On the simulation of iterated Itô integrals

Abstract:
We consider algorithms for simulation of iterated Itô integrals with

application to simulation of stochastic differential equations. The

fact that the iterated Itô integral

I_{ij}(t_n,t_n+h)=\int_{t_n}^{t_n+h} \int_{t_n}^{s} dW_{i}(u)dW_{j}(s)

conditioned on W_i(t_n+h)-W_i(t_n) and W_j(t_n+h)-W_j(t_n), has an

infinitely divisible distribution is utilised for the simultaneous

simulation of $I_{ij}(t_n,t_n+h)$,W_{i}(t_n+h)-W_{i}(t_n) and

W_j(t_n+h)-W_j(t_n). Different simulation methods for the iterated

Itô integrals are investigated. We show mean square convergence rates

for approximations of shot-noise type and asymptotic normality of the

remainder of the approximations. This together with the fact that the

conditional distribution of I_{ij}(t_n,t_n+h), apart from an additive

constant, is a Gaussian variance mixture is used to achieve an

improved convergence rate. This is done by a coupling method for the

remainder of the approximation.