Terminal Invariance Sufficient Conditions for Jump Stochastic Systems

In this work, we present conditions of terminal invariance for nonlinear controllable dynamical stochastic systems with jumps. Jump component has the form of integral over Poisson random measure. We assume that the measure parameters (intensity and distribution of jump values) are time-dependent. Thus, the systems under consideration describe piecewise continuous stochastic processes with additional randomness in times and sizes of gaps. The initial condition is fixed. Terminal invariance means that some given functional (terminal criterion) is constant with probability 1. Here we formulate sufficient conditions for terminal invariance, which allow us to calculate this value explicitly. In model examples we show how these conditions can be used. The examples demonstrate the key property of terminal invariant control: to parry any possible realization of the random jump process.