Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes

Abstract

This paper is concerned with the numerical approximation of stochastic ordinary differential equations, which satisfy a global monotonicity condition. This condition includes several equations with super-linearly growing drift and diffusion coefficient functions such as the stochastic Ginzburg-Landau equation and the 3/2-volatility model from mathematical finance. Our analysis of the mean-square error of convergence is based on a suitable generalization of the notions of C-stability and B-consistency known from deterministic numerical analysis for stiff ordinary differential equations. An important feature of our stability concept is that it does not rely on the availability of higher moment bounds of the numerical one-step scheme. While the convergence theorem is derived in a somewhat more abstract framework, this paper also contains two more concrete examples of stochastically C-stable numerical one-step schemes: the split-step backward Euler method from Higham et al. (2002) and a newly proposed explicit variant of the Euler-Maruyama scheme, the so called projected Euler-Maruyama method. For both methods the optimal rate of strong convergence is proven theoretically and verified in a series of numerical experiments.

31 pages, 3 figures

References in corpus (2)

• Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition
• On Tamed Milstein Schemes of SDEs Driven by Lévy Noise

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• Well-posedness and tamed Euler schemes for McKean-Vlasov equations driven by Lévy noise
• Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients
• A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations
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• Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients
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• On hard quadrature problems for marginal distributions of SDEs with bounded smooth coefficients
• Projected Euler method for stochastic delay differential equation under a global monotonicity condition
• Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equations
• Strongly Asymptotically Optimal Schemes for the Strong Approximation of Stochastic Differential Equations with respect to the Supremum Error
• The random periodic solution of a stochastic differential equation with a monotone drift and its numerical approximation