The truncated Milstein method for super-linear stochastic differential equations with Markovian switching

Weijun Zhan; Qian Guo; Yuhao Cong

doi:10.3934/dcdsb.2021201

Article Contents

2022, Volume 27, Issue 7: 3663-3682. Doi: 10.3934/dcdsb.2021201

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The truncated Milstein method for super-linear stochastic differential equations with Markovian switching

1.
Department of Mathematics, Anhui Normal University, Wuhu 241000, China
2.
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
3.
Shanghai Customs College, Shanghai 201204, China

^* Corresponding author: Qian Guo
^* Corresponding author: Qian Guo

Received: August 31, 2019

Revised: September 30, 2020

Published: July 2022

The second author is supported by NSFC of China (No:11871343). The third author is supported by NSFC of China (No:11971303)

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Abstract

In this paper, to approximate the super-linear stochastic differential equations modulated by a Markov chain, we investigate a truncated Milstein method with convergence order 1 in the mean-square sense. Under Khasminskii-type conditions, we establish the convergence result by employing a relationship between local and global errors. Finally, we confirm the convergence rate by a numerical example.

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Mathematics Subject Classification: Primary: 60H35; Secondary: 65C30.

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Figure 1. The strong convergence order at the terminal time $ T = 1 $. The red dashed line is the reference line with the slope of 1

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References

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