On construction of the comparison function of program motion in probable statement

Abstract

In the class of ordinary differential equations the following modification of the inverse problem of differential systems was previously considered: to construct both a set of systems of differential equations and a set of comparison functions for the given program motion. In this article, the modification of the inverse problem is considered in Stochastic case. In this problem it is assumed that random perturbations are from the class of processes with independent increments. By the given program of motion, two sets are constructed: the set of first-order Itoˆ stochastic differential equations and the set of comparison functions. It is proved that there is a stability in probability of the given program motion with respect to the constructed comparison functions. To solve the problem Lyapunov functions method is used. Using Lyapunov’s second method makes it is possible to weaken the conditions imposed on the components of the constructed comparison functions, in contrast to the application of the Lyapunov characteristic numbers method for solving the inverse problem in the class of ordinary differential equations. The following cases are considered: 1) comparison functions obviously not depending on time; 2) the set of comparison vector functions depends on y and t; 3) the set of comparison vector functions has the form Q(λ, t, where λ(y, t) describes an analytically given program motion; 4) the set of comparison vector functions has the form C(t)λ.