Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients

By Martin Hutzenthaler, Arnulf Jentzen, Peter E. Kloeden. 2010

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On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.

Subfields: Demography, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/620fe96fbb082b68250688d3591701020f8f1ccc

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Citations in the corpus

Adaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drift (2016, unknown): “However, as reported in [14], the cost of the implicit method per timestep is much higher.”
Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients (2013, unknown): “1 Introduction Motivated by the work of [11] and [6] on explicit Euler-type schemes which approximate (in an Lp sense) SDEs with superlinearly growing drift coefficients, the author extends the techniques developed in [11] and [3] to obtain, under very mild assumptions, convergence results for the c”
Simulation of McKean–Vlasov SDEs with super-linear growth (2018, methodology): “To solve this problem an explicit scheme was then developed in [17], a so-called Tamed Euler scheme.”
Local Lipschitz Continuity in the Initial Value and Strong Completeness for Nonlinear Stochastic Differential Equations (2013, unknown): “, [20, 18, 22, 33, 25] and the references mentioned therein.”
A note on tamed Euler approximations (2013, methodology): “One however observes in [7] that for the “taming" of the drift coefficient, the term n−1 is used when it is known from classical literature that the standard strong convergence rate is one-half.”
A numerical method for SDEs with discontinuous drift (2015, methodology): “[7] present an explicit numerical method for which they are able to prove strong convergence in case of an SDE with non-globally Lipschitz coefficient.”
On the Euler-Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients (2015, background): “Note that this one-sided Lipschitz condition also plays an indispensable role to establish the rate of convergence for SDEs with super-linear growth coefficients (see [16], [8]).”
On tamed milstein schemes of SDEs driven by Lévy noise (2014, methodology): “The proof is an immediate consequence of Lemmas [11, 12].”
Loss of regularity for Kolmogorov equations (2012, background): “1 in Hutzenthaler, Jentzen & Kloeden [36], Theorem 3.”
Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: truncation methods, convergence in pth moment, and stability (2019, methodology): “Using a similar condition to the study by Higham et al. (2002), the rate for the tamed Euler was obtained (Hutzenthaler et al., 2012).”
The truncated Euler-Maruyama method for stochastic differential equations (2015, methodology): “For example, the tamed EM method was developed in [14] to approximate SDEs with one-sided Lipschitz drift coefficient and the linear growth diffusion coefficient.”
Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities (2015, background): “[14]), because the solution of a nonlinear equation has to be computed approximatively at each time step.”