We define a generalized Brownian bridge and we provide some information about its filtration. Two decompositions of this process as a semi-martingale are given. The first one is a Volterra decomposition and the second one is its canonical decomposition in…

We present a class of processes which enjoy an exponential analogue of Pitman's $2M-X$ theorem, improving hence some works of H. Matsumoto and M. Yor.

The product of subsequent partial sums of independent, identically distributed, square integrable, positive random variables is asymptotically lognormal. The result extends in a rather routine way to non-degenerate U-statistics.

The inequality of Vapnik and Chervonenkis controls the expectation of the function  by its sample average uniformly over a VC-major class of functions taking into account the size of the expectation. Using Talagrand's kernel method we prove a similar result…

The purpose of this note is to give a PDE satisfied by a call option when the price process is a semimartingale. The main result generalizes the PDE in the case when the stock price is a diffusion. Its proof…

We present a reordered version of the inclusion-exclusion principle, which is useful when computing the probability of a union of events which are close to independent. The advantages of this formulation are demonstrated in the context of 3 classic problems…

Let $X, X_1, X_2, \dots $ be a sequence of nondegenerate, independent and identically distributed random variables and set $S_n=X_1+\dots +X_n$,  $V_n^2=X_1^2+\dots +X_n^2$. We answer a question of Gotze, Gine and Mason by providing a simple necessary and sufficient condition…

We prove that random walks in i.i.d. random environments which oscillate in a given direction have velocity zero with respect to that direction. This complements existing results thus giving a general law of large numbers under the only assumption of…

Consider a martingale $M$ with bounded jumps and two sequences $a_n, b_n \to \infty$. We show that if the rescaled martingales $$ M^n_t =\frac{1}{\sqrt{a_n}}M_{b_n t}$$ converge weakly, then the limit is necessarily a continous Ocone martingale. Necessary and sufficient conditions…

We consider Markov chains on the space of (countable) partitions of the interval $[0,1]$, obtained first by size biased sampling twice (allowing repetitions) and then merging the parts with probability $\beta_m$ (if the sampled parts are distinct) or splitting the…