Time-reversion of a hybrid state stochastic difference system with a jump-linear smoothing application

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 nonGaussian 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