On the simulation of iterated Itô integrals

Tobias Rydén and Magnus Wiktorsson

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,

ISSN 1403-9338
We consider algorithms for simulation of iterated Itô integrals with application to simulation of multi-dimensional stochastic differential equations. The fact that the iterated Itô integral
conditioned on Wi(tn + h) - Wi(tn) and Wj(tn + h) - Wj(tn), has an infinitely divisible distribution is utilised for the simultaneous simulation of Iij(tn,tn + h), Wi(tn + h) - Wi(tn) and Wj(tn + h) - Wj(tn). 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 Iij(tn,tn + 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.
Key words:
iterated Itô integral, infinitely divisible distribution, multi-dimensional stochastic differential equation, numerical approximation, class G distribution, variance mixture, coupling