A strong order 3/4 method for SDEs with discontinuous drift coefficient. (20th November 2020)

• Record Type: Journal Article
• Authors: Müller-Gronbach, Thomas; Yaroslavtseva, Larisa
• Is Part Of: IMA journal of numerical analysis. Volume 42:Number 1(2022)
• Page Start: 229
• Page End: 259
• Publication Date: 2020-11-20
• Subjects: Stochastic differential equations -- discontinuous drift coefficient -- strong approximation -- Milstein scheme; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/draa078
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: In this paper we study strong approximation of the solution of a scalar stochastic differential equation (SDE) at the final time in the case when the drift coefficient may have discontinuities in space. Recently, it has been shown in Müller-Gronbach & Yaroslavtseva (2020, On the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficient. Ann. Inst. Henri Poincaré Probab. Stat ., 56, 1162–1178) that for scalar SDEs with a piecewise Lipschitz drift coefficient, and a Lipschitz diffusion coefficient that is nonzero at the discontinuity points of the drift coefficient the classical Euler–Maruyama scheme achieves an $L_p$ -error rate of at least $1/2$ for all $p\in [1, \infty )$ . Up to now this was the best $L_p$ -error rate available in the literature for equations of that type. In the present paper we construct a method based on finitely many evaluations of the driving Brownian motion that even achieves an $L_p$ -error rate of at least $3/4$ for all $p\in [1, \infty )$ under additional piecewise smoothness assumptions on the coefficients. This is the first higher order method known in the literature for SDEs of that type. We add that the $L_p$ -error rate $3/4$ cannot be improved in general. To obtain the upper error bound for our new method, we prove in particular that a quasi-Milstein scheme achieves an $L_p$ -error rate of at least $3/4$ in the case of coefficients that are both Lipschitz continuous and piecewise differentiable with Lipschitz continuous derivatives, which is of interest in itself. The latter error rates are obtained via a detailed analysis of the average size of increments of the time-continuous quasi-Milstein scheme over time intervals in which the scheme crosses a point of nondifferentiability of the coefficients.