Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition

Abstract

In this paper the numerical approximation of stochastic differential equations satisfying a global monotonicity condition is studied. The strong rate of convergence with respect to the mean square norm is determined to be $\frac{1}{2}$ for the two-step BDF-Maruyama scheme and for the backward Euler-Maruyama method. In particular, this is the first paper which proves a strong convergence rate for a multi-step method applied to equations with possibly superlinearly growing drift and diffusion coefficient functions. We also present numerical experiments for the $\frac{3}{2}$-volatility model from finance, which verify our results in practice and indicate that the BDF2-Maruyama method offers advantages over Euler-type methods if the stochastic differential equation is stiff or driven by a noise with small intensity.

31 pages, 9 tables, 2 figures, accepted for publication in BIT, num. math

References in corpus (2)

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

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• $L^p$-Convergence Rate of Backward Euler Schemes for Monotone SDEs
• Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients
• Perturbation estimates for order-one strong approximations of SDEs without globally monotone coefficients
• The BDF2-Maruyama Scheme for Stochastic Evolution Equations with Monotone Drift
• Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equations
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