Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients

By Martin Hutzenthaler, Arnulf Jentzen, Peter E. Kloeden. Proceedings of the Royal Society A, 2009

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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.

Subfields: Computational Mechanics, Demography, Finance

View Paper: https://www.semanticscholar.org/paper/33742f3504e55d6ca8fd89a0965937fb43320c1a

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Citations in the corpus

The truncated Euler-Maruyama method for stochastic differential equations (2015, methodology): “Although the authors in [13] showed the strong and weak divergence in finite time of the EM method for SDEs with non-globally Lipschitz continuous coefficients, some modified EM methods have recently been developed for the nonlinear SDEs without the linear growth condition.”
Adaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drift (2016, unknown): “For example, the Ginzburg-Landau equation, which describes a phase transition from the theory of superconductivity [13, 19], is”
Convergence of tamed Euler schemes for a class of stochastic evolution equations (2014, background): “Hutzenthaler, Jentzen and Kloeden [9] have observed that the absolute moments of explicit Euler approximations for stochastic differential equations with superlinearly growing coefficients may diverge to infinity at finite time.”
On the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficient (2018, background): “Note that the recent literature on numerical approximation of SDEs contains a number of examples of SDEs with coefficients that are not Lipschitz continuous and such that the EulerMaruyama scheme at the final time does not achieve an Lp-error rate of 1/2, see [2, 6, 9, 11, 12, 22, 29].”
A note on tamed Euler approximations (2013, background): “Submitted to ECP on May 21, 2013, final version accepted on May 31, 2013.”
Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: truncation methods, convergence in pth moment, and stability (2019, background): “Hutzenthaler et al. (2011) showed that the pth moments of the EM approximation for a large class of SDEs with coefficients satisfying super-linear growth diverge to infinity for all p ∈ [1, ∞).”
Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients (2010, unknown): “More precisely, in the case d=m= 1, it has recently been shown in [16] that the root mean square distance of the exact solution of the SDE (1) and of the explicit Euler approximation (4) diverges to infinity lim N→∞ (E[‖XT − Ỹ N N ‖(2)]) =∞, (5) if the drift coefficient μ of the SDE (1) grows superl”
Simulation of McKean–Vlasov SDEs with super-linear growth (2018, background): “It is well known for standard SDEs that the explicit Euler scheme runs into difficulties in the super-linear growth setting, see [16], even though the SDE is known to have a unique strong solution.”
Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities (2015, background): “[13]) proves that the classical explicit Euler scheme (also known as Euler-scheme or Euler-Maruyama scheme; see Maruyama [24]) diverges strongly and numerically weakly in finite time time when applied to a SODE with superlinearly growing (and hence non-globally Lipschitz continuous) nonlinearities.”
Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients (2013, background): “[17] Those authors proved that in the case of super-linearly growing coefficients the Euler–Maruyama approximation may not converge in the strong L sense, nor in the numerically weak sense, to the exact solution.”
The partially truncated Euler–Maruyama method and its stability and boundedness (2017, methodology): “, [9, 11]) that the EM method may not preserve this asymptotic boundedness.”