Poisson processes directed by subordinators, stuttering poisson and pseudo-poisson processes, with applications to actuarial mathematics

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Poisson processes directed by subordinators, stuttering poisson and pseudo-poisson processes, with applications to actuarial mathematics. / Rusakov, O.; Yakubovich, Yu.

In: Journal of Physics: Conference Series, Vol. 2131, No. 2, 022107, 29.12.2021.

Research output: Contribution to journal › Conference article › peer-review

BibTeX

@article{26774c30d2034584b0f75d70fd54e72c,

title = "Poisson processes directed by subordinators, stuttering poisson and pseudo-poisson processes, with applications to actuarial mathematics",

abstract = "Weconsider a PSI-process, that is a sequence of random variables (ζi), i = 0.1, ..., which is subordinated by a continuous-time non-decreasing integer-valued process N(t): φ(t) = ζN(t). Our main example is when N(t) itself is obtained as a subordination of the standard Poisson process Π(s) by a non-decreasing L{\'e}vy process S(t): N(t) = Π(S(t)).The values (ζi)one interprets as random claims, while their accumulated intensity S(t) is itself random. We show that in this case the process N(t) is a compound Poisson process of the stuttering type and its rate depends just on the value of theLaplace exponent of S(t) at 1. Under the assumption that the driven sequence (ζi) consists of i.i.d. random variables with finite variance we calculate a correlation function of the constructed PSI-process. Finally, we show that properly rescaled sums of such processes converge to the Ornstein-Uhlenbeck process in the Skorokhod space. We suppose that the results stated in the paper mightbe interesting for theorists and practitioners in insurance, in particular, for solution of reinsurance tasks.",

author = "O. Rusakov and Yu Yakubovich",

note = "Publisher Copyright: {\textcopyright} 2021 Institute of Physics Publishing. All rights reserved.; Intelligent Information Technology and Mathematical Modeling 2021, IITMM 2021 ; Conference date: 31-05-2021 Through 06-06-2021",

year = "2021",

month = dec,

day = "29",

doi = "10.1088/1742-6596/2131/2/022107",

language = "English",

volume = "2131",

journal = "Journal of Physics: Conference Series",

issn = "1742-6588",

publisher = "IOP Publishing Ltd.",

number = "2",

}

RIS

TY - JOUR

T1 - Poisson processes directed by subordinators, stuttering poisson and pseudo-poisson processes, with applications to actuarial mathematics

AU - Rusakov, O.

AU - Yakubovich, Yu

PY - 2021/12/29

Y1 - 2021/12/29

N2 - Weconsider a PSI-process, that is a sequence of random variables (ζi), i = 0.1, ..., which is subordinated by a continuous-time non-decreasing integer-valued process N(t): φ(t) = ζN(t). Our main example is when N(t) itself is obtained as a subordination of the standard Poisson process Π(s) by a non-decreasing Lévy process S(t): N(t) = Π(S(t)).The values (ζi)one interprets as random claims, while their accumulated intensity S(t) is itself random. We show that in this case the process N(t) is a compound Poisson process of the stuttering type and its rate depends just on the value of theLaplace exponent of S(t) at 1. Under the assumption that the driven sequence (ζi) consists of i.i.d. random variables with finite variance we calculate a correlation function of the constructed PSI-process. Finally, we show that properly rescaled sums of such processes converge to the Ornstein-Uhlenbeck process in the Skorokhod space. We suppose that the results stated in the paper mightbe interesting for theorists and practitioners in insurance, in particular, for solution of reinsurance tasks.

AB - Weconsider a PSI-process, that is a sequence of random variables (ζi), i = 0.1, ..., which is subordinated by a continuous-time non-decreasing integer-valued process N(t): φ(t) = ζN(t). Our main example is when N(t) itself is obtained as a subordination of the standard Poisson process Π(s) by a non-decreasing Lévy process S(t): N(t) = Π(S(t)).The values (ζi)one interprets as random claims, while their accumulated intensity S(t) is itself random. We show that in this case the process N(t) is a compound Poisson process of the stuttering type and its rate depends just on the value of theLaplace exponent of S(t) at 1. Under the assumption that the driven sequence (ζi) consists of i.i.d. random variables with finite variance we calculate a correlation function of the constructed PSI-process. Finally, we show that properly rescaled sums of such processes converge to the Ornstein-Uhlenbeck process in the Skorokhod space. We suppose that the results stated in the paper mightbe interesting for theorists and practitioners in insurance, in particular, for solution of reinsurance tasks.

UR - http://www.scopus.com/inward/record.url?scp=85123598858&partnerID=8YFLogxK

U2 - 10.1088/1742-6596/2131/2/022107

DO - 10.1088/1742-6596/2131/2/022107

M3 - Conference article

AN - SCOPUS:85123598858

VL - 2131

JO - Journal of Physics: Conference Series

JF - Journal of Physics: Conference Series

SN - 1742-6588

IS - 2

M1 - 022107

T2 - Intelligent Information Technology and Mathematical Modeling 2021, IITMM 2021

Y2 - 31 May 2021 through 6 June 2021

ER -

ID: 92425814