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Marc Yor

Publications and source records attributed to Marc Yor.

At least 19 recordsLinked to original sources
arXiv2018

A guide to Brownian motion and related stochastic processes

This is a guide to the mathematical theory of Brownian motion and related stochastic processes, with indications of how this theory is related to other branches of mathematics, most notably the classical theory of partial differential equations associated with the Laplace and heat operators, and various generalizations thereof. As a typical reader, we have in mind a student, familiar with the basic concepts of probability based on measure theory, at the level of the graduate texts of Billingsley and Durrett , and who wants a broader perspective on the theory of Brownian motion and related stochastic processes than can be found in these texts.

math.PR↗
arXiv2016

Exercices sur les temps locaux de semi-martingales continues et les excursions browniennes

Depuis le tout début du XX${}^\text{e}$ siècle, l'étude des processus stochastiques est un domaine très actif de la recherche en mathématiques. Parmi ces processus, le mouvement brownien --- dont l'étude mathématique a été initiée dès 1900, avec la thèse de Bachelier, entre autres travaux --- a joué, et joue encore, un rôle primordial. Ceci peut s'expliquer par le fait que le mouvement brownien est l'objet limite quasi-universel qui apparaît dans le théorème central limite, lorsqu'on fait agir le temps. Depuis la fin de la seconde guerre mondiale et les travaux d'Itô, Meyer, Tanaka et bien d'autres, les temps locaux et les excursions sont devenus des outils essentiels pour étudier ce processus. Les exercices de ce volume ont été élaborés, année après année, par le second auteur, soit à la suite de lectures d'articles présentant, parfois avec des méthodes très différentes, telle ou telle propriété brownienne, soit simplement pour illustrer le contenu de son cours de DEA (anciennement), de M2 aujourd'hui. Le premier auteur en a organisé la synthèse, de façon économique et néanmoins --- espérons-le --- très lisible. Les chapitres ont été conçus pour créer un aller-retour permanent entre les principaux résultats du cours et les exercices corrigés, afin que la compréhension des uns renforce celle des autres. C'est ainsi que de nombreuses solutions d'exercices données ici offrent un aperçu de la façon de prouver certains des théorèmes rappelés plus haut.

math.PR↗
arXiv2016

The maximal drawdown of the Brownian meander

Motivated by evaluating the limiting distribution of randomly biased random walks on trees, we compute the exact value of a negative moment of the maximal drawdown of the standard Brownian meander.

math.PR↗
arXiv2013

Some explicit formulas for the Brownian bridge, Brownian meander and Bessel process under uniform sampling

We show that simple explicit formulas can be obtained for several relevant quantities related to the laws of the uniformly sampled Brownian bridge, Brownian meander and three dimensional Bessel process. To prove such results, we use the distribution of a triplet of random variables associated to the pseudo-Brownian bridge together with various relationships between the laws of these four processes.

math.PR↗
arXiv2013

On the law of a triplet associated with the pseudo-Brownian bridge

We identify the distribution of a natural triplet associated with the pseudo-Brownian bridge. In particular, for $B$ a Brownian motion and $T_1$ its first hitting time of the level one, this remarkable law allows us to understand some properties of the process $(B_{uT_1}/\sqrt{T_1}, u\leq 1)$ under uniform random sampling.

math.PR↗
arXiv2013

On the Mellin transforms of the perpetuity and the remainder variables associated to a subordinator

Results about the laws of the perpetuity and remainder variables associated to a subordinator are presented, with particular emphasis on their Mellin transforms, and multiplicative infinite divisibility property. Previous results by Bertoin-Yor (Electron. Commun. Probab. 6 (2001) 95-106) are incorporated in our discussion; important examples when the subordinator is the inverse local time of a diffusion are exhibited. Results of Urbanik (Probab. Math. Statist. 15 (1995) 493-513) are also discussed in detail; they appear to be too little known, despite the fact that quite a few of them have priority upon other works in this area.

