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: dX(t) = b(X-t)dt + dB(t), dY(t)(epsilon) = gamma(X-t)dt + epsilon dV(t), where B., V. are standard independent Brownian motions. Denote by q(1)(epsilon)(.) the density of the law of Xi(1) conditioned on sigma(Y-t(epsilon) : 0 <= t <= 1). We provide "quenched" large deviation estimates for the random family of measures q(1)(epsilon)(x)dx: there exists a continuous, explicit mapping (J) over bar : R-2 -> R such that for almost all B., V., (J) over bar(., X-1) is a good rate function, and for any measurable G subset of R, -inf(x is an element of Go) (J) over bar (x, X-1) <= lim(epsilon -> 0) inf epsilon log integral(G) q(1)(epsilon)(x)dx <= lim(epsilon -> 0) sup epsilon log integral(G) q(1)(epsilon)(x)dx <= - inf(x is an element of(G) over bar)(J) over bar (x, X-1).