Time-Reversion of a Hybrid State Stochastic Difference System with a Jump-Linear Smoothing Application

Abstract

The reversion in time of a stochastic difference equation in a hybrid space with a Markovian solution is presented. The reversion is obtained by a Martingale approach, which previously led to reverse time forms for stochastic equations with Gauss-Markov or diffusion solutions. The reverse time equations follow from a particular noncanonical Martingale decomposition, while the reverse time equations for Gauss-Markov and diffusion solutions followed from the canonical Martingale decomposition. The need for this noncanonical decomposition stems from the hybrid state space situation. Moreover, the non-
Gaussian discrete time situation leads to reverse time equations that incorporate a Bayesian estimation step. The latter step is carried out for linear systems with Markovian switching coefficients, and the result is shown to provide the solution to the problem of fixed-interval smoothing. For an application of this smoothing approach to a trajectory with sudden maneuvers, simulation results are given to illustrate the practical use of the reverse time equations obtained.