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In the theory of progressive enlargements of filtrations, the supermartingale Zt = P (g > t | Ft) associated with an honest time g, and its additive (Doob-Meyer) decomposition, play an essential role. In this paper, we propose an alternative approach, using a multiplicative representation for the supermartingale Zt, based on Doob's maximal identity. We thus give new examples of progressive enlargements. Moreover, we give, in our setting, a proof of the decomposition formula for martingales , using initial enlargement techniques, and use it to obtain some path decompositions given the maximum or minimum of some processes.

We introduce a new type of convergence in probability theory, which we call "mod-Gaussian convergence". It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of characteristic polynomials or zeta functions. We study this type of convergence in detail in the framework of infinitely divisible distributions, and exhibit some unconditional occurrences in number theory, in particular for families of L-functions over function fields in the KatzSarnak framework. A similar phenomenon of "mod-Poisson convergence" turns out to also appear in the classical Erdős-Kac Theorem.

Recently, D. Williams [19] gave an explicit example of a random time ρ associated with Brownian motion such that ρ is not a stopping time but EMρ = EM0 for every bounded martingale M . The aim of this paper is to give some characterizations for such random times, which we call pseudo-stopping times, and to construct further examples, using techniques of progressive enlargements of filtrations.

Abstract. We propose a general framework for studying last passage times, suprema, and drawdowns of a large class of continuous-time stochastic processes. Our approach is based on processes of class Sigma and the more general concept of two processes, one of which moves only when the other is at the origin. After investigating certain transformations of such processes and their convergence properties, we provide three general representation results. The first allows the recovery of a process of class Sigma from its final value and the last time it visited the origin. In many situations this gives access to the distribution of the last time a stochastic process attains a certain level or is equal to its running maximum. It also leads to recently discovered formulas expressing option prices in terms of last passage times. Our second representation result is a stochastic integral representation that will allow us to price and hedge options on the running maximum of an underlying that are triggered when the underlying drops to a given level or, alternatively, when the drawdown or relative drawdown of the underlying attains a given height. The third representation gives conditional expectations of certain functionals of processes of class Sigma. It can be used to deduce the distributions of a variety of interesting random variables such as running maxima, drawdowns, and maximum drawdowns of suitably stopped processes.

Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analog of the Jacobi ensemble:with Reδ > −1/2. If e is a cyclic vector for a unitary n × n matrix U , the spectral measure of the pair (U , e) is well parameterized by its Verblunsky coefficients (α 0 , . . . , α n−1 ). We introduce here a deformation (γ 0 , . . . , γ n−1 ) of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product r(γ 0 ) · · · r(γ n−1 ) of elementary reflections parameterized by these coefficients. If γ 0 , . . . , γ n−1 are independent random variables with some remarkable distributions, then the eigenvalues of the GGT matrix follow the circular Jacobi distribution above.These deformed Verblunsky coefficients also allow us to prove that, in the regime δ = δ(n) with δ(n)/n → βd/2, the spectral measure and the empirical spectral distribution

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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 in [7], 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 in [7] 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.2000 Mathematics Subject Classification. 15A52, 60F05, 60F15.