Larbi Alili

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Gerber and Shiu [Astin Bull. 24 (1994) 195–220], Boyarchenko and Levendorskii [Working paper series EERS 98/02 (1998), Unpublished manuscript (1999), SIAM J. Control Optim. 40 (2002) 1663–1696], Chan [Original unpublished manuscript (2000)], Avram, Chan and Usabel [Stochastic Process.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.

Representations of the first hitting time density of an Ornstein-Uhlenbeck process

ABSTRACT Three expressions are provided for the first hitting time density of an Ornstein-Uhlenbeck process to reach a fixed level. The first hinges on an eigenvalue expansion involving zeros of the parabolic cylinder functions. The second is an integral representation involving some special functions whereas the third is given in terms of a functional of a 3-dimensional Bessel bridge. The expressions are used for approximating the density. 1Research supported by RiskLab, Switzerland, funded by Credit Suisse Group, Swiss Re and UBS AG. The third author was supported by a Steno grant from the Danish Natural Science Research Council.

Wiener-Hopf factorization revisited and some applications

A reformulation of the classical Wiener-Hopf factorization for random walks is given; this is applied to the study of the asymptotic behaviour of the ladder variables, the distribution of the maximum and the renewal mass function in the bivariate renewal process of ladder times and heights

A new fluctuation identity for Lévy processes and some applications

Let τ and H be respectively the ladder time and ladder height processes associated with a given Levy process X. We give an identity in law between (τ,H) and (X,H*), H* being the right-continuous inverse of the process H. This allows us to obtain a relationship between the entrance law of X and the entrance law of the excursion measure away from 0 of the reflected process (Xt- infs≤tXs, t ≥0). In the stable case, some explicit calculations are provided. These results also lead to an explicit form of the entrance law of the Levy process conditioned to stay positive.

Martin boundaries associated with a killed random walk

We start by studying the connection between the full Martin boundary associated with a space time version of a random walk which is killed on entering the negative half-line, and that associated with the bivariate renewal process of weak increasing ladder heights and times in the random walk. We show that although the corresponding spatial boundaries are isomorphic, the space time boundaries are not. The rest of the paper is devoted to determining these boundaries explicitly in the special case that the moment generating function of the step distribution exists in a non-empty interval.

On inversions and Doob h-transforms of linear diffusions

Let X be a regular linear diffusion whose state space is an open interval \(E \subseteq \mathbb{R}\). We consider the dual diffusion X∗ whose probability law is obtained as a Doob h-transform of the law of X, where h is a positive harmonic function for the infinitesimal generator of X on E. We provide a construction of X∗ as a deterministic inversion I(X) of X, time changed with some random clock. Such inversions generalize the Euclidean inversions that intervene when X is a Brownian motion. The important case where X∗ is X conditioned to stay above some fixed level is included. The families of deterministic inversions are given explicitly for the Brownian motion with drift, Bessel processes and the three-dimensional hyperbolic Bessel process.

Boundary crossing identities for Brownian motion and some nonlinear ode’s

We start by introducing a nonlinear involution operator which maps the space of solutions of Sturm-Liouville equations into the space of solutions of the associated equations which turn out to be nonlinear ordinary differential equations. We study some algebraic and analytical properties of this involution operator as well as some properties of a two-parameter family of operators describing the set of solutions of Sturm-Liouville equations. Next, we show how a specific composition of these mappings allows us to connect, by means of a simple analytical expression, the law of the first passage time of a Brownian motion over a curve to a two-parameter family of curves. We offer three different proofs of this fact which may be of independent interests. In particular, one is based on the construction of parametric time-space harmonic transforms of the law of some Gauss-Markov processes. Another one, which is of algebraic nature, relies on the Lie group symmetry methods applied to the heat equation and reveals that our two-parameter transformation is the unique nontrivial one.

On a fluctuation identity for random walks and lévy processes

In this paper, some identities in laws involving ladder processes for random walks and Levy processes are extended and unified.

Müntz linear transforms of Brownian motion

We consider a class of Volterra linear transforms of Brownian motion associated to a sequence of Muntz Gaussian spaces and determine explicitly their kernels; the kernels take a simple form when expressed in terms of Muntz-Legendre polynomials. These are new explicit examples of progressive Gaussian enlargement of a Brownian filtration. We give a necessary and sufficient condition for the existence of kernels of infinite order associated to an infinite dimensional Muntz Gaussian space; we also examine when the transformed Brownian motion remains a semimartingale in the filtration of the original process. This completes some already obtained partial answers to the aforementioned problems in the infinite dimensional case.

Quelques nouvelles identités de fluctuation pour les processus de Lévy

Let τ and H be the ladder time and ladder height processes of a Levy process X. We give an identity in law between (τ,H) and (X,H*), H* being the right continuous inverse of the process H. The later allows us to get a relationship between the entrance law of X and the entrance law of the excursion measure away from 0 of the reflected process (Xt- infs≤ Xs >- 0). In the stable case, some explicit calculations are provided.