Fixed lag smoothing of scalar diffusions. Part I. The filtering-smoothing equation (Q1374637)

This paper is devoted to the estimation of a random signal \(x_s\) simultaneously at time \(t\) and at an earlier time \(t-\tau\), given its measurements in a nonlinear noisy channel \(y_s\), up to time \(t\). Denoting by \(Y^t\) the measurements of \(y_s\) for \(s\) in the interval \((0,t)\), the nonlinear filtering and fixed lag smoothing problem is equivalent to the calculation of the joint conditional probability distribution function \(P(x_t\leq u,x_{t-\tau} \leq v\big|Y^t)\), for \(t>\tau\) and \(u\), \(v\in \mathbb R\). The main result is a derivation of evolution equations for the normalized and unnormalized conditional joint probability density functions. The former is a stochastic integro-partial-differential equation of Kushner type and the latter is a linear stochastic partial differential equation of Zakai type. The fixed lag smoothing probability density function is obtained as a marginal of the joint probability density function.