Stochastic Functional Differential Equations and Procedure of Constructing Lyapunov Functionals

Abstract

In Chap. 2 short introduction to stochastic functional differential equations is presented, in particular, the definitions of the Wiener process, the Ito integral, the Ito stochastic differential equations and the Ito formula are given. Special algorithm for the Wiener process numerical simulation is described and the trajectories of the Wiener process obtained by this algorithm are represented in figure. Definitions of asymptotic mean square stability and stability in probability for stochastic functional differential equations are also considered, basic Lyapunov-type stability theorems and description of the procedure of Lyapunov functionals construction for stability investigation. In this section some useful statements about stability for linear stochastic differential equation and system of two linear stochastic differential equations are included, some useful inequality and some unsolved stability problems are included too.