Sharp lower error bounds for strong approximation of SDEs with discontinuous drift coefficient by coupling of noise

Abstract

In the past decade, an intensive study of strong approximation of stochastic differential equations (SDEs) with a drift coefficient that has discontinuities 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 non-degenerate at the discontinuities of the drift coefficient. Under this type of assumptions the best Lp-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 Lp-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 Lp-error of an arbitrary approximation based on evaluation of the driving Brownian motion at finitely many times to the analysis of the Lp-distance of two solutions of the same equation that are driven by Brownian motions that are coupled at the given time-points and independent, conditioned on their values at these points. To obtain lower bounds for the latter quantity, we prove a new quantitative version of positive association for bivariate normal random variables (Y, Z) by providing explict 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 1 0 f (Wt) dt of a Brownian motion W , which substantially extends known results for the case of f being an indicator function.

Regular conditional distributions for semimartingale SDEs

“…In the end we give an example how this result can be applied for proving Markov properties for solutions of semimartingale SDEs. In the special case of a classical Brownian motion-driven SDE such a result has been proven in [8] using similar arguments and referring to the functional representation proven in [3].…”

“…This kind of result is needed, for example, in computational stochastics as it implies the Markov property. For an example where the classical result for the Brownian case has been applied, see [8]. The class of SDEs we consider if very general and therefore covers a wide range of applications.…”

Stochastic differential equations with irregular coefficients:~mind the gap!

“…Theorem 5.1 (Müller-Gronbach and Yaroslavtseva [26]). For scalar SDEs with additive noise the convergence order of any numerical method on a finite deterministic grid is at most 3/4.…”

Optimal L2 -approximation of occupation and local times for symmetric stable processes

“…Rate optimality of the Riemann sum estimators in the case of Brownian motion (with drift) can be obtained from [20,13,2], but it is unclear if their methods extend to jump processes, or if Riemann estimators are asymptotically efficient in the sense of reaching the minimal asymptotic error. More recently, there is also some interest in numerical analysis for the L p -approximation error in the context of analysing Euler schemes with non-degenerate coefficients ( [18], [17]), see also [8].…”