Privat, Meziane

In this short article, we will focus on the different links between some stochastic processes resulting from Brownian motion and two notions of probability theory (proportional increments and last hitting times).

Schulmann, Viktor

Die Dissertation beschäftigt sich mit dem folgenden Problem: Sei X ein bekannter stochastischer Prozess und T eine unbekannte von X unabhängige Stoppzeit. Das Ziel ist es, auf Grundlage einer Stichprobe von X zur Zeit T die Verteilung von T zurückzugewinnen. Insbesondere sollen nichtparametrische Schätzer für die Dichte f von T konstruiert werden. ...

Salminen, Paavo Vostrikova, Lioudmila

Let X = (X t) t≥0 be a real-valued additive process, i.e., a process with independent increments. In this paper we study the exponential integral functionals of X, namely, the functionals of the form I s,t = t s exp(−X u)du, 0 ≤ s

Zubkov, Andrey M. Savelov, Maksim P.
Published in
Discrete Mathematics and Applications

It is shown that, with suitable time change, the finite-dimensional distributions of the process formed by successive values of the Pearson statistics for an expanding sample converge to finite-dimensional distributions of the stationary random process, namely, the normalized square of the Bessel process. The results obtained earlier on the limit j...

Siorpaes, P

In this paper, after generalizing the pathwise Burkholder–Davis–Gundy (BDG) inequalities from discrete time to cadlag semimartingales, we present several applications of the pathwise inequalities. In particular we show that they allow to extend the classical BDG inequalities 1. to the Bessel process of order α ≥ 1 2. to the case of a random exponen...

Miclo, Laurent

A necessary and sufficient condition is obtained for the existence of strong stationary times for ergodic one-dimensional diffusions, whatever the initial distribution. The strong stationary times are constructed through intertwinings with dual processes, in the Diaconis-Fill sense, taking values in the set of segments of the extended line $\mathbb...

Lagnoux, Agnès Mercier, Sabine Vallois, Pierre

Probability that the maximum of the reflected Brownian motion over a finite interval [0, t] is achieved by its last zero before t Abstract We calculate the probability pc that the maximum of a reflected Brownian motion U is achieved on a complete excursion, i.e. pc := P U (t) = U * (t) where U (t) (respectively U * (t)) is the maximum of the proces...

Hu, Yueyun Shi, Zhan Yor, Marc

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.

Elie, Romuald Rosenbaum, Mathieu Yor, Marc

Let B be a Brownian motion and T1 its first hitting time of the level 1. For U a uniform random variable independent of B, we study in depth the distribution of B UT1/√T1, that is the rescaled Brownian motion sampled at uniform time. In particular, we show that this variable is centered.

Rosenbaum, Mathieu Yor, Marc

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.