Almost sure exponential stability of solutions to highly nonlinear neutral stochastic differential equations with time‐dependent delay and the Euler–Maruyama approximation

This paper may be considered as a natural sequel to the paper [M. Miloševic, Highly nonlinear neutral stochastic differential equations with time‐dependent delay and the Euler–Maruyama method, Mathematical and Computer Modelling 54 (2011) 2235–2251]. In the present paper, global almost sure (a.s.) asymptotic exponential stability of the equilibrium solution for a class of neutral stochastic differential equations with time‐dependent delay is considered, under nonlinear growth conditions. Additionally, the moment estimates are established for solutions of equations of this type. Under more restrictive conditions, including the linear growth condition, we show that the appropriate Euler–Maruyama equilibrium solution is globally a.s. asymptotically exponentially stable. As expected, the whole consideration is affected by the presence and properties of the delay function. In that sense, the delayed terms are explicitly treated as arguments of the coefficients of the equation and, particularly, under the derivative of the state variable. Additionally, some requirements related to the rate of change of the delay function are imposed in order to provide the main results of the paper.