Discrete Razumikhin-type technique and stability of the Euler-Maruyama method to stochastic functional differential equations (Q1946329)

Theorems are proved that establish conditions for the \(p\)th-moment exponential stability and almost sure exponential stability of Euler-Maruyama numerical solutions of stochastic functional differential equations of the form \[ dx(t)= f(t, x_t)\,dt+ g(t, x_t)\,dw(t),\quad t\geq 0. \] Then conditions yielding exponential stability of exact and numerical solutions are established for stochastic delay differential equations with variable delay and for stochastically perturbed equations including, as a special case, a stochastic Volterra delay-integrodifferential equation.