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

Minimal asymptotic error for one-point approximation of SDEs with time-irregular coefficients

“…In order to show an algorithm with, at least in some cases, optimal convergence properties we recall after [13] the definition of the randomized Euler algorithmX RE (T ) for SDEs (1). For n ∈ N, let…”

“…The error and optimality in the worst case setting [15] of the randomized Euler algorithmX RE (T ) was investigated in [13].…”

“…(1), with coefficients a and b having finite number of unknown discontinuities with respect to the time variable t, have been shown in [12]. In [13] the authors investigated the error of the randomized Euler scheme applied to (1) with non-smooth a and b.…”

“…Moreover, for upper bound, due to low regularity of a and b, we cannot use the algorithms developed there. In order to approximate the solution of (1) we use here the randomized Euler algorithmX RE (T ) = {X RE n (T )}, whose worst case error in the context of approximating SDEs was investigated in [13]. (See also [17,[1][2][3] where the authors used suitable versions ofX RE (T ) for approximation of solutions of ODEs.)…”

“…In [13] the problem (1) was investigated in the worst-case setting with respect to a, b and η. The error of an algorithm was meant as the largest value of the averaged difference between the actual and computed solutions over a certain class of (a, b, η), and the rate of convergence of the error to zero was of interest.…”