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

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.