Integral invariants of stochastic systems: the kernel functions of the invariants and their application

The stochastic systems, which are described by the Ito equations with the Wiener and the Poisson perturbations (SDEI), have invariant functionals. These functionals are defined by integrals, which are connected with solutions of the SDEI and they introduce non-random and random functions, which are called the kernel functions of these invariants. The kernel functions are defined as a solutions of the same Partial SDEI. In the talk I will tell about an application of this SDEI for obtaining the Kolmogorov equations and the generalized Ito-Wentzell formula for SDEI; for exploring the existence problem for the first and the stochastic first integral for the SDEI and defining the necessary and sufficient conditions for their existence; and it allows to investigate the program control problem with probability one for the stochastic systems with perturbations, which are commensurable with parameters of the system.