Taming singular stochastic differential equations: A numerical method

We consider a generic and explicit tamed Euler–Maruyama scheme for multidimensional time-inhomogeneous stochastic differential equations with multiplicative Brownian noise. The diffusive coefficient is uniformly elliptic, Hölder continuous and weakly differentiable in the spatial variables while the drift satisfies the strict Ladyzhenskaya–Prodi–Serrin condition, as considered by Krylov and Röckner (Probab. Theory Related Fields 131 (2005) 154–196). In the discrete scheme, the drift is tamed by replacing it by an approximation. A strong rate of convergence of the scheme is provided in terms of the approximation error of the drift in a suitable and possibly very weak topology. A few examples of approximating drifts are discussed in detail. The parameters of the approximating drifts can vary and—under suitable conditions—be fine-tuned to achieve a strong convergence rate, which is arbitrarily close to the benchmark 0.5 rate. The result is then applied to provide numerical solutions for stochastic transport equations with singular vector fields satisfying the aforementioned condition.