Abstract
The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of $\alpha$-H\"older drift in the recent literature the rate $\alpha/2$ was proved in many related situations. By exploiting the regularising effect of the noise more efficiently, we show that the rate is in fact arbitrarily close to $1/2$ for all $\alpha>0$. The result extends to Dini continuous coefficients, while in $d=1$ also to all bounded measurable coefficients.
Highlights
Introduction and main resultsWe consider stochastic differential equations (SDEs) dXt = b(Xt) dt + dWt, X0 = x0, (1.1)on a fixed time horizon [0, T ], driven by a d-dimensional Wiener process W = (W 1, . . . , W d) on a filtered probability space (Ω, F, (Ft)t∈[0,T ], P) with (F )t∈[0,T ] satisfying the usual conditions, where x0 is an Rd-valued F0-measurable random variable with finite variance, and the drift coefficient b = (b1, . . . , bd) is a measurable function on Rd with values in Rd
It is well known that equation (1.1) has the remarkable property that even if b is only known to be bounded and measurable a unique strong solution exists [Zvo[74], Ver80], for further developments see among others [GM01, KR05, Dav[07], FF11, Sha16]
Both In2 and In3 are integrals over domains whose size is bounded by 2T n−1 with an integrand that is bounded by sup |f |2
Summary
Going beyond (locally) Lipschitz b, the first result goes back to [GK96], who established convergence in probability (without rate) of the Euler-Maruyama scheme. Outside of this exceptional set the usual Lipschitz condition in assumed Under this condition, rate 1/2 is achieved for Euler-type schemes, and in the scalar case recently the improved rate 3/4 was shown [MY19] for a modified Milstein-type scheme. A few months after the current work was available on the ArXiv, the article [NS19] appeared on the ArXiv. In the one-dimensional setting the authors show an interesting complementary result to Theorem 1.2: if in addition to our assumption b possesses regularity of order α ∈ (0, 1) in a Sobolev-Slobodecki scale, the rate of convergence improves. We would like to thank the authors of [NS19] for bringing to our knowledge that a time-homogeneous version of the estimate (2.1) below has appeared in [Alt17]
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