Optimality of Euler-type algorithms for approximation of stochastic differential equations with discontinuous coefficients

Abstract

We consider the pointwise approximation of solutions of scalar stochastic differential equations with discontinuous coefficients. We assume the singularities of coefficients to be unknown. We show that any algorithm which does not locate the discontinuities of a diffusion coefficient has the error at least Ω(n−min{1/2, ϱ}), where ϱ∈(0, 1] is the Hölder exponent of the coefficient. In order to obtain better results, we consider algorithms that adaptively locate the unknown singularities. In the additive noise case, for a single discontinuity of a diffusion coefficient, we define an Euler-type algorithm based on adaptive mesh which obtains an error of order n−ϱ. That is, this algorithm preserves the optimal error known from the Hölder continuous case. In the case of multiple discontinuities we show, both for the additive and the multiplicative noise case, that the optimal error is Θ(n−min{1/2, ϱ}), even for the algorithms locating unknown singularities.