Almost sure exponential stability of the Euler–Maruyama approximations for stochastic functional differential equations

By the continuous and discrete nonnegative semimartingale convergence theorems,
this paper investigates conditions under which the Euler–Maruyama (EM) approximations of stochastic functional differential equations (SFDEs) can share the almost sure exponential stability of the exact solution. Moreover, for sufficiently small stepsize, the decay rate as measured by the Lyapunov exponent can be reproduced arbitrarily accurately.