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Session I.7 - Stochastic Computation

Tuesday, June 13, 18:00 ~ 18:30

Sharp lower error bounds for strong approximation of SDEs with a drift coefficient of Sobolev regularity $s\in (1/2,1)$

Thomas Müller-Gronbach

University of Passau, Germany   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We study strong approximation of scalar SDEs $dX_t = \mu(X_t)\, dt + dW_t$ at time $t=1$ in the case that $\mu$ is bounded and has fractional Sobolev regularity $s\in (0,1)$. Recently, it has been shown in [1] that in this case the equidistant Euler scheme achieves a root mean squared error of order $(1+s)/2$, up to an arbitrary small $\epsilon$, in terms of the number of evaluations of the driving Brownian motion $W$. In this talk we show that, for $s\in (1/2,1)$, this order can not be improved in general.

References [1] K. Dareiotis, M. Gerencsér and K. Lê. Quantifying a convergence theorem of Gy\"ongy and Krylov. arXiv:2101.12185v2 (2022).

Joint work with Simon Ellinger (University of Passau, Germany) and Larisa Yaroslavtseva (University of Graz, Austria).

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