Perturbation estimates for order-one strong approximations of SDEs without globally monotone coefficients

Abstract

To obtain strong convergence rates of numerical schemes, an overwhelming majority of existing works impose a global monotonicity condition on coefficients of stochastic differential equations (SDEs). Nevertheless, there are still many SDEs from applications that do not have globally monotone coefficients. As a recent breakthrough, the authors of (Hutzenthaler and Jentzen 2020, Ann. Prob., 48, 53–93) originally presented a perturbation theory for SDEs, which is crucial to recovering strong convergence rates of numerical schemes in a non-globally monotone setting. However, only a convergence rate of order $1/2$ was obtained there for time-stepping schemes such as a stopped increment-tamed Euler–Maruyama (SITEM) method. An interesting question arises, also raised by the aforementioned work, as to whether a higher convergence rate than $1/2$ can be obtained when higher order schemes are used. The present work attempts to give a positive answer to this question. To this end, we develop some new perturbation estimates that are able to reveal the order-one strong convergence of numerical methods. As the first application of the newly developed estimates, we identify the expected order-one pathwise uniformly strong convergence of the SITEM method for additive noise driven SDEs and multiplicative noise driven second-order SDEs with non-globally monotone coefficients. As the other application, we propose and analyze a positivity preserving explicit Milstein-type method for Lotka–Volterra competition model driven by multi-dimensional noise, with a pathwise uniformly strong convergence rate of order one recovered under mild assumptions. These obtained results are completely new and significantly improve the existing theory. Numerical experiments are also provided to confirm the theoretical findings.