Abstract: The classical nonlinear filtering problem deals with the question of finding an optimal estimate (filter) for an unobservable signal $X_t$ based on observation sigma-field ${\cal F}^Y_t = \{ Y_s : 0\leq s\leq t\}$, where $Y_s = \int_0^s h(X_u)du + W_s$. Here $W$ is assumed to be a Brownian Motion. The innovations process $I_t$ is defined by $$ I_t = Y_t - \int_0^t \hat{h}_sds$$ where $\hat{h}_s = E[h(X_s)|{\cal F}^Y_s]$. It is clear that the corresponding innovations sigma-field ${\cal F}^I_t = \{ I_s: 0\leq s\leq t\}$ is contained in ${\cal F}^Y_t$ for all $t$. The innovations conjecture states that one actually has equality of the sigma-fields.

The innovations conjecture was shown to be false by Benes (1977) in this ``minimal'' set-up. The continuing challenge in the theory of non-linear filtering is the identification of natural conditions under which the conjecture is valid.

In this lecture we will present some recent results in this direction obtained by A. J. Heunis and V. M. Lucic (2008).