Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4626259 | Applied Mathematics and Computation | 2015 | 17 Pages |
Abstract
We investigate strong approximation of solutions of scalar stochastic differential equations (SDEs) with irregular coefficients. In PrzybyÅowicz (2015) [23], an approximation of solutions of SDEs at a single point is considered (such kind of approximation is also called a one-point approximation). Comparing to that article, we are interested here in a global reconstruction of trajectories of the solutions of SDEs in a whole interval of existence. We assume that a drift coefficient a:[0,T]ÃRâR is globally Lipschitz continuous with respect to a space variable, but only measurable with respect to a time variable. A diffusion coefficient b:[0,T]âR is only piecewise Hölder continuous with Hölder exponent ϱ â (0, 1]. The algorithm and results concerning lower bounds from PrzybyÅowicz (2015) [23] cannot be applied for this problem, and therefore we develop a suitable new technique. In order to approximate solutions of SDEs under such assumptions we define a discrete type randomized Euler scheme. We provide the error analysis of the algorithm, showing that its error is O(nâmin{ϱ,1/2}). Moreover, we prove that, roughly speaking, the error of an arbitrary algorithm (for fixed a and b) that uses n values of the diffusion coefficient, cannot converge to zero faster than nâmin{ϱ,1/2} as nâ+â. Hence, the proposed version of the randomized Euler scheme achieves the established best rate of convergence.
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
PaweÅ PrzybyÅowicz,