Almost sure stability of the Euler-Maruyama method with random variable stepsize for stochastic differential equations (Q509650)

When using the Euler-Maruyama method to approximate solutions of \(n\)-dimensional stochastic differential equations (SDEs) of the form \[ dx(t)= f(x(t))\,dt+ g(x(t))\,dB(t), \] it is proved that the (variable) stepsize can be chosen to be a function of the last approximation of \(x\) in order to make the time variable a stopping time and insure that almost sure stability of the SDEs will be retained. In this way weaker sufficient conditions for almost sure stability of the numerical solution are established. Examples are given that illustrate the efficacy of this approach.