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 $L_p$-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 motion at finitely many times to the analysis of the $L_p$-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 $\text{Cov}(f(Y),g(Z))$ 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 $\int_0^1 f(W_t)\, dt$ of a Brownian motion $W$, which substantially extends known results for the case of $f$ being an indicator function.