Strong convergence of the truncated Euler–Maruyama method for stochastic functional differential equations

ABSTRACT

In this paper, we establish the truncated Euler–Maruyama (EM) method for stochastic functional differential equation (SFDE) $\mathrm{d}y\left(t\right)=f\left(y_t\right)\mathrm{d}t+g\left(y_t\right)\mathrm{d}B\left(t\right)$ and consider the strong convergence theory for the numerical solutions of SFDEs under the local Lipschitz condition plus Khasminskii-type condition instead of the linear growth condition. The type of convergence specifically addressed in this paper is strong-$L^q$ convergence for $2\le q<p$, and p is a parameter in Khasminskii-type condition. We also discussed the rates of $L^q$-convergence for the truncated EM method.

Keywords:

• Stochastic functional differential equation
• truncated Euler–Maruyama method
• Local Lipschitz condition
• Khasminskii-type condition
• strong convergence

2010 AMS Subject Classification:

• 65L05
• 65L60

Acknowledgements

The authors wish to thank the referees for their helpful comments which improve this paper significantly. The authors would also like to thank Prof. Xuerong Mao's help.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China NSF of P.R. China (No. 11671113), the Heilongjiang University Youth Science Fund project (No. QL201308) and Basic scientific research in colleges and universities of Heilongjiang province (special fund project of Heilongjiang university No. HDJCCX-201619).