Abstract
The numerical approximation of exponential Euler method is constructed for semilinear stochastic differential equations (SDEs). The convergence and mean‐square (MS) stability of exponential Euler method are investigated. It is proved that the exponential Euler method is convergent with the strong order 1/2 for semilinear SDEs. A mean‐square linear stability analysis shows that the stability region of exponential Euler method contains that of EM method and stochastic Theta method (0 ≤ θ < 1) and also contains that of the scale linear SDE, that is, exponential Euler method is analogue mean‐square A‐stable. Then the exponential stability of the exponential Euler method for scalar semi‐linear SDEs is considered. Under the conditions that guarantee the analytic solution is exponentially stable in mean‐square sense, the exponential Euler method can reproduce the mean‐square exponential stability for any nonzero stepsize. Numerical experiments are given to verify the conclusions.
Highlights
Stochastic differential equations are utilized as mathematical models for physical application that possesses inherent noise and uncertainty
One surprising observation is that the exponential Euler as a kind of explicit numerical method has a good stability property, that is, the MS stability region of this method contains that of the test equation and contains that of EM method and stochastic Theta method 0 ≤ θ < 1
We show the convergence result of exponential Euler method for semi-linear SDE 2.1
Summary
Stochastic differential equations are utilized as mathematical models for physical application that possesses inherent noise and uncertainty. One surprising observation is that the exponential Euler as a kind of explicit numerical method has a good stability property, that is, the MS stability region of this method contains that of the test equation and contains that of EM method and stochastic Theta method 0 ≤ θ < 1. In another word, if the test equation is MS stable, so is the exponential Euler method applied to the systems for any stepsize.
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