Some remarks on first passage of Levy processes, the American put and pasting principles

The purpose of this article is to provide, with the help of a fluctuation identity, a generic link between a number of known identities for the first passage time and overshoot above/below a fixed level of a Levy process and the solution of Gerber and Shiu [Astin Bull. 24 (1994) 195-220], Boyarchenko and Levendorskii [Working paper series EERS 98/02 (1998), Unpublished manuscript (1999), SIAM J. Control Optim. 40 (2002) 1663-1696], Chan [Original unpublished manuscript (2000)], Avram, Chan and Usabel [Stochastic Process. Appl. 100 (2002) 75-107], Mordecki [Finance Stoch. 6 (2002) 473-493], Asmussen, Avram and Pistorius [Stochastic Process. Appl. 109 (2004) 79-111] and Chesney and Jeanblanc [Appl. Math. Fin. 11 (2004) 207-225] to the American perpetual put optimal stopping problem. Furthermore, we make folklore precise and give necessary and sufficient conditions for smooth pasting to occur in the considered problem.Comment: Published at http://dx.doi.org/10.1214/105051605000000377 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Branching processes in random environment die slowly

Let $Z_{n,}n=0,1,...,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f_{0}(s),f_{1}(s),...,$ and let $S_{0}=0,S_{k}=X_{1}+...+X_{k},k\geq 1,$ be the associated random walk with $X_{i}=\log f_{i-1}^{\prime}(1),$ $\tau (m,n)$ be the left-most point of minimum of $\left\{S_{k},k\geq 0\right\}$ on the interval $[m,n],$ and $T=\min \left\{k:Z_{k}=0\right\}$. Assuming that the associated random walk satisfies the Doney condition $P(S_{n}>0) \to \rho \in (0,1),n\to \infty ,$ we prove (under the quenched approach) conditional limit theorems, as $n\to \infty$, for the distribution of $Z_{nt},$ $Z_{\tau (0,nt)},$ and $Z_{\tau (nt,n)},$ $t\in (0,1),$ given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt.

Overshoots and undershoots of L\'{e}vy processes

We obtain a new fluctuation identity for a general L\'{e}vy process giving a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the undershoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Kl\"{u}ppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766--1801] concerning asymptotic overshoot distribution of a particular class of L\'{e}vy processes with semi-heavy tails and refine some of their main conclusions. In particular, we explain how different types of first passage contribute to the form of the asymptotic overshoot distribution established in the aforementioned paper. Applications in insurance mathematics are noted with emphasis on the case that the underlying L\'{e}vy process is spectrally one sided.Comment: Published at http://dx.doi.org/10.1214/105051605000000647 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Ciesielski-Taylor type identity for positive self-similar Markov processes

The aim of this note is to give a straightforward proof of a general version of the Ciesielski-Taylor identity for positive self-similar Markov processes of the spectrally negative type which umbrellas all previously known Ciesielski-Taylor identities within the latter class. The approach makes use of three fundamental features. Firstly a new transformation which maps a subset of the family of Laplace exponents of spectrally negative L\'evy processes into itself. Secondly some classical features of fluctuation theory for spectrally negative L\'evy processes as well as more recent fluctuation identities for positive self-similar Markov processes

The extended hypergeometric class of L\'evy processes

With a view to computing fluctuation identities related to stable processes, we review and extend the class of hypergeometric L\'evy processes explored in Kuznetsov and Pardo (arXiv:1012.0817). We give the Wiener-Hopf factorisation of a process in the extended class, and characterise its exponential functional. Finally, we give three concrete examples arising from transformations of stable processes.Comment: 22 page

Ruin Probabilities and Overshoots for General Levy Insurance Risk Processes

We formulate the insurance risk process in a general Levy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to -\infty a.s. and the positive tail of the Levy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Kluppelberg [Stochastic Process. Appl. 64 (1996) 103-125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207-226] for ruin probabilities and the overshoot in random walk and compound Poisson models are shown to have analogues in the general setup. The identities we derive open the way to further investigation of general renewal-type properties of Levy processes.Comment: Published at http://dx.doi.org/10.1214/105051604000000927 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org