A strong order 3/4 method for SDEs with discontinuous drift coefficient

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.