Factorization Property of Generalized <i>s</i>-Selfdecomposable Measures and Class $L^f$ Distributions

Czyżewska-Jankowska, Agnieszka; Jurek, Zbigniew J.

doi:10.1137/s0040585x97985169

Theory Probab. Appl.

2011

DOI: 10.1137/s0040585x97985169

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Factorization Property of Generalized s-Selfdecomposable Measures and Class $L^f$ Distributions

Agnieszka Czyżewska-Jankowska¹,

Zbigniew J. Jurek²

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Cited by 3 publications

(2 citation statements)

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“…and hence from [14], Lemma 1, which can be verified directly by taking derivatives with respect to ω of…”

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confidence: 85%

Quantile clocks

James¹,

Zhang²

2011

Ann. Appl. Probab.

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Quantile clocks are defined as convolutions of subordinators $L$, with quantile functions of positive random variables. We show that quantile clocks can be chosen to be strictly increasing and continuous and discuss their practical modeling advantages as business activity times in models for asset prices. We show that the marginal distributions of a quantile clock, at each fixed time, equate with the marginal distribution of a single subordinator. Moreover, we show that there are many quantile clocks where one can specify $L$, such that their marginal distributions have a desired law in the class of generalized $s$-self decomposable distributions, and in particular the class of self-decomposable distributions. The development of these results involves elements of distribution theory for specific classes of infinitely divisible random variables and also decompositions of a gamma subordinator, that is of independent interest. As applications, we construct many price models that have continuous trajectories, exhibit volatility clustering and have marginal distributions that are equivalent to those of quite general exponential L\'{e}vy price models. In particular, we provide explicit details for continuous processes whose marginals equate with the popular VG, CGMY and NIG price models. We also show how to perfectly sample the marginal distributions of more general classes of convoluted subordinators when $L$ is in a sub-class of generalized gamma convolutions, which is relevant for pricing of European style options.Comment: Published in at http://dx.doi.org/10.1214/10-AAP752 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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“…and hence from [14], Lemma 1, which can be verified directly by taking derivatives with respect to ω of…”

mentioning

confidence: 85%

Quantile clocks

James¹,

Zhang²

2011

Ann. Appl. Probab.

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“…However, using the formal integration by parts we are able to define random integrals for larger class of integrands h and Lévy processes Y . Moreover, still having the crucial equality (4).…”

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confidence: 99%

Invariant measures under random integral mappings and marginal distributions of fractional Lévy processes

Jurek

2013

Statistics & Probability Letters

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It is shown that some convolution semigroups of infinitely divisible measures are invariant under the random integral mappings I h,r (a,b] defined in (⋆) below. The converse implication is specified for the semigroups of generalized s-selfdecomposable and selfdecomposable distributions. Some application are given to the moving average fractional Lévy process (MAFLP). (2000): Primary 60F05 , 60E07, 60B11; Secondary 60H05, 60B10. Mathematics Subject Classifications

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Remarks on compositions of some random integral mappings

Jurek

2018

Statistics & Probability Letters

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Factorization Property of Generalized s-Selfdecomposable Measures and Class $L^f$ Distributions

Cited by 3 publications

References 6 publications

Quantile clocks

Quantile clocks

Invariant measures under random integral mappings and marginal distributions of fractional Lévy processes

Remarks on compositions of some random integral mappings

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