Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise

Abstract

We study the strong rate of convergence of the Euler-Maruyama scheme for a multidimensional stochastic differential equation (SDE)with irregular β-Hölder drift, β > 0, driven by a Lévy process with exponent α ∈ (0, 2]. For α ∈ [2/3, 2] we obtain strong L p and almost sure convergence rates in the whole range β > 1 − α/2, where the SDE is known to be strongly well-posed. This significantly improves the current state of the art both in terms of convergence rate and the range of α. In particular, the obtained convergence rate does not deteriorate for large p and is always at least n −1/2 ; this allowed us to show for the first time that the the Euler-Maruyama scheme for such SDEs converges almost surely and obtain explicit convergence rate. Furthermore, our results are new even in the case of smooth drifts. Our technique is based on a new extension of the stochastic sewing arguments.

Stochastic equations with singular drift driven by fractional Brownian motion

“…Thus, parameter H ∈ (1/2, 1) "morally" corresponds to 1/α. Therefore, if one repeats the strategy of the proof of Theorem 2.6(i) for an α-stable process instead of fractional Brownian motion (with appropriate modifications due to the discontinuity of L α , see, e.g., [BDG22]) one would get the following condition for weak existence of a solution to Eq(x; b) with the driving noise L α in place of (ii) let (X, W H ) be any weak regularized solution to equation (1.1) in the class BV.…”

A higher order approximation method for jump-diffusion SDEs with discontinuous drift coefficient

“…In the case of presence of jumps, under standard assumptions, classical and jump-adapted Itô-Taylor approximations and Runge-Kutta methods are studied, e.g., in [11,12,44,2]. Approximation results for jump-diffusion SDEs under non-standard assumptions can be found, e.g., in [18,19,20,7,10,9,56,5,29,4,24,36]. Asymptotically optimal approximation rates are proven in [46,23,47,48,52,22].…”