SHARP LOWER ERROR BOUNDS FOR STRONG APPROXIMATION OF SDES WITH DISCONTINUOUS DRIFT COEFFICIENT BY COUPLING OF NOISE

• 成果类型: Article
• 署名作者: Mueller-Gronbach, Thomas; Yaroslavtseva, Larisa
• 署名单位: University of Passau
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1837
• 发表日期: 2023
• 页码: 902-935
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; euler-maruyama scheme; multidimensional sdes; occupation time; CONVERGENCE; EXISTENCE

摘要

In the past decade, an intensive study of strong approximation of stochas-tic differential equations (SDEs) with a drift coefficient that has discontinu-ities in space has begun. In the majority of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and that the diffusion coefficient is globally Lipschitz continuous and nondegenerate at the discon-tinuities of the drift coefficient. Under this type of assumptions the best L-p-error rate obtained so far for approximation of scalar SDEs at the final time is 3/4 in terms of the number of evaluations of the driving Brownian motion. In the present article, we prove for the first time in the literature sharp lower error bounds for such SDEs. We show that for a huge class of additive noise driven SDEs of this type the L-p-error rate 3/4 can not be improved. For the proof of this result we employ a novel technique by studying equations with coupled noise: we reduce the analysis of the L-p-error of an arbitrary approximation based on evaluation of the driving Brownian mo-tion at finitely many times to the analysis of the L-p-distance of two so-lutions of the same equation that are driven by Brownian motions that are coupled at the given time-points and independent, conditioned on their val-ues at these points. To obtain lower bounds for the latter quantity, we prove a new quantitative version of positive association for bivariate normal ran-dom variables (Y, Z) by providing explicit lower bounds for the covariance Cov(f (Y), g(Z)) in case of piecewise Lipschitz continuous functions f and g. In addition it turns out that our proof technique also leads to lower error bounds for estimating occupation time functionals f(0)(1) f(W-t) dt of a Brownian motion W, which substantially extends known results for the case of f being an indicator function.