## Electronic Communications in Probability

### The Law of the Hitting Times to Points by a Stable Lévy Process with No Negative Jumps

Goran Peskir

#### Abstract

Let $X=(X_t)_{t \ge 0}$ be a stable Lévy process of index $\alpha \in$ with the Lévy measure $\nu(dx) = (c/x^{1+\alpha}) I_{ dx$ for $c<0$, let $x<0$ be given and fixed, and let $\tau_x = \inf\{ t<0 : X_t=x \}$ denote the first hitting time of $X$ to $x$. Then the density function $f_{\tau_x}$ of $\tau_x$ admits the following series representation: $$f_{\tau_x}(t) = \frac{x^{\alpha-1}}{\pi ( \Gamma(-\alpha) t)^{2-1/\alpha}} \sum_{n=1}^\infty \bigg[(-1)^{n-1} \sin(\pi/\alpha) \frac{\Gamma(n-1/\alpha)}{\Gamma(\alpha n-1)} \Big(\frac{x^\alpha}{c \Gamma(-\alpha)t} \Big)^{n-1}$$ $$- \sin\Big(\frac{n \pi}{\alpha}\Big) \frac{\Gamma(1+n/\alpha)}{n!} \Big(\frac{x^\alpha}{c \Gamma(-\alpha)t}\Big)^{(n+1)/\alpha-1} \bigg]$$ for $t<0$. In particular, this yields $f_{\tau_x}(0+)=0$ and $$f_{\tau_x}(t) \sim \frac{x^{\alpha-1}}{\Gamma(\alpha-1), \Gamma(1/\alpha)} (c \Gamma(-\alpha)t)^{-2+1/\alpha}$$ as $t \rightarrow \infty$. The method of proof exploits a simple identity linking the law of $\tau_x$ to the laws of $X_t$ and $\sup_{0 \le s \le t} X_s$ that makes a Laplace inversion amenable. A simpler series representation for $f_{\tau_x}$ is also known to be valid when $x<0$.

#### Article information

Source
Electron. Commun. Probab., Volume 13 (2008), paper no. 60, 653-659.

Dates
Accepted: 19 December 2008
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ecp/1465233487

Digital Object Identifier
doi:10.1214/ECP.v13-1431

Mathematical Reviews number (MathSciNet)
MR2466193

Zentralblatt MATH identifier
1193.60066

Rights

#### Citation

Peskir, Goran. The Law of the Hitting Times to Points by a Stable Lévy Process with No Negative Jumps. Electron. Commun. Probab. 13 (2008), paper no. 60, 653--659. doi:10.1214/ECP.v13-1431. https://projecteuclid.org/euclid.ecp/1465233487

#### References

• Bernyk, V. Dalang, R. C. and Peskir, G. (2008). The law of the supremum of a stable Lévy process with no negative jumps. Ann. Probab. 36 (1777-1789). (A review for this item is in process).
• Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press.
• Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Academic Press.
• Borovkov, K. and Burq, Z. (2001). Kendall's identity for the first crossing time revisited. Electron. Comm. Probab. 6 (91-94).
• Braaksma, B. L. J. (1964). Asymptotic expansions and analytic continuations for a class of Barnes-integrals. Compositio Math. 15 (239-341). not available.
• Doney, R. A. (1991). Hitting probabilities for spectrally positive Lévy processes. J. London Math. Soc. 44 (566-576).
• Doney, R. A. (2008). A note on the supremum of a stable process. Stochastics 80 (151-155). (A review for this item is in process).
• Erdélyi, A. (1954). Tables of Integral Transforms, Vol. 1. McGraw-Hill.
• Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer-Verlag.
• Monrad, D. (1976). Lévy processes: absolute continuity of hitting times for points. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 37 (43-49).
• Peskir, G. (2002). On integral equations arising in the first-passage problem for Brownian motion. J. Integral Equations Appl. 14 (397-423).
• Pollard, H. (1946). The representation of $e^{-x^lambda$ as a Laplace integral. Bull. Amer. Math. Soc. 52 (908-910).
• Pollard, H. (1948). The completely monotonic character of the Mittag-Leffler function $E_a(-x)$. Bull. Amer. Math. Soc. 54 (1115-1116).
• Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
• Schneider, W. R. (1986). Stable distributions: Fox functions representation and generalization. Proc. Stoch. Process. Class. Quant. Syst. (Ascona 1985), Lecture Notes in Phys. 262, Springer (497-511).