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 $\tfrac32$-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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2015
- DOI:
- 10.48550/arXiv.1509.00609
- arXiv:
- arXiv:1509.00609
- Bibcode:
- 2015arXiv150900609A
- Keywords:
-
- Mathematics - Numerical Analysis;
- Mathematics - Probability;
- 65C60;
- 65L06;
- 65L20
- E-Print:
- 31 pages, 9 tables, 2 figures, accepted for publication in BIT, num. math