Factorization property of generalized $s$-selfdecomposable measures and class $L^F$ distributions

Czyżewska-Jankowska, Agnieszka; Jurek, Zbigniew J.

doi:10.4213/tvp4286

Cited by 4 publications

(4 citation statements)

References 7 publications

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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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“…where < •, • > denotes the scalar product and the triple: a vector z ∈ (5) and the corresponding random integral mapping…”

Section: Proofsmentioning

confidence: 99%

“…[Term: random integral emphasizes that h in ( 5) is a deterministic function ( not a stochastic process).] (b) Integrals over intervals (a,b) or (a,∞) or [a,b] and others are defined as weak limits of integrals over intervals (a,b] in (5). Thus, the random integral (a,∞) h(t)dY ν (r(t)) is well-defined if and only if the function…”

Section: Proofsmentioning

confidence: 99%

Remarks on compositions of some random integral mappings

Jurek

2018

Statistics & Probability Letters

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“…(3) cf. Jurek (1985), Theorem 4.5 , Corollary 4.6 andIksanov, Jurek andSchreiber (2004), Theorem 1, Proposition 3; more on the class L f is in Czyżewska and Jurek (2011).…”

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confidence: 98%

On background driving distribution functions (BDDF) for some selfdecomposable variables

Jurek¹

2021

Preprint

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Many classical variables (statistics) are selfdecomposable. They admit the random integral representations via Lévy processes. In this note are given formulas for their background driving distribution functions (BDDF). This may be used for a simulation of those variables. Among the examples discussed are: gamma variables, hyperbolic characteristic functions, Student t-distributions, stochastic area under planar Brownian motions, inverse Gaussian variable, logistic distributions, non-central chi-square, Bessel densities and Fisher z-distributions. Found representations might be of use in statistical applications.

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Factorization property of generalized $s$-selfdecomposable measures and class $L^F$ distributions

Cited by 4 publications

References 7 publications

Quantile clocks

Quantile clocks

Remarks on compositions of some random integral mappings

On background driving distribution functions (BDDF) for some selfdecomposable variables

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