2019/04/18 by Thomas Müller-Gronbach, Müller-Gronbach, Thomas, Larisa Yaroslavtseva +1
Economics, Econometrics and Finance · Social Sciences · #FOS: Mathematics #Insurance, Mortality, Demography, Risk Management #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1904.09178
openalex publication_date 2019/04/18 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
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, T., and Yaroslavtseva, L., On the performance of the Euler-Maruyama scheme for SDEs with discontinuous drift coefficient, arXiv:1809.08423 (2018)] that for scalar SDEs with a piecewise Lipschitz drift coefficient and a Lipschitz diffusion coefficient that is non-zero at the discontinuity points of the drift coefficient the classical Euler-Maruyama scheme achieves an Lp-error rate of at least 1/2 for all p∈ [1,∞). Up to now this was the best Lp-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 Lp-error rate of at least 3/4 for all p∈ [1,∞) under additional piecewise smoothness assumptions on the coefficients. To obtain this result we prove in particular that a quasi-Milstein scheme achieves an Lp-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.