Fixed Interval Smoothing for Nonlinear Continuous Time Systems

Here, dr~dr and dw/dt are independent, zero mean, ganssian white noise processes, with covariances I 3 ( t r ) and R ( t ) 8 ( t T), respectively. The matrix R(t) is positive definite for all t. An a priori distribution for the density of x 0 is assumed known, and it is assumed that f , G, h and R all have sufficient smoothness properties to guarantee the usual existence and uniqueness requirements on solutions of (1) and (2), together with such other quantities as will be introduced. In particular, we assume that the conditional density p(x~]Zto,~l) of xt , given the measurements z~ over [0, t], exists and satisfies the conditional Fokker-Planck equation; see, e.g., Jazwinski (1970). In this paper, we aim to give a differential equation for the probability density p(xt I ZL0,T]), with T fixed. This is the probability density associated with the fixed-interval smoothing problem.