On applying comparison theorems to studying stability with probability 1 of stochastic differential equations

Abstract

In the work we obtain two results concerning trajectory-wise properties of stochastic differential equations (SDE) with Stratonovich integral. First, we prove comparison theorems for SDE with Stratonovich integral with respect to the standard Wiener process, that is, we obtain the conditions for the coefficients of SDE, under which the solutions of one equation for a fixed trajectory of the Wiener process is always located above or below a solution to another equations for the same trajectory. At that, the drift and diffusion coefficients of the studied equations can be different. Second, on the base of the proved theorems we establish the conditions for the stability with probability 1 for perturbed solutions to scalar SDE with Stratonovich integral with respect to the trivial solution. The stability with probability 1 implies the Lyapunov stability for almost all solutions to SDE. It should be noted that, as a rule, the stability for SDE is treated in weaker sense: stability in probability, $p$-stability, exponential stability. Employing the formula of passage between Ito integral and Stratonovich integral, which is valid for sufficiently smooth coefficients of SDE, these results can be extended to SDE with Ito integral. The approach of the work is based on the fact that a solution to SDE can be represented as a deterministic function of a random variable solving, in its turn, a chain of ordinary differential equations with a random right hand side. Since this technique is trajectory-wise, the presented results can be also reformulated for deterministic analogues of SDE, namely, for equations with symmetric integrals.