Weak approximation errors for stochastic differential equations with non-regular drift

Arturo Kohatsu-Higa (1) , Antoine Lejay (2, 3) , Kazuhiro Yasuda (4)

1 Department of Mathematical Sciences, Ritsumeikan Universtiy
2 TOSCA - TO Simulate and CAlibrate stochastic models
3 Probabilités et statistiques
4 Faculty of Science and Engineering, Hosei University

We consider an Euler-Maruyama type approximation methods for a Stochastic Differential Equation (SDE) with a discontinuous drift and regular diffusion coefficient. The method regularizes the drift coefficient within a certain class of functions and then the Euler-Maruyama scheme for the regularized scheme is used as an approximation. This methodology gives two errors, the first is the error of regularization of the drift coefficient within a given class of functions and the second is the error of the regularized Euler-Maruyama scheme. This second error rate will be determined by the method of proof which takes into account that the regularized stochastic differential equation have coefficients in a given class of functions. After an optimization procedure with respect to this parameter we obtain various rates, which improve other known results but all of them of order less than one, which depend on the approximation class used. Finally, we consider a particular example of a diffusion with an explicit discontinuous drift and we show that the weak rate of convergence of the Euler-Maruyama scheme is the same as in the classical Euler-Maruyama scheme with smooth coefficients.

Mots clés: Stochastic Differential Equation; Euler-Maruyama scheme; discontinuous drift; weak rate of convergence; Malliavin calculus

Domaines: Probabilités [math.PR]

Arturo Kohatsu-Higa, Antoine Lejay, Kazuhiro Yasuda. Weak approximation errors for stochastic differential equations with non-regular drift. 2013. ⟨hal-00840211v1⟩