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Stochastic complex integrals associated with homogeneous independently scattered random measures on the line

Yamamuro Kouji

Probability and Mathematical Statistics

2019

Vol. 39, Fasc. 1

219--236

Complex integrals associated with homogeneous independently scattered random measures on the line are discussed. Theorems corresponding to Cauchy’s theorem and the residue theorem are given. Furthermore, the converse of Cauchy’s theorem is discussed.

Several forms of stochastic integral representations of gamma random variables and related topics

Aoyama T., Maejima M., Ueda Y.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 1

99--118

Gamma distributions can be characterized as the laws of stochastic integrals with respect to many different Lévy processes with different nonrandom integrands. A Lévy process corresponds to an infinitely divisible distribution. Therefore, many infinitely divisible distributions can yield a gamma distribution through stochastic integral mappings with different integrands. In this paper, we pick up several integrands which have appeared in characterizing well-studied classes of infinitely divisible distributions, and find inverse images of a gamma distribution through each stochastic integral mapping. As a by-product of our approach to stochastic integral representations of gamma random variables, we find a remarkable new general characterization of classes of infinitely divisible distributions, which were already considered by James et al. (2008) and Aoyama et al. (2010) in some special cases.

Computing VaR and AVaR in infinitely divisible distributions

Kim Y. S., Rachev S. T., Bianchi M. L., Fabozzi F. J.

Probability and Mathematical Statistics

2010

Vol. 30, Fasc. 2

223--245

In this paper we derive closed-form solutions for the cumulative distribution function and the average value-at-risk for five subclasses of the infinitely divisible distributions: classical tempered stable distribution, Kim–Rachev distribution, modified tempered stable distribution, normal tempered stable distribution, and rapidly decreasing tempered stable distribution. We present empirical evidence using the daily performance of the S&P 500 for the period January 2, 1997 through December 29, 2006.

Monotonicity and non-monotonicity of domains of stochastic integral operators

Sato K.

Probability and Mathematical Statistics

2006

Vol. 26, Fasc. 1

23--39

A Lévy process on Rd with distribution }μ at time 1 is denoted by Xμ = {Xμt}, If the improper stochastic integral [formula] of f with respect to Xμ is definable, its distribution is denoted by Ф∫(μ). The class of all infinitely divisible distributions μ on Rd such that Ф∫(μ) is definable is denoted by D(Φ∫). The class D(Φ∫), its two extensions Dc(Φ∫) and Des(Φ∫) (compensated and essential), and its restriction D0(Φ∫)(absolutely definable) are studied. It is shown that Des(Φ∫) is monotonic with respect to ∫, which means that |f2| ≤ |f1| implies Des(Φ∫1) ⊂ Des(Φ∫2). Further D0Φf is monotonic with respect to ∫ but neither D(Φ∫) nor D0(Φ∫)is monotonic with respect to ∫. Furthermore, there exist μ, ∫1 and ∫2 such that 0 ≤∫2 ∫1, and μ∈D(Φ∫1), and μ ∉D(Φ∫2) An explicit example for this is related to some properties of a class of martingale Levy processes.