# On the simulation of iterated Itô integrals

## Tobias Rydén and Magnus Wiktorsson

### Centre for Mathematical Sciences Mathematical Statistics Lund Institute of Technology, Lund University, 1999

##### ISSN 1403-9338
Abstract:
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
I_{ij}(t_n,t_n+h)=\int_{t_n}^{t_n+h}
\int_{t_n}^{s}\,dW_{i}(u)\,dW_{j}(s),\nonumber

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