Taming singular stochastic differential equations: A numerical method

Abstract

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 (2005). 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.

Updated error formula. To appear on Annals of Probability

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Cited by in corpus (5)

• Quantifying a convergence theorem of Gyöngy and Krylov
• An averaged space-time discretization of the stochastic $p$-Laplace system
• Numerical approximation of SDEs with fractional noise and distributional drift
• Path-by-path uniqueness for stochastic differential equations under Krylov-Röckner condition
• Strong convergence rate of Euler-Maruyama approximations in temporal-spatial Hölder-norms