A. We consider a generic and explicit tamed Euler-Maruyama scheme for multidimensional time-inhomogeneous stochastic di erential equations with multiplicative Brownian noise. The di usive coe cient is uniformly elliptic, Hölder continuous and weakly di erentiable in the spatial variables while the drift satis es the 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 ne-tuned to achieve the standard 1/2-strong convergence rate with a logarithmic factor.

We investigate the space-time regularity of the local time associated with Volterra–Lévy processes, including Volterra processes driven by $$\alpha $$ α -stable processes for $$\alpha \in (0,2]$$ α ∈ ( 0 , 2 ] . We show that the spatial regularity of the local time for Volterra–Lévy process is $${\mathbb {P}}$$ P -a.s. inverse proportional to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturbation of ordinary differential equations by a Volterra–Lévy process which has sufficiently regular local time. Following along the lines of Harang and Perkowski (2020), we show existence, uniqueness and differentiability of the flow associated with such equations.

No abstract

No abstract