Tohoku Mathematical Journal
2018
March
SECOND SERIES VOL. 70, NO. 1

Tohoku Math. J.
70 (2018), 65-95

Title THE RATES OF THE $L^P$-CONVERGENCE OF THE EULER-MARUYAMA AND WONG-ZAKAI APPROXIMATIONS OF PATH-DEPENDENT STOCHASTIC DIFFERENTIAL EQUATIONS UNDER THE LIPSCHITZ CONDITION

Author Shigeki Aida, Takanori Kikuchi and Seiichiro Kusuoka

(Received January 5, 2015, revised December 24, 2015)
Abstract. We consider the rates of the $L^p$-convergence of the Euler-Maruyama and Wong-Zakai approximations of path-dependent stochastic differential equations under the Lipschitz condition on the coefficients. By a transformation, the stochastic differential equations of Markovian type with reflecting boundary condition on sufficiently good domains are to be associated with the equations concerned in the present paper. The obtained rates of the $L^p$-convergence are the same as those in the case of the stochastic differential equations of Markovian type without boundaries.

Mathematics Subject Classification. Primary 60H10; Secondary 65C30.
Key words and phrases. Stochastic differential equation, reflecting boundary condition, path-dependent coefficient, Euler-Maruyama approximation, Wong-Zakai approximation, rate of convergence.

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