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 $L^2$, 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 $\R_+$ 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 $\R_+.$ 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

Quelques approximations du temps local brownien

We give some approximations of the local time process $(L_t^x)_{t\geqslant 0}$ at level $x$ of the real Brownian motion $(X_t)$. 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 $L_t^0$, as $\epsilon \to 0$. We show that \frac{1}{\epsilon}\int_0^t (\indi_{\{x goes to $L_t^x$ in $L^2(\Omega)$ as $\epsilon \to 0$, and that the rate of convergence is of order $\epsilon^\alpha$, for any $\alpha < {1/4}$.Comment: Soumis dans les Comptes rendus - Math\'ematiqu

Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index H>=1/4

Given a locally bounded real function g, we examine the existence of a 4-covariation $[g(B^H), B^H, B^H, B^H]$, where $B^H$ is a fractional Brownian motion with a Hurst index $H \ge \tfrac{1}{4}$. We provide two essential applications. First, we relate the 4-covariation to one expression involving the derivative of local time, in the case $H = \tfrac{1}{4}$, generalizing an identity of Bouleau--Yor type, well known for the classical Brownian motion. A second application is an Itô formula of Stratonovich type for $f(B^H)$. The main difficulty comes from the fact $B^H$ has only a finite 4-variation