Fixed lag smoothing of scalar diffusions. Part I. The filtering-smoothing equation

The problem of optimal fixed lag smoothing of a diffusion process, xt is to estimate xt − τ for t > τ, given the output of a nonlinear noisy sensor up to time t. The nonlinear filtering-smoothing problem is to estimate both xt and xt − τ. The optimal estimators are the conditional expectations of the processes, given the measurements. We derive evolution equations for both the normalized and unnormalized versions of the joint probability density function of xt and xt − τ, given the noisy measurements. The former is an equation of Kushnerʹs type and the latter is of Zakaiʹs type.