A stable Cox-Ingersoll-Ross model with restart
type: Item
instance of: scholarly article
title: A stable Cox-Ingersoll-Ross model with restart
zbMATH Open document ID: 1355.60113
author: Jun Peng
published in: Journal of Mathematical Analysis and Applications
publication date: 2016-08-18T00:00:00Z
review text: A stable Cox-Ingersoll-Ross (CIR) model \(\{X_t: t\geq0\}\) defined by \(dX_t = (a-bX_t)dt +\sigma\root\alpha\of{X_{t-}}dZ_t\), where \(\{Z_t: t\geq0\}\) is a spectrally positive stable (\(\mathcal{F}_t\))-Lévy process with index \(1<\alpha\leq2\) is considered. Let the operator \(L\) be defined by \(f\in C_0^2(R+)\) \[ Lf(x)=(a-bx)f'(x) + \frac{\sigma^\alpha x}{\alpha\Gamma( -\alpha)} \int_0^\infty[f(x+y)-f(x)-yf'(x)]\frac {dy} {y^{\alpha+1}}. \] Consider a Markov process \(X(t)\) in \(D = [0,\infty)\) which performs a stable CIR process in \(D\) generated by the operator \(L\) and stopped at 0 and \(\infty\), and which while running, jumps instantaneously at an exponentially distributed random time with intensity \(\lambda>0\) to a new point, according to a distribution \(\nu\). Proposition 2.1 states: Let \(p(t, x, y)\) be the transition density of the stable CIR process \(X(t)\), then the Markov process \(\{\hat X(t): t\in[0,\infty)\}\) that restart from a distribution \(\nu\) after independent exponentially distributed random times with mean \(1/\lambda\) has the following transition density \[ \hat p_\nu(t,x,y) = e^{-\lambda t}p(t,x,y) + \int_0^t\lambda e^{-\lambda s}p(s,\nu,y)\,ds. \] In Section 2, the infinitesimal generator of the process \(\{\hat X(t): t\in[0,\infty)\}\) (Theorem 2.2) and the uniqueness of the martingale problem (Proposition 2.4) are given. In Section 3, the exponential decay of a stable CIR model with restart as \(t\to\infty\) is considered. The compactness of the corresponding semigroup is proved in Proposition 3.1 and thus the spectrum of the generator consists of exclusively of eigenvalues. Further, in Theorem 3.2 it is shown that the principal eigenvalue gives the exponential rate of decay of not exiting the domain by time \(t\). Finally, in Proposition 3.3 some properties of the principal eigenvalue are studied.
reviewed by: Valery V. Karachik
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 91G20
zbMATH DE Number: 6618080
zbMATH Keywords: stable Cox-Ingersoll-Ross model
zbMATH Keywords: restart
zbMATH Keywords: infinitesimal generator
zbMATH Keywords: decay rate
zbMATH Keywords: Markov process
MaRDI profile type: Publication
full work available at URL: j.jmaa.2016.07.010
OpenAlex ID: W2493976400
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DOI: 10.1016/J.JMAA.2016.07.010
description: scientific article; zbMATH DE number 6618080