IJPAM: Volume 24, No. 3 (2005)

MEASURE ORDER OF CONVERGENCE WITHOUT
AN EXACT SOLUTION, EULER VS MILSTEIN SCHEME

Klaus Schmitz Abe$^{1}$Room 221, 25 University of Oxford, Students Accomodation Office, 25 Wellington Squire, Oxford, Oxfordshire, OX1 2Jh, UK, William T. Shaw$^{2}$
$^{1,2}$Oxford Centre for Industrial and Applied Mathematics (OCIAM)
Mathematical Institute
University of Oxford
Oxford, OX1 2JH, UK
$^{1}$e-mail: schmitz@maths.ox.ac.uk
$^{2}$e-mail: shaww@maths.ox.ac.uk


Abstract.The purpose of this paper is to measure the strong and weak order of convergence of both the Euler and Milstein schemes using a stochastic volatility model and an $N-$dimensional Exponential Brownian Motion Process (EBM). An exact solution is normally required to calculate the order of convergence, however there are none available for this volatility process. We propose a method to solve this problem. We also show numerically that when we apply the Milstein scheme to an $N-$dimensional stochastic process, there is a need to take into account the correlation between the systems.

Received: August 10, 2005

AMS Subject Classification: 60H10, 91B70, 62P05

Key Words and Phrases: Euler and Milstein scheme, strong convergence, weak convergence, Exponential Brownian Motion, stochastic volatility models

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 24
Issue: 3