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Tanaka formula for strictly stable processes

100%

Tsukada H.

2019

tom Vol. 39, Fasc. 1

39--60

For symmetric Lévy processes, if the local times exist, the Tanaka formula has already been constructed via the techniques in the potential theory by Salminen and Yor (2007). In this paper, we study the Tanaka formula for arbitrary strictly stable processes with index α ϵ (1, 2), including spectrally positive and negative cases in a framework of Itô’s stochastic calculus. Our approach to the existence of local times for such processes is different from that of Bertoin (1996).

Existence and non-existence of solutions of one-dimensional stochastic equations

100%

Engelbert H. J.

tom Vol. 20, Fasc. 2

343--358

We consider the one-dimensional stochastic equation [formula] for a continuous local martingale M with square variation [M] and measurable drift and diffusion coefficients b and σ. The main purpose of this paper is to derive a necessary condition for the existence of a solution X starting from x0. As a result, we construct a diffusion coefficient σ such that the above stochastic equation has no solution X whatever the initial value x0 and the non-zero, say, continuous drift coefficient b might be.

Multifractal properties of the sets of zeroes of Brownian paths

88%

Dolgopyat D. , Sidorov V.

nr 2

157-171

We study Brownian zeroes in the neighborhood of which one can observe a non-typical growth rate of Brownian excursions. We interpret the multifractal curve for the Brownian zeroes calculated in [6] as the Hausdorff dimension of certain sets. This provides an example of the multifractal analysis of a statistically self-similar random fractal when both the spacing and the size of the corresponding nested sets are random.

On the local time of multiparameter symmmetric stable processes. Regularity and limit theorem in Besov spaces

75%

Boufoussi B. , Dozzi M.

Probability and Mathematical Statistics

2003

tom Vol. 23, Fasc. 2

369--387

Let X = (Xz, z ϵ TN = [0, l]N) be a symmetric α-stable process, 1 < α ≤ 2. Based on a Kolmogorov type continuity theorem we show Hölder conditions in Lp-norms for the local time of X with respect to the space and time variables, by distinguishing the cases where the time variables do or do not meet the axes. Weak convergence of the occupation integral is proved.