Quenched Large Deviations for One Dimensional Nonlinear Filtering

Consider the standard, one dimensional, nonlinear filtering problem for diffusion processes observed in small additive white noise: dXt = b(Xt)dt + dBt , dY ε t = γ(Xt)dt + εdVt , where B·, V· are standard independent Brownian motions. Denote by qε 1(·) the density of the law of Ξ1 conditioned on σ(Y ε t : 0 ≤ t ≤ 1). We provide “quenched” large deviation estimates for the random family of measures qε 1(x)dx: there exists a continuous, explicit mapping J̄ : IR → IR such that for almost all B·, V·, J̄ (·, X1) is a good rate function and for any measurable G ⊂ IR, − inf x∈Go J̄ (x,X1) ≤ lim inf ε→0 ε log ∫ G q 1(x)dx ≤ lim sup ε→0 ε log ∫ G q 1(x)dx ≤ − inf x∈Ḡ J̄ (x,X1) .