Particle representation for the solution of the filtering problem. Application to the error expansion of filtering discretizations

We introduce a weighted particle representation for the solution of the filtering problem based on a suitably chosen variation of the classical de Finetti theorem. This representation has important theoretical and numerical applications. In this paper, we explore some of its theoretical consequences. The first is to deduce the equations satisfied by the solution of the filtering problem in three different frameworks: the signal independent Brownian measurement noise model, the spatial observations with additive white noise model and the cluster detection model in spatial point processes. Secondly we use the representation to show that a suitably chosen filtering discretisation converges to the filtering solution. Thirdly we study the leading error coefficient for the discretisation. We show that it satisfies a stochastic partial differential equation by exploiting the weighted particle representation for both the approximation and the limiting filtering solution. MSC 2010: 60G35, 60F05, 60F25, 60H35, 60H07, 93E11.

[1]  Zhengxiao Wu.  A cluster identification framework illustrated by a filtering model for earthquake occurrences , 2009, 0906.2099.

[2]  Dan Crisan,et al.  Particle representations for stochastic partial differential equations with boundary conditions , 2016, 1607.08909.

[3]  H. Kunita.  Cauchy problem for stochastic partial differential equations arizing in nonlinear filtering theory , 1981 .

[4]  H. Kunita.  Non linear filtering for the system with general noise , 1979 .

[5]  P. Brémaud.  Point Processes and Queues , 1981 .

[6]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[7]  T. Kurtz,et al.  Conditional distributions, exchangeable particle systems, and stochastic partial differential equations , 2014 .

[8]  D. Crisan,et al.  A high order time discretization of the solution of the non-linear filtering problem , 2017, Stochastics and Partial Differential Equations: Analysis and Computations.

[9]  P. Donnelly.  MARKOV PROCESSES Characterization and Convergence (Wiley Series in Probability and Mathematical Statistics) , 1987 .

[10]  G. Kallianpur,et al.  Arbitrary system process with additive white noise observation errors , 1968 .

[11]  H. Kunita,et al.  Stochastic differential equations for the non linear filtering problem , 1972 .

[12]  T. Kurtz,et al.  Particle representations for a class of nonlinear SPDEs , 1999 .

[13]  Philip Protter,et al.  Weak convergence of stochastic integrals and differential equations II: Infinite dimensional case , 1996 .

[14]  Dan Crisan,et al.  On a robust version of the integral representation formula of nonlinear filtering , 2005 .

[15]  H. Kunita.  Ergodic Properties of Nonlinear Filtering Processes , 1991 .

[16]  D. Mayne,et al.  Monte Carlo techniques to estimate the conditional expectation in multi-stage non-linear filtering† , 1969 .

[17]  J. Stoyanov.  The Oxford Handbook of Nonlinear Filtering , 2012 .

[18]  Thomas G. Kurtz,et al.  Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type , 2008 .

[19]  P. Protter,et al.  Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations , 1991 .

[20]  P. Protter.  Stochastic integration and differential equations , 1990 .

[21]  Dan Crisan,et al.  Pathwise approximations for the solution of the non-linear filtering problem , 2021, ArXiv.

[22]  H. Kunita.  Asymptotic behavior of the nonlinear filtering errors of Markov processes , 1971 .