Optimal global approximation of SDEs with time-irregular coefficients in asymptotic setting

Paweł Przybyłowicz

Applied Mathematics and Computation, 2015, vol. 270, issue C, 441-457

Abstract: We investigate strong approximation of solutions of scalar stochastic differential equations (SDEs) with irregular coefficients. In Przybyłowicz (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]×R→R 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]→R is only piecewise Hölder continuous with Hölder exponent ϱ ∈ (0, 1]. The algorithm and results concerning lower bounds from Przybyłowicz (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(n−min{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 n−min{ϱ,1/2} as n→+∞. Hence, the proposed version of the randomized Euler scheme achieves the established best rate of convergence.

Keywords: Non-standard assumptions; Global approximation; Adaptive/nonadaptive standard information; Minimal error algorithm; Monte Carlo algorithms (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:270:y:2015:i:c:p:441-457

DOI: 10.1016/j.amc.2015.08.055

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