442 research outputs found
Exponential functionals of Levy processes
This text surveys properties and applications of the exponential functional
of real-valued L\'evy processes .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 and we obtain a Central
Limit Theorem for a sequence of such Brownian motions. We also generalize the
results to the case of the -dimensional Ornstein-Uhlenbeck processes
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
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 a Brownian motion and its
first hitting time of the level one, this remarkable law allows us to
understand some properties of the process
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
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 , 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 and its
supremum . A complete characterization of -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
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
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