On the convergence rate of Euler scheme for SDE with Lipschitz drift and constant diffusion (Q1415886)

The author investigates weak approximations of stochastic ordinary differential equations\break (SODEs) with a constant diffusion term by the Euler method. The SODEs he considers are of the form \[ X_t = x + \int_0^t b(X_s)ds + \sigma W_t,\quad t\in [0,T].\tag{1} \] The Euler scheme for some step-size \(h\) is then given by \[ X_0^h=x,\quad X_t^h= X_{kh}^h+b(X_{kh})(t-kh)+\sigma(W_t - W_{kh}) \] and the global error of the method is defined as \(\text{Err}(f,T,h):= Ef(X_T) -Ef(X_T^{h}).\) It is well-known that under sufficient smoothness conditions one has \(\text{Err}(f,T,h)= O(h)\) as \(h\rightarrow 0\), i.e. the weak order of the Euler method is one. The purpose of this article is to relax the smoothness conditions. In particular, the author's result says, that the weak order one of the Euler method is still achieved, if \(b\) is only Lipschitz continuous and \(f\) is three times continuously differentiable with all the derivatives of polynomial growth.