial and Applied Mathematics
N2  - Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows us to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in [D. Crisan and J. Xiong, Stochastics, 82 (2010), pp. 53-68; J. M. Clark and D. Crisan, Probab. Theory Related Fields, 133 (2005), pp. 43-56]. We consider three filters that have been proposed in the literature and use this framework to derive Ito representations of their limiting forms as the approximation parameter delta -> 0. All filters require the solution of a Poisson equation defined on R-d, for which existence and uniqueness of solutions can be a nontrivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.
KW  - data assimilation
KW  - feedback particle filter
KW  - Poincare inequality
KW  - well-posedness
KW  - nonlinear filtering
KW  - McKean-Vlasov
KW  - mean-field equations
Y1  - 2022
U6  - https://doi.org/10.1137/20M1355197
SN  - 0363-0129
SN  - 1095-7138
VL  - 59
IS  - 6
SP  - 4188
EP  - 4215
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Cattiaux, Patrick
A1  - Fradon, Myriam
A1  - Kulik, Alexei M.
A1  - Roelly, Sylvie
T1  - Long time behavior of stochastic hard ball systems
JF  - Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability
N2  - We study the long time behavior of a system of n = 2, 3 Brownian hard balls, living in R-d for d >= 2, submitted to a mutual attraction and to elastic collisions.
KW  - hard core interaction
KW  - local time
KW  - Lyapunov function
KW  - normal reflection
KW  - Poincare inequality
KW  - reversible measure
KW  - stochastic differential equations
Y1  - 2016
U6  - https://doi.org/10.3150/14-BEJ672
SN  - 1350-7265
SN  - 1573-9759
VL  - 22
SP  - 681
EP  - 710
PB  - International Statistical Institute
CY  - Voorburg
ER  -