Almost sure exponential stability of the backward Euler–Maruyama discretization for highly nonlinear stochastic functional differential equation

Stability of numerical solutions to stochastic differential equations have received an increasing attention, but there is so far no work on stability of numerical solutions to nonlinear stochastic functional differential equations (SFDEs). To close the gap, the paper develops a criterion on stability of numerical solutions to highly nonlinear SFDEs. By using of the discrete semi-martingale convergence theorem, we show that backward Euler–Maruyama method can reproduce almost sure exponential stability of the exact solutions to highly nonlinear SFDEs. Several high order examples are provided to illustrate the main results, which implies that a wide class of nonlinear systems obeys the new criterion.