A note on optimal nonlinear prediction for independent increment processes

Abstract

In this note, some earlier results of Fisher and Stear on optimal nonlinear filtering for independent increment processes are extended to include the problem of optimal non-linear prediction. Specifically, it is shown that the prediction problem requires solution of the filtering problem plus extrapolation of the last filter output through the nonlinear system dynamics, and it is shown that the conditional density function of the system state during the extrapolation operation satisfies the Fokker-Planck-Kolmogorov-Feller equation. Finally, some earlier results obtained by Fisher and Stear on the approximate solution of the optimal non-linear filtering problem for independent increment processes are extended to the case of prediction and, as a by-product, an earlier result of Kalman is extended to the case of independent increment processes.