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1

Tanaka formula for strictly stable processes

Tsukada Hiroshi

Probability and Mathematical Statistics

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2019

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Vol. 39, Fasc. 1

39--60

EN

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).

2

Occupation time problem for multifractional Brownian motion

Ouahra Mohamed Ait, Guerbaz Raby, Ouahhabi Hanae, Sghir Aissa

Probability and Mathematical Statistics

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2019

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Vol. 39, Fasc. 1

99--113

EN

In this paper, by using a Fourier analytic approach, we investigate sample path properties of the fractional derivatives of multifractional Brownian motion local times. We also show that those additive functionals satisfy a property of local asymptotic self-similarity. As a consequence, we derive some local limit theorems for the occupation time of multifractional Brownian motion in the space of continuous functions.

3

Fractional derivatives of local times of stable Lévy processes as the limits of the occupation time problem in Besov space

Ouhara M. A., Eddahbi M., Ouali M.

Probability and Mathematical Statistics

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2004

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Vol. 24, Fasc. 2

263--279

EN

In this paper, we firstly study the Besov regularity of the local time of symmetric stable processes and of its fractional derivative. Secondly, we establish limit theorems for occupation times of α-symmetric stable processes with 1 < α ≤ 2 in some Besov spaces. Finally, we give the strong approximation version of our limit theorems.

4

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

Boufoussi B., Dozzi M.

Probability and Mathematical Statistics

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2003

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Vol. 23, Fasc. 2

369--387

EN

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.

5

Asymptotics of the supremum of scaled Brownian motion

Dębicki K.

Probability and Mathematical Statistics

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2001

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Vol. 21, Fasc. 1

199--212

EN

We consider the problem of estimating the tail of the distribution of the supremum of scaled Brownian motion B(ƒ(t)) processes with linear drift.Using the local time technique we obtain asymptotics and bounds of Pt≥t0(sup(B(ƒ(t))−t)> u), which are expressed in terms of the expected value of thelocal timeof B(ƒ(t))−tprocesses at levelu.As an application we obtain upper bounds for the tail of distribution of the supremum for some Gaussian processes with stationary increments.

6

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

Engelbert H. J.

Probability and Mathematical Statistics

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2000

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Vol. 20, Fasc. 2

343--358

EN

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.