Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
SIAM Journal on Numerical Analysis
Physically consistent simulation of mesoscale chemical kinetics: The non-negative FIS-α method
Journal of Computational Physics
Journal of Computational and Applied Mathematics
Computers & Mathematics with Applications
Journal of Computational and Applied Mathematics
Asymptotic moment boundedness of the numerical solutions of stochastic differential equations
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
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Relatively little is known about the ability of numerical methods for stochastic differential equations (SDEs) to reproduce almost sure and small-moment stability. Here, we focus on these stability properties in the limit as the timestep tends to zero. Our analysis is motivated by an example of an exponentially almost surely stable nonlinear SDE for which the Euler-Maruyama (EM)method fails to reproduce this behavior for any nonzero timestep. We begin by showing that EM correctly reproduces almost sure and small-moment exponential stability for sufficiently small timesteps on scalar linear SDEs. We then generalize our results to multidimensional nonlinear SDEs. We show that when the SDE obeys a linear growth condition, EM recovers almost surely exponential stability very well. Under the less restrictive condition that the drift coefficient of the SDE obeys a one-sided Lipschitz condition, where EM may break down, we show that the backward Euler method maintains almost surely exponential stability.