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 14039338

Abstract:

We consider algorithms for simulation of iterated Itô integrals with
application to simulation of multidimensional stochastic differential equations.
The fact that the iterated Itô integral

\begin{equation}I_{ij}(t_n,t_n+h)=\int_{t_n}^{t_n+h}

\int_{t_n}^{s}\,dW_{i}(u)\,dW_{j}(s),\nonumber\end{equation}


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 shotnoise 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.


Key words:

iterated Itô integral, infinitely divisible distribution, multidimensional
stochastic differential equation, numerical approximation, class G distribution,
variance mixture, coupling