Moment decay rates of stochastic differential equations with time-varying delay

Abstract

One of the most important questions, especially in applications, is how to choose a decay function in the study of stability for a concrete equation. Motivated by the fact that the coefficients of the considered equation mainly suggest the choice of the decay function, the point of analysis in the paper is to carry out the Lyapunov function approach and to state coercivity conditions dependent on decay in the study of the pth moment stability with a general decay rate for a certain stochastic differential equation with variable time delay. Some criteria including the usual exponential decay are discussed, particularly the ones with a concave decay, special cases of which are polynomial and logarithmic decays. Some consequences and examples are given to illustrate the theory.

Caratheodory's approximate solution to stochastic differential delay equation

“…n } stands for the Lebesgue measure on R. For some more details on stochastic differential equations, refer to [1][2][3][5][6][7][8][9][10][11] and references therein. By using the nonlinear growth condition and nonlinear growth condition, in 2015, Kim [4] studied the difference between the approximate solution and the accurate solution to the stochastic differential delay equation (shortly, SDEs).…”