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On the simulation of iterated Itô integrals

Wiktorsson, Magnus LU and Rydén, Tobias LU (2001) In Stochastic Processes and their Applications 91(1). p.151-168
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... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Iterated Itô integral, Infinitely divisible distribution, Multi-dimensional stochastic differential equation, Numerical approximation, Class G distribution, Variance mixture, Coupling
in
Stochastic Processes and their Applications
volume
91
issue
1
pages
151 - 168
publisher
Elsevier
external identifiers
  • scopus:0007336796
ISSN
1879-209X
DOI
language
English
LU publication?
yes
id
b8cb2298-7ebe-4696-98cf-ff1a298f15bc (old id 1478506)
date added to LUP
2009-09-21 10:14:53
date last changed
2018-06-10 04:40:29
@article{b8cb2298-7ebe-4696-98cf-ff1a298f15bc,
  abstract     = {We consider algorithms for simulation of iterated Itô integrals with<br/><br>
application to simulation of stochastic differential equations. The<br/><br>
fact that the iterated Itô integral<br/><br>
I_{ij}(t_n,t_n+h)=\int_{t_n}^{t_n+h} \int_{t_n}^{s} dW_{i}(u)dW_{j}(s)<br/><br>
conditioned on W_i(t_n+h)-W_i(t_n) and W_j(t_n+h)-W_j(t_n), has an<br/><br>
infinitely divisible distribution is utilised for the simultaneous<br/><br>
simulation of $I_{ij}(t_n,t_n+h)$,W_{i}(t_n+h)-W_{i}(t_n) and<br/><br>
W_j(t_n+h)-W_j(t_n). Different simulation methods for the iterated<br/><br>
Itô integrals are investigated. We show mean square convergence rates<br/><br>
for approximations of shot-noise type and asymptotic normality of the<br/><br>
remainder of the approximations. This together with the fact that the<br/><br>
conditional distribution of I_{ij}(t_n,t_n+h), apart from an additive<br/><br>
constant, is a Gaussian variance mixture is used to achieve an<br/><br>
improved convergence rate. This is done by a coupling method for the<br/><br>
remainder of the approximation.},
  author       = {Wiktorsson, Magnus and Rydén, Tobias},
  issn         = {1879-209X},
  keyword      = {Iterated Itô integral,Infinitely divisible distribution,Multi-dimensional stochastic differential equation,Numerical approximation,Class G distribution,Variance mixture,Coupling},
  language     = {eng},
  number       = {1},
  pages        = {151--168},
  publisher    = {Elsevier},
  series       = {Stochastic Processes and their Applications},
  title        = {On the simulation of iterated Itô integrals},
  url          = {http://dx.doi.org/},
  volume       = {91},
  year         = {2001},
}