Exponential functionals of Levy processes

This text surveys properties and applications of the exponential functional $\int_0^t\exp(-\xi_s)ds$ of real-valued L\'evy processes $\xi=(\xi_t,t\geq0)$.Comment: Published at http://dx.doi.org/10.1214/154957805100000122 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

A Central Limit Theorem for a sequence of Brownian motions in the unit sphere in Rn

We use a Stochastic Differential Equation satisfied by Brownian motion taking values in the unit sphere $S_{n-1}subsetmathbb{R}^{n}$ and we obtain a Central Limit Theorem for a sequence of such Brownian motions. We also generalize the results to the case of the $n$-dimensional Ornstein-Uhlenbeck processes

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

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

On local martingale and its supremum: harmonic functions and beyond

We discuss certain facts involving a continuous local martingale $N$ and its supremum $\bar{N}$. A complete characterization of $(N,\bar{N})$-harmonic functions is proposed. This yields an important family of martingales, the usefulness of which is demonstrated, by means of examples involving the Skorokhod embedding problem, bounds on the law of the supremum, or the local time at 0, of a martingale with a fixed terminal distribution, or yet in some Brownian penalization problems. In particular we obtain new bounds on the law of the local time at 0, which involve the excess wealth order

A note on a.s. finiteness of perpetual integral functionals of diffusions

In this note, with the help of the boundary classification of diffusions, we derive a criterion of the convergence of perpetual integral functionals of transient real-valued diffusions. In the particular case of transient Bessel processes, we note that this criterion agrees with the one obtained via Jeulin's convergence lemma

Burkholder's submartingales from a stochastic calculus perspective

We provide a simple proof, as well as several generalizations, of a recent result by Davis and Suh, characterizing a class of continuous submartingales and supermartingales that can be expressed in terms of a squared Brownian motion and of some appropriate powers of its maximum. Our techniques involve elementary stochastic calculus, as well as the Doob-Meyer decomposition of continuous submartingales. These results can be used to obtain an explicit expression of the constants appearing in the Burkholder-Davis-Gundy inequalities. A connection with some balayage formulae is also established.Comment: 7 page