Asymptotic Properties of Extremal Markov Processes Driven by Kendall Convolution

Arendarczyk, Marek; Jasiulis-Gołdyn, Barbara; Omey, Edward

doi:10.1007/s10959-023-01285-2

J Theor Probab

2023

DOI: 10.1007/s10959-023-01285-2

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Asymptotic Properties of Extremal Markov Processes Driven by Kendall Convolution

Marek Arendarczyk,

Barbara Jasiulis-Gołdyn,

Edward Omey

Abstract: This paper is devoted to the analysis of the finite-dimensional distributions and asymptotic behavior of extremal Markov processes connected with the Kendall convolution. In particular, we provide general formulas for the finite dimensional distributions of the random walk driven by the Kendall convolution for a large class of step size distributions. Moreover, we prove limit theorems for random walks and associated continuous-time stochastic processes.

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Cited by 2 publications

References 32 publications

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How Exceptional is the Extremal Kendall and Kendall-Type Convolution

Jasiulis-Gołdyn,

Misiewicz,

Omey

et al. 2023

Results Math

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This paper deals with the generalized convolutions connected with the Williamson transform and the maximum operation. We focus on such convolutions which can define transition probabilities of renewal processes. They should be monotonic since the described time or destruction does not go back, it should admit existence of a distribution with a lack of memory property because the analog of the Poisson process shall exist. Another valuable property is the simplicity of calculating and inverting the corresponding generalized characteristic function (in particular Williamson transform) so that the technique of generalized characteristic function can be used in description of our processes. The convex linear combination property (the generalized convolution of two point measures is the convex combination of several fixed measures), or representability (which means that the generalized convolution can be easily written in the language of independent random variables)—they also facilitate the modeling of real processes in that language. We describe examples of generalized convolutions having the required properties ranging from the maximum convolution and its simplest generalization—the Kendall convolution (associated with the Williamson transform), up to the most complicated here—Kingman convolution. It is novel approach to apply in the extreme value theory. Stochastic representation of the Kucharczak-Urbanik in the order statistics terms is proved, which open new paths to investigate Archimedean copulas. This paper open the door to solve an old open problem of the relationship between copulas and generalized convolutions mentioned by B. Schweizer and A. Sklar in 1983. This indicates the path of further research towards extremes and dependency modelling.

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How Exceptional is the Extremal Kendall and Kendall-Type Convolution

Jasiulis-Gołdyn,

Misiewