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With the help of two Skorokhod embeddings, we construct martin-gales which enjoy the Brownian scaling property and the (inhomogeneous) Markov property. The second method necessitates randomization, but allows to reach any law with finite moment of order 1, centered, as the distribution of such a martingale at unit time. The first method does not necessitate… (More)

- Simona Dobre, Thierry Bastogne, Christophe Profeta, Muriel Barberi-Heyob, Alain Richard
- Automatica
- 2012

We compute the persistence exponent of the integral of a stable Lévy process in terms of its self-similarity and positivity parameters. This solves a problem raised by Z. Shi (2003). Along the way, we investigate the law of the stable process L evaluated at the first time its integral X hits zero, when the bivariate process (X, L) starts from a coordinate… (More)

We study some limit theorems for the normalized law of integrated Brownian motion perturbed by several examples of functionals: the first passage time, the n th passage time, the last passage time up to a finite horizon and the supremum. We show that the penalization principle holds in all these cases and give descriptions of the conditioned processes. In… (More)

The model consists of a signal process X which is a general Brownian diffusion process and an observation process Y , also a diffusion process, which is supposed to be correlated to the signal process. We suppose that the process Y is observed from time 0 to s > 0 at discrete times and aim to estimate, conditionally on these observations, the probability… (More)

J o u r n a l o f P r o b a b i l i t y Electron. Abstract We present some limit theorems for the normalized laws (with respect to functionals involving last passage times at a given level a up to time t) of a large class of null recurrent diffusions. Our results rely on hypotheses on the Lévy measure of the diffusion inverse local time at 0. As a special… (More)

Using three hypergeometric identities, we evaluate the harmonic measure of a finite interval and of its complementary for a strictly stable real Lévy process. This gives a simple and unified proof of several results in the literature, old and recent. We also provide a full description of the corresponding Green functions. As a by-product, we compute the… (More)

We investigate the windings around the origin of the two-dimensional Markov process (X, L) having the stable Lévy process L and its primitive X as coordinates, in the non-trivial case when |L| is not a subordinator. First, we show that these windings have an almost sure limit velocity, extending McKean's result [8] in the Brownian case. Second, we evaluate… (More)

Let (E t , t ≥ 0) be a geometric Brownian motion. In this paper, we compute the law of a generalization of Dufresne's translated perpetuity (following the terminology of Salminen-Yor) : +∞ 0 E 2 s (E 2 s + 2aE s + b) 2 ds, and show that, in some cases, this perpetuity is identical in law with the first hitting time of a three-dimensional Bessel process with… (More)

The aim of this paper is to study the law of the last passage time of a linear diffusion to a curved boundary. We start by giving a general expression for the density of such a random variable under some regularity assumptions. Following Robbins & Siegmund, we then show that this expression may be computed for some implicit boundaries via a martingale… (More)