On the Innovations Conjecture of Nonlinear Filtering with Dependent Data

Abstract

We establish the innovations conjecture for a nonlinear filtering problem in which the signal to be estimated is conditioned by the observations. The approach uses only elementary stochastic analysis, together with a variant due to J.M.C. Clark of a theorem of Yamada and Watanabe on pathwise-uniqueness and strong solutions of stochastic differential equations.

On partially observed jump diffusions I. The filtering equations

“…It is called the innovation process in the case when the observation process does not have a stochastic integral component with respect to Poisson measures, i.e., when ν 1 " 0. In this case it was conjectured that p Vs q sPr0,ts together with Y 0 carry the same information as the observation pY s q sPr0,ts , i.e., that the σ-algebra generated by p Vs q sPr0,ts and Y 0 coincides with the σ-algebra generated by pY s q sPr0,ts for every t. An affirmative result concerning this conjecture, under quite general conditions on the filtering models (but without jump components) was proved in [17] and [13]. For our filtering model we conjecture that p Vs q sPr0,ts , together with Y 0 and t Ñ pp0, ss ˆΓq : s P r0, ts, Γ P Z 1 u carry the same information as the observation pY s q sPr0,ts , if Assumption 2.1 holds and the coefficients of (1.1) satisfy an appropriate Lipschitz condition.…”