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1

Probability on matrix-cone hypergroups: limit theorems and structural properties

Hazod W.

Journal of Applied Analysis

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2009

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Vol. 15, nr 2

205-245

EN

Recent investigations of M. Rösler [13] and M. Voit [17] provide examples of hypergroups with properties similar to the group-or vector space case and with a sufficiently rich structure of automorphisms, providing thus tools to investigate the limit theory of normalized random walks and the structure of the corresponding limit, laws. The investigations are parallel to corresponding investigations for vector spaces and simply connected nilpotent Lie groups.

2

On a particular class of self-decomposable random variables : the durations of Bessel excursions straddling independent exponential times

Bertoin J., Fujita T., Roynette B., Yor M.

Probability and Mathematical Statistics

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2006

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Vol. 26, Fasc. 2

315--366

EN

The distributional properties of the duration of a recurrent Bessel process straddling an independent exponential time are studied in detail. Although our study may be considered as a particular case of Winkel’s in [25], the infinite divisibility structure of these Bessel durations is particularly rich and we develop algebraic properties for a family of random variables arising from the Lévy measures of these durations.

3

Lévy processes and self-decomposability in finance

Bingham N. H.

Probability and Mathematical Statistics

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2006

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Vol. 26, Fasc. 2

367--378

EN

The main theme of Urbanik’s work was infinite divisibility and its ramifications. The aim of this memorial article is to trace the application of this theme in mathematical finance, one of the main growth areas in contemporary probability theory. We begin in Section 1 with a discussion of the nature of prices. In particular, we focus on whether (or when) prices may be taken as continuous, with a view to using Lévy processes to model the case of prices with jumps. We turn in Section 2 to asset return distributions; prime candidates for modelling here include the normal, hyperbolic and Student t cases. In Section 3, we turn to distributions of type G, in particular, those in which the mixing law is not only infinitely divisible but also self-decomposable (i.e. in the class SD), which includes all three cases above. Then in Section 4 we turn to the dynamic counterpart of this, in which the law of class SD occurs as the limit law of a stochastic process of Ornstein-Uhlenbeck type, with Lévy driving noise. Finally, in Section 5 we discuss stochastic volatility models.

4

Semistable selfdecomposable laws on groups

Hazod W., Shah R.

Journal of Applied Analysis

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2001

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Vol. 7, nr 1

1-22

EN

The most prominent examples of (operator-) selfdecomposable laws on vector spaces are (operator-) stable laws. In the past (operator-) semistability — a natural generalisation — had been intensively investigated, hence the description of the intersection of the classes of semistable and selfdecomposable laws turned out to be a challenging problem, which was finally solved by A. Łuczak's investigations [17]. For probabilities on groups, in particular on simply connected nilpotent Lie groups there exists meanwhile a satisfying theory of decomposability and semistability. Consequently it is possible to obtain a description of the intersection of these classes of measures — under additional commutativity assumptions — leading finally to partial extensions of the above-mentioned results for vector spaces to the group case.