Interacting Particle Filtering with Discrete-Time Observations: Asymptotic Behaviour in the Gaussian Case

Abstract

In this paper we are concerned with a very particular case of the following general filtering problem. The state process Xis the solution of a stochastic differential equation of the form

$$\begin{array}{*{20}{c}} {d{X_t} = \alpha ({X_t})dt + \beta ({X_t})d{W_t},}&{\mathcal{L}({X_0}) = {\pi _0},} \end{array}$$

where π₀ is a known distribution on ℝ^d, and α,β are known functions, and W is a d-dimensional Wiener process. We have noisy observations Y ₁,...,Y _N at N regularly spaced times, and without loss of generality we will assume that these times are. That is, at each time i ∈ ℕ* we have an ℝ^d-valued observation Y _i given by

$${Y_i} = h({X_i},{\varepsilon _i}),$$

where the ε _i are i.i.d. q′-dimensional variables, independent of X and with a law having a known density g, and h is a known function from ℝ^d × ℝ^q′ into ℝq. We denote by π _Y,N the filter for X _N , that is a regular version of the conditional distribution of the random variable X _N knowing Y ₁,…,Y _N .