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We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) with $alphain(0,2]$, to include both linear multifractional Brownian sheets ($alpha=2$) and linear multifractional stable sheets ($alpha<2$). The purpose of the present paper is to study the existence and joint continuity of the local times of LMSS. The main results are Theorems 2.9 and 2.11, which provide a sufficient and necessary condition for the existence of local times and a weaker sufficient condition for the joint continuity of local times of LMSS, respectively. We also prove a local Holder condition for the local time in the set variable in Theorem 3.1. All these theorems improve significantly the existing results for the local times of multifractional Brownian sheets and linear fractional stable sheets in the literature.
The L2-approximation of occupation and local times of a symmetric $alpha$-stable L{e}vy process from high frequency discrete time observations is studied. The standard Riemann sum estimators are shown to be asymptotically efficient when 0 < $alpha$ $
We propose a new multifractional stochastic process which allows for self-exciting behavior, similar to what can be seen for example in earthquakes and other self-organizing phenomena. The process can be seen as an extension of a multifractional Brow
We consider a family of Bessel Processes that depend on the starting point $x$ and dimension $delta$, but are driven by the same Brownian motion. Our main result is that almost surely the first time a process hits $0$ is jointly continuous in $x$ and
In this paper we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes. We also show that a fractional Brownian motion and the related Riemann-Liouville proce
We consider the linear stochastic wave equation driven by a Gaussian noise. We show that the solution satisfies a certain form of strong local nondeterminism and we use this property to derive the exact uniform modulus of continuity for the solution.