Strong approximation of solutions of stochastic differential equations with time-irregular coefficients via randomized Euler algorithm

We investigate pointwise approximation of the solution of a scalar stochastic differential equation in case when drift coefficient is a Carathéodory mapping and diffusion coefficient is only piecewise Hölder continuous with Hölder exponent ϱ∈(0,1]ϱ∈(0,1]. Since under imposed assumptions drift is only measurable with respect to the time variable, the classical Euler algorithm does not converge in general to the solution of such equation. We give a construction of the randomized Euler scheme and prove that it has the error O(n−min{ϱ,1/2})O(n−min{ϱ,1/2}), where n is the number of discretization points. We also investigate the optimality of the defined algorithm.