math.ST↗
arXiv2013

Unifying the Dynkin and Lebesgue-Stieltjes formulae

We establish a local martingale $M$ associate with $f(X,Y)$ under some restrictions on $f$, where $Y$ is a process of bounded variation (on compact intervals) and either $X$ is a jump diffusion (a special case being a L\'evy process) or $X$ is some general (c\'adl\'ag metric space valued) Markov process. In the latter case $f$ is restricted to the form $f(x,y)=\sum_{k=1}^K\xi_k(x)\eta_k(y)$. This local martingale unifies both Dynkin's formula for Markov processes and the Lebesgue-Stieltjes integration (change of variable) formula for (right continuous) functions of bounded variation. For the jump diffusion case, when further relatively easily verifiable conditions are assumed then this local martingale becomes an $L^2$ martingale. Convergence of the product of this Martingale with some deterministic function (of time) to zero both in $L^2$ and a.s. is also considered and sufficient conditions for functions for which this happens are identified.

math.PR↗
arXiv2013

Local times for functions with finite variation: two versions of Stieltjes change of variables formula

We introduce two natural notions for the occupation measure of a function $V$ with finite variation. The first yields a signed measure, and the second a positive measure. By comparing two versions of the change-of-variables formula, we show that both measures are absolutely continuous with respect to Lebesgue measure. Occupation densities can be thought of as local times of $V$, and are described by a Meyer-Tanaka like formula.

math.PR↗
arXiv2013

Increasing processes and the change of variables formula for non-decreasing functions

Given an increasing process $(A_t)_{t\geq 0}$, we characterize the right-continuous non-decreasing functions $f: \R_+\to \R_+$ that map $A$ to a pure-jump process. As an example of application, we show for instance that functions with bounded variations belong to the domain of the extended generator of any subordinators with no drift and infinite Lévy measure.

math.PR↗
arXiv2013

How to make Dupire's local volatility work with jumps

There are several (mathematical) reasons why Dupire's formula fails in the non-diffusion setting. And yet, in practice, ad-hoc preconditioning of the option data works reasonably well. In this note we attempt to explain why. In particular, we propose a regularization procedure of the option data so that Dupire's local vol diffusion process recreates the correct option prices, even in manifest presence of jumps.

q-fin.PR↗
arXiv2013

Last-Hitting Times and Williams' Decomposition of the Bessel Process of Dimension 3 at its Ultimate Minimum

In this note we shortly recall the importance of last-hitting times in theory and applications of optimal stopping. As a small contribution to this domain we then propose a concise proof of David Williams' decomposition of the Bessel Process of dimension 3 (BES(3)), starting from r > 0 at its ultimate minimum. This discussion is strongly motivated by our interest in properties of last hitting times in general, and here specifically, directly linked with the forthcoming reading guide of Nikeghbali and Platen on this subject.

math.PR↗
arXiv2012

A scaling proof for Walsh's Brownian motion extended arc-sine law

We present a new proof of the extended arc-sine law related to Walsh's Brownian motion, known also as Brownian spider. The main argument mimics the scaling property used previously, in particular by D. Williams in the 1-dimensional Brownian case, which can be generalized to the multivariate case. A discussion concerning the time spent positive by a skew Bessel process is also presented.

math.PR↗
arXiv2011

The Mean First Rotation Time of a planar polymer

We estimate the mean first time, called the mean rotation time (MRT), for a planar random polymer to wind around a point. This polymer is modeled as a collection of n rods, each of them being parameterized by a Brownian angle. We are led to study the sum of i.i.d. imaginary exponentials with one dimensional Brownian motions as arguments. We find that the free end of the polymer satisfies a novel stochastic equation with a nonlinear time function. Finally, we obtain an asymptotic formula for the MRT, whose leading order term depends on the square root of n and, interestingly, depends weakly on the mean initial configuration. Our analytical results are confirmed by Brownian simulations.Our analytical results are confirmed by Brownian simulations.

math.PR↗