## 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.

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.

Original language | English |
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Pages (from-to) | 151-168 |

Journal | Stochastic Processes and their Applications |

Volume | 91 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2001 |

## Subject classification (UKÄ)

- Probability Theory and Statistics

## Keywords

- Iterated Itô integral
- Infinitely divisible distribution
- Multi-dimensional stochastic differential equation
- Numerical approximation
- Class G distribution
- Variance mixture
- Coupling