Weak approximations of nonlinear SDEs with non-globally Lipschitz continuous coefficients
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As opposed to an overwhelming number of works on strong approximations, weak approximations of stochastic differential equations (SDEs), sometimes more relevant in applications, are less studied in the literature. Most of the weak error analysis among them relies on a fundamental weak approximation theorem originally proposed by Milstein in 1986, which requires the coefficients of SDEs to be globally Lipschitz continuous. However, SDEs from applications rarely obey such a restrictive condition and the study of weak approximations in a non-globally Lipschitz setting turns out to be a challenging problem. This paper aims to carry out the weak error analysis of discrete-time approximations for SDEs with non-globally Lipschitz coefficients. Under certain board assumptions on the analytical and numerical solutions of SDEs, a general weak convergence theorem is formulated for one-step numerical approximations of SDEs. Explicit conditions on coefficients of SDEs are also offered to guarantee the aforementioned board assumptions, which allows coefficients to grow super-linearly. As applications of the obtained weak convergence theorems, we prove the expected weak convergence rate of two well-known types of schemes such as the tamed Euler method and the backward Euler method, in the non-globally Lipschitz setting. Numerical examples are finally provided to confirm the previous findings.
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