Taming singular stochastic differential equations: A numerical method

Abstract

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.

Sharp convex generalizations of stochastic Gronwall inequalities

“…We extend the results of Le and Ling [20,Lemma 3.8] in this section. Under similar assumptions this inequality has also been studied by Makasu [22,Theorem 2.2].…”

“…Then, for θ ∈ (0, 1), a continuous local martingale M and otherwise similar assumptions as before, upper bounds on E[sup t∈[0,T ] X p t ], p ∈ (0, θ) which do not depend on the martingale M can be derived, see Makasu [22,Theorem 2.2]. If in addition X is assumed to be nondecreasing, estimates of E[sup t∈[0,T ] X p t ] can by obtained all θ ∈ (0, ∞) by Le and Ling [20,Lemma 3.8], where the range of p depends on whether A is deterministic or random.…”

Strong and weak convergence for averaging principle of DDSDE with singular drift

“…Please see Section 3 below for more details. We mention that similar estimates are obtained in [30] by stochastic sewing techniques.…”

Concave and other generalizations of stochastic Gronwall inequalities

“…◮ H predictable or ∆M ≥ 0: [22,Lemma 3.8] studied a non-linear generalization where in the assumption the term (0,t] X s − dA s is replaced by (0,t] (X s ) θ dA s 1/θ for θ > 0 and obtained estimates for E[sup t∈[0,T ] X p t ]. A further nonlinear extension of the stochastic Gronwall inequalities has been studied by Mekki, Nieto and Ouahab [27,Theorem 2.4]: For continuous local martingales a stochastic Henry Gronwall's inequality with upper bounds that do not depend on the local martingale M can be proven.…”