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Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

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Abstract

A class of implicit Milstein type methods is introduced and analyzed in the present article for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. By incorporating a pair of method parameters \(\theta , \eta \in [0, 1]\) into both the drift and diffusion parts, the new schemes are indeed a kind of drift-diffusion double implicit methods. Within a general framework, we offer upper mean-square error bounds for the proposed schemes, based on certain error terms only getting involved with the exact solution processes. Such error bounds help us to easily analyze mean-square convergence rates of the schemes, without relying on a priori high-order moment estimates of numerical approximations. Putting further globally polynomial growth condition, we successfully recover the expected mean-square convergence rate of order one for the considered schemes with \(\theta \in [\tfrac{1}{2}, 1], \eta \in [0, 1]\). Also, some of the proposed schemes are applied to solve three SDE models evolving in the positive domain \((0, \infty )\). More specifically, the particular drift-diffusion implicit Milstein method (\( \theta = \eta = 1 \)) is utilized to approximate the Heston \(\frac{3}{2}\)-volatility model and the stochastic Lotka-Volterra competition model. The semi-implicit Milstein method (\(\theta =1, \eta = 0\)) is used to solve the Ait-Sahalia interest rate model. Thanks to the previously obtained error bounds, we reveal the optimal mean-square convergence rate of the positivity preserving schemes under more relaxed conditions, compared with existing relevant results in the literature. Numerical examples are also reported to confirm the previous findings.

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Funding

This work was supported by Natural Science Foundation of China (12071488, 11971488) and Natural Science Foundation of Hunan Province for Distinguished Young Scholars (2020JJ2040).

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  1. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, China

    Xiaojie Wang

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  1. Xiaojie Wang
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Communicated by: Ivan Oseledets

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The original online version of this article was revised: the affiliation detail for Xiaojie Wang was incorrectly given as “School of Mathematics and Statistics, HNP-LAMA Institute for Metal Physics, National Academy of Sciences” but should have been “School of Mathematics and Statistics, HNP-LAMA, Central South University”.

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Wang, X. Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients. Adv Comput Math 49, 37 (2023). https://doi.org/10.1007/s10444-023-10034-2

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