1 Introduction
Type: R functions for ‘Exact simulation of tempered stable Ornstein-Uhlenbeck processes’
Version: 2.02
Date: 2012-2-9
Author: Shibin Zhang
Maintainer: Shibin Zhang <sbzhang@shmtu.edu.cn>
Description: Companion functions to the paper ‘Exact simulation of tempered
stable Ornstein-Uhlenbeck processes’.
This document provides R functions for the exact simulation method
of tempered stable Ornstein-Uhlenbeck processes using the parametrisation
described by Zhang (2011).
Usage: To use the software, you will need to download the file
exact_simu_TSOU.R (in R_exact_simu_TSOUv2.02.rar) into a suitable directory on your computer. This contains
the functions listed below and various supporting functions. You should
not need to look at the R code in this file unless you want to see the details of
what’s going on.
2 The functions
rTS(n,kappa,delta,gamma)
rIG(n,delta,gamma)
rPS(n,kappa,delta)
TSOU(x=2*kappa*delta*gamma^ (1-1/kappa), lambda=1, kappa=1/2, delta=1, gamma=1, T=1, N=100)
IGOU(x=delta/gamma, lambda=1, delta=1, gamma=1, T=1, N=100)
PSOU(x=1, lambda=1, kappa=1/2, delta=1, T=1, N=100)

The functions rTS, rIG and rPS are used to sample n points from the TS (κ,δ,γ),
IG(δ,γ) and S (κ,δ) distributions, respectively. And the function rTS employs the
double rejection method in Devroye (2009); rIG employs the method in Michael et
al. (1976); and rPS employs the method in Chambers et al. (1976) (see also Zhang
(2008).

The functions TSOU, IGOU and PSOU are the simulations of the O-U process with
marginals TS (κ,δ,γ), IG(δ,γ) and S (κ,δ), respectively. lambda is the intensity parameter
of the O-U process. x is the initial value of the process at time t0. T is the
final time. N is the number of intervals in which to split [t0,T].  These three functions employ the methods in Zhang and Zhang (2008),
Zhang and Zhang (2009) and Zhang (2011).

References

Chambers, J. M., Mallows, C. L., Stuck, B.W., 1976. A method for simulating
stable random variables. J. Amer. Statist. Assoc. 71, 340-344.
Devroye, L., 2009. Random variate generation for exponentially and polynomially
tilted stable distributions. ACM Transactions on Modeling and Computer Simu-
lation (TOMACS) 19(4), Article No. 18.
Michael J. R., Schucany W. R., Haas, R.W., 1976. Generating random variates us-
ing transformations with multiple roots. American Statistician 30, 88C90.
R Development Core Team, 2009. R: A language and environment for statistical
computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-
900051-07-0, URL http://www.R-project.org.
Zhang, S., 2008. Simulation of non-Gaussian OU-based stochastic volatility mod-
els. In: Ai, C.,Wu, D. (Eds.), Proceedings of International Symposium on Finan-
cial Engineering and Risk Management 2008: 234-238.
Zhang, S., Zhang, X., 2008a. Exact simulation of IG-OU processes. Methodol.
Comput. Appl. Probab. 10(3), 337-355.
Zhang, S., Zhang, X., 2009. On the transition law of tempered stable Ornstein-
Uhlenbeck processes. J. Appl. Prob. 46, 721-731.
Zhang S., 2011. Exact simulation of tempered stable Ornstein-Uhlenbeck pro-
cesses. J. Stat. Comput. Simul. 81(11), 1533-1544.