Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions

Abstract

We consider all two times iterated Ito integrals obtained by pairing m independent standard Brownian motions. First we calculate the conditional joint characteristic function of these integrals, given the Brownian increments over the integration interval, and show that it has a form entirely similar to what is obtained in the univariate case. Then we propose an algorithm for the simultaneous simulation of the m^2 integrals conditioned on the Brownian increments that achieves a mean square error of order 1/n^2, where n is the number of terms in a truncated sum. The algorithm is based on approximation of the tail-sum distribution, which is a multivariate normal variance mixture, by a multivariate normal distribution.