158 research outputs found
Approximation via regularization of the local time of semimartingales and Brownian motion
Through a regularization procedure, few approximation schemes of the local
time of a large class of one dimensional processes are given. We mainly
consider the local time of continuous semimartingales and reversible
diffusions, and the convergence holds in ucp sense. In the case of standard
Brownian motion, we have been able to determine a rate of convergence in ,
and a.s. convergence of some of our schemes.Comment: Accept\'e conditionnelement par Stochastic processes and their
application
Limiting laws associated with Brownian motion perturbed by its maximum, minmum and local time II
We obtain probability measures on the canonical space penalizing the Wiener
measure by a function of its maximum (resp. minimum, local time). We study the
law of the canonical process under these new probability measures
Limiting laws associated with Brownian motion perturbated by normalized exponential weights I
We determine the rate of decay of the expectation Z(t) of some multiplicative
functional related to Brownian motion up to time t. This permits to prove that
the Wiener measure, penalized by this multiplicative functional, converges as t
goes to infinity to a probability measure (p.m.) . We obtain the law of the
canonical process under this new p.m
On the excursion theory for linear diffusions
We present a number of important identities related to the excursion theory
of linear diffusions. In particular, excursions straddling an independent
exponential time are studied in detail. Letting the parameter of the
exponential time tend to zero it is seen that these results connect to the
corresponding results for excursions of stationary diffusions (in stationary
state). We characterize also the laws of the diffusion prior and posterior to
the last zero before the exponential time. It is proved using Krein's
representations that, e.g., the law of the length of the excursion straddling
an exponential time is infinitely divisible. As an illustration of the results
we discuss Ornstein-Uhlenbeck processes
On subexponentiality of the L\'evy measure of the diffusion inverse local time; with applications to penalizations
For a recurrent linear diffusion on we study the asymptotics of the
distribution of its local time at 0 as the time parameter tends to infinity.
Under the assumption that the L\'evy measure of the inverse local time is
subexponential this distribution behaves asymtotically as a multiple of the
L\'evy measure. Using spectral representations we find the exact value of the
multiple. For this we also need a result on the asymptotic behavior of the
convolution of a subexponential distribution and an arbitrary distribution on
The exact knowledge of the asymptotic behavior of the distribution of
the local time allows us to analyze the process derived via a penalization
procedure with the local time. This result generalizes the penalizations
obtained in Roynette, Vallois and Yor \cite{rvyV} for Bessel processes
Statistical and renewal results for the random sequential adsorption model applied to a unidirectional multicracking problem
AbstractWe work out a stationary process on the real line to represent the positions of the multiple cracks which are observed in some composites materials submitted to a fixed unidirectional stress ɛ. Our model is the one-dimensional random sequential adsorption. We calculate the intensity of the process and the distribution of the inter-crack distance in the Palm sense. Moreover, the successive crack positions of the one-sided process (denoted by Xiɛ, i⩾1) are described. We prove that the sequence {(Xiɛ,Yiɛ),1⩽i⩽n} is a “conditional renewal process”, where Yiɛ is the value of the stress at which Xiɛ forms. The approaches “in the Palm sense” and “one-sided process” merge when n→+∞. The saturation case (ɛ=+∞) is also investigated
Quelques approximations du temps local brownien
We give some approximations of the local time process at level of the real Brownian motion . We prove that
\frac{2}{\epsilon}\int_0^{t} X_{(u+\epsilon)\wedge t}^+ \indi_{\{X_u \leqslant
0\}} du + \frac{2}{\epsilon}\int_0^{t} X_{(u+\epsilon) \wedge t}^-
\indi_{\{X_u>0\}} du and \frac{4}{\epsilon}\int_0^{t} X_u^-
\indi_{\{X_{(u+\epsilon) \wedge t} > 0\}} du converge in the ucp sense to
, as . We show that \frac{1}{\epsilon}\int_0^t
(\indi_{\{x
goes to in as , and that the rate of
convergence is of order , for any .Comment: Soumis dans les Comptes rendus - Math\'ematiqu
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