Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · 2010 · 413 citations · 20 references
Numerical AnalysisWeak DivergenceLipschitz Continuous DriftEngineeringFinite TimeStochastic Euler SchemeStochastic ProcessesNatural SciencesPartial Differential EquationsStochastic CalculusStochastic Dynamical SystemStochastic AnalysisStochastic PhenomenonStochastic Differential EquationStochastic Differential EquationsLipschitz Continuous CoefficientsNumerical Method For Partial Differential Equation
The stochastic Euler scheme is known to converge for globally Lipschitz coefficients, recent work extends this to linearly growing coefficients, but finite‑time convergence for superlinearly growing coefficients remains unresolved. The study demonstrates that for a broad class of SDEs with non‑globally Lipschitz coefficients, Euler’s method fails to converge in either strong mean‑square or numerically weak sense at any finite time. The difference between the exact solution and the Euler approximation diverges to infinity in both the strong mean‑square and numerically weak senses at finite times.
The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation (SDE) with globally Lipschitz continuous drift and diffusion coefficients. Recent results extend this convergence to coefficients that grow, at most, linearly. For superlinearly growing coefficients, finite-time convergence in the strong mean-square sense remains. In this article, we answer this question to the negative and prove, for a large class of SDEs with non-globally Lipschitz continuous coefficients, that Euler’s approximation converges neither in the strong mean-square sense nor in the numerically weak sense to the exact solution at a finite time point. Even worse, the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean-square sense and in the numerically weak sense.
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