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

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.