Hitting times for sticky skew CIR process

Abstract

In this paper, we consider an extended skew CIR processes with sticky points, which is referred to as the sticky skew CIR process. We first calculate the infinitesimal generator and its domain. To explore its trajectory properties, we compute the Laplace transforms and the expectations of first hitting times over a constant boundary. The solutions of Laplace transforms are expressed in terms of Tricomi and Kummer confluent hypergeometric functions.

Keywords:

• Sticky skew diffusion
• CIR process
• time change
• infinitesimal generator
• first hitting time

AMS Subject Classifications:

• 60J55
• 60J60

Acknowledgements

Authors would like to give our thanks to Professor Renming Song at Department of Mathematics, University of Illinois at Urbana-Champaign (UIUC), and gratefully acknowledge the participants in probability theory and financial engineering seminar in Nankai University for their stimulating comments and suggestions in earlier version of this article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 With the exception of the recent results on sticky Brownian motion [4,7], few such pathwise representation is known in the case of one-dimensional processes that may have sticky points. In this paper, we address this question.

2 We take the notation that $\frac{\mathrm{d}{f}^{-}}{\mathrm{d}{S}_X}(x)={lim}_{\epsilon \downarrow 0}\frac{f(x)-f(x-\epsilon )}{S_X(x)-S_X(x-\epsilon ).}$

3 $\mathrm{\Gamma }(\cdot )$ is the Gamma function denoted by $\mathrm{\Gamma }(z)={\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{z-1}\mathrm{d}t,\mathrm{Re}(z)>0.$

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (No. 12101602) and the Counterpart Fund of National Natural Science Foundation of China (No. 3122022PT17).