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

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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

              \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 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.

Key words:

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