Weak Approximation For Stochastic Differential Equations With Lipschitz Conditions

It is well known that the weak Euler approximation of a stochastic differential equation has order one,provided the cofficients of the equation are sufficiently smooth.It seems that,in general,when the coefficients are not smooth but only satisfy the Lipschitz condition,the order of the Euler approximation does not remain equal to one.The positive answer is given by Mackevicius[23]in 2003.Mackevicius[23]proved that the order of the approximation is still one in the case that the drift coefficient is a lipschitz function and the diffusinon cofficient is constant.The method of proof in Mackevicius[23] is centered on two points.One is the use of the Girsanov formula to express the error as a certain functional of the Brownian process.Another is to rewrite the expression in terms of u which is the solution u(t,x) of a parabolic PDE by adapting the essential technique used to obtain weak convergence rate.In this paper,we extend the results of Mackevicius[23]on two aspects.Firstly,we consider the weak Euler approximation of stochastic differential equations with additive noise.By means of the formula for integrating by parts of the Malliavin calculus,we prove that the convergence rate still has order one when the drift coefficient is a Lipschitz function and the test function satisfy the condition f∈C_p~2,which relaxes conditions of Mackevicius[23].Then,we study the convergence rate of Euler Scheme for stochastic hyperbolic differential equations with additive white noise.We prove that the weak error is of order one when the drift coefficient is a Lipschitz function.In fact,the equations that we concern here are two-parameter version of the equations in Mackevicius[23].The problem comes from the fact that there is no partial differential equation associated to the solutions of our equations.Therefore,the essential technique used in[23]doe not work any more here.To handle this difficulty,we use the formula for integrating by parts of the Malliavin calculus instead of the partial differential equation,then we estimate the error by the method of the conditional expectation.Moreover,with the help of these tools we also weaken the assumption on the test function f.