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Marc Yor

Publications and source records attributed to Marc Yor.

At least 37 records · Page 2Linked to original sources
arXiv2010

On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes

Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary $t\mapsto a+bt,\ a\geq 0,\,b\in \R,$ by a reflecting Brownian motion. The main tool hereby is Doob's formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter $δ>0$ and show that the probability that such a Bessel bridge crosses an affine boundary is equal to the probability that this Bessel bridge stays below some fixed value.

math.PR↗
arXiv2010

Selfdecomposable Laws Associated with Hyperbolic Functins

It is shown that the hyperbolic functions can be associated with selfdecomposable distributions (in short: SD probability distributions or Lévy class L probability laws). Consequently, they admit associated background driving Lévy processes $Y$ (BDLP Y). We interpret the distributions of Y(1) via Bessel squared processes, Bessel bridges and local times.

math.PR↗
arXiv2010

A new formula for some linear stochastic equations with applications

We give a representation of the solution for a stochastic linear equation of the form $X_t=Y_t+\int_{(0,t]}X_{s-} \mathrm {d}{Z}_s$ where $Z$ is a cádlág semimartingale and $Y$ is a cádlág adapted process with bounded variation on finite intervals. As an application we study the case where $Y$ and $-Z$ are nondecreasing, jointly have stationary increments and the jumps of $-Z$ are bounded by 1. Special cases of this process are shot-noise processes, growth collapse (additive increase, multiplicative decrease) processes and clearing processes. When $Y$ and $Z$ are, in addition, independent Lévy processes, the resulting $X$ is called a generalized Ornstein-Uhlenbeck process.

math.PR↗
arXiv2010

Retrieving information from subordination

We show that if $(X_s, s\geq 0)$ is a right-continuous process, $Y_t=\int_0^t\d s X_s$ its integral process and $τ= (τ_{\ell}, \ell \geq 0)$ a subordinator, then the time-changed process $(Y_{τ_{\ell}}, \ell\geq 0)$ allows to retrieve the information about $(X_{τ_{\ell}}, \ell\geq 0)$ when $τ$ is stable, but not when $τ$ is a gamma subordinator. This question has been motivated by a striking identity in law involving the Bessel clock taken at an independent inverse Gaussian variable.

math.PR↗
arXiv2010

Around Tsirelson's equation, or: The evolution process may not explain everything

We present a synthesis of a number of developments which have been made around the celebrated Tsirelson's equation (1975), conveniently modified in the framework of a Markov chain taking values in a compact group $ G $, and indexed by negative time. To illustrate, we discuss in detail the case of the one-dimensional torus $ G=\bT $.

math.PR↗
arXiv2009

A global view of Brownian penalisations

In this monograph, we construct and study a sigma-finite measure on continuous functions from R_+ to R, strongly related to many probability measures obtained by penalisation of Brownian motion, i.e. as limits of probabilities which are absolutely continuous with respect to Wiener measure. This remarkable sigma-finite measure can be generalized in three other cases: one can start from a two-dimensional Brownian motion, from a recurrent diffusion with values in R_+, and from a discrete, recurrent Markov chain.

math.PR↗
arXiv2009

Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples

In Section 1, we present a number of classical results concerning the Generalized Gamma Convolution (:GGC) variables, their Wiener-Gamma representations, and relation with the Dirichlet processes.To a GGC variable, one may associate a unique Thorin measure. Let $G$ a positive r.v. and $Γ_t(G)$ (resp. $Γ_t(1/G))$ the Generalized Gamma Convolution with Thorin measure $t$-times the law of $G$ (resp. the law of $1/G$). In Section 2, we compare the laws of $Γ_t(G)$ and $Γ_t(1/G)$.In Section 3, we present some old and some new examples of GGC variables, among which the lengths of excursions of Bessel processes straddling an independent exponential time.

math.PR↗
arXiv2008

Measuring the "non-stopping timeness" of ends of previsible sets

In this paper, we propose several "measurements" of the "non-stopping timeness" of ends g of previsible sets, such that g avoids stopping times, in an ambiant filtration. We then study several explicit examples, involving last passage times of some remarkable martingales.

math.PR↗
arXiv2008

Call option prices based on Bessel processes

As a complement to some recent work by Pal and Protter, "Strict local martingales, bubbles, and no early exercise", we show that the call option prices associated with the Bessel strict local martingales are integrable over time, and we discuss the probability densities obtained thus.

math.PR↗
arXiv2008

Penalising symmetric stable Lévy paths

Limit theorems for the normalized laws with respect to two kinds of weight functionals are studied for any symmetric stable Lévy process of index $ 1 < α\le 2 $. The first kind is a function of the local time at the origin, and the second kind is the exponential of an occupation time integral. Special emphasis is put on the role played by a stable Lévy counterpart of the universal $ σ$-finite measure, found in [9] and [10], which unifies the corresponding limit theorems in the Brownian setup for which $ α=2 $.

math.PR↗
arXiv2008

Renewal series and square-root boundaries for Bessel processes

We show how a description of Brownian exponential functionals as a renewal series gives access to the law of the hitting time of a square-root boundary by a Bessel process. This extends classical results by Breiman and Shepp, concerning Brownian motion, and recovers by different means, extensions for Bessel processes, obtained independently by Delong and Yor.

math.PR↗
arXiv2008

On the time to reach maximum for a variety of constrained Brownian motions

We derive P(M,t_m), the joint probability density of the maximum M and the time t_m at which this maximum is achieved for a class of constrained Brownian motions. In particular, we provide explicit results for excursions, meanders and reflected bridges associated with Brownian motion. By subsequently integrating over M, the marginal density P(t_m) is obtained in each case in the form of a doubly infinite series. For the excursion and meander, we analyse the moments and asymptotic limits of P(t_m) in some detail and show that the theoretical results are in excellent accord with numerical simulations. Our primary method of derivation is based on a path integral technique; however, an alternative approach is also outlined which is founded on certain "agreement formulae" that are encountered more generally in probabilistic studies of Brownian motion processes.

cond-mat.stat-mech↗
arXiv2007

The barnes G function and its relations with sums and products of generalized Gamma convolution variables

We give a probabilistic interpretation for the Barnes G-function which appears in random matrix theory and in analytic number theory in the important moments conjecture due to Keating-Snaith for the Riemann zeta function, via the analogy with the characteristic polynomial of random unitary matrices. We show that the Mellin transform of the characteristic polynomial of random unitary matrices and the Barnes G-function are intimately related with products and sums of gamma, beta and log-gamma variables. In particular, we show that the law of the modulus of the characteristic polynomial of random unitary matrices can be expressed with the help of products of gamma or beta variables, and that the reciprocal of the Barnes G-function has a Lévy-Khintchin type representation. These results lead us to introduce the so called generalized gamma convolution variables.

math.PR↗
arXiv2007

Quasi-invariance properties of a class of subordinators

We study absolute-continuity properties of a class of stochastic processes, including the gamma and the Dirichlet processes. We prove that the laws of a general class of non-linear transformations of such processes are locally equivalent to the law of the original process and we compute explicitly the associated Radon-Nikodym densities. This work unifies and generalizes to random non-linear transformations several previous results on quasi-invariance of gamma and Dirichlet processes.

math.PR↗
arXiv2007

The characteristic polynomial of a random unitary matrix: a probabilistic approach

In this paper, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin Fourier transform of such a random polynomial, first obtained by Keating and Snaith, using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of independent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith is now obtained from the classical central limit theorems of Probability Theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm type results.

math.PR↗