Numerical solution of quadratic SDE with measurable drift

Abstract

In this paper we are interested in solving numerically quadratic SDEs with non-necessary continuous drift of the from Xt = x + ? t 0 b(s,Xs)ds + ? t 0 f (Xs)?2(Xs)ds + ? t 0 ?(Xs)dWs, where, x is the initial data b and ? are given coefficients that are assumed to be Lipschitz and bounded and f is a measurable bounded and integrable function on the whole space R. Numerical simulations for this class of SDE of quadratic growth and measurable drift, induced by the singular term f (x)?2(x), is implemented and illustrated by some examples. The main idea is to use a phase space transformation to transform our initial SDEs to a standard SDE without the discontinuous and quadratic term. The Euler-Maruyama scheme will be used to discretize the new equation, thus numerical approximation of the original equation is given by taking the inverse of the space transformation. The rate of convergence are shown to be of order 1/2.