An adaptive Euler-Maruyama scheme for SDEs: convergence and stability

J. C. Mattingly ^{The understanding of adaptive algorithms for stochastic differentialequations (SDEs) is an open area, where many issues relatedto both convergence and stability (long-time behaviour) of algorithmsare unresolved. This paper considers a very simple adaptivealgorithm, based on controlling only the drift component ofa time step. Both convergence and stability are studied. Theprimary issue in the convergence analysis is that the adaptivemethod does not necessarily drive the time steps to zero withthe user-input tolerance. This possibility must be quantifiedand shown to have low probability. The primary issue in thestability analysis is ergodicity. It is assumed that the noiseis nondegenerate, so that the diffusion process is elliptic,and the drift is assumed to satisfy a coercivity condition.The SDE is then geometrically ergodic (averages converge tostatistical equilibrium exponentially quickly). If the driftis not linearly bounded, then explicit fixed time step approximations,such as the Euler–Maruyama scheme, may fail to be ergodic.In this work, it is shown that the simple adaptive time-steppingstrategy cures this problem. In addition to proving ergodicity,an exponential moment bound is also proved, generalizing a resultknown to hold for the SDE itself.}