Pathwise convergence of the Euler scheme for rough and stochastic differential equations

Andrew L. Allan; Anna P. Kwossek; Chong Liu; David J. Prömel

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arXiv · 2309.16489

Pathwise convergence of the Euler scheme for rough and stochastic differential equations

Andrew L. Allan·Anna P. Kwossek·Chong Liu·David J. Prömel

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Abstract

The convergence of the first order Euler scheme and an approximative variant thereof, along with convergence rates, are established for rough differential equations driven by c\`adl\`ag paths satisfying a suitable criterion, namely the so-called Property (RIE), along time discretizations with vanishing mesh size. This property is then verified for almost all sample paths of Brownian motion, It\^o processes, L\'evy processes and general c\`adl\`ag semimartingales, as well as the driving signals of both mixed and rough stochastic differential equations, relative to various time discretizations. Consequently, we obtain pathwise convergence in p-variation of the Euler--Maruyama scheme for stochastic differential equations driven by these processes.

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Andrew L. Allan, Anna P. Kwossek, Chong Liu, David J. Prömel. 2023-09-28. Pathwise convergence of the Euler scheme for rough and stochastic differential equations. https://arxiv.org/abs/2309.16489

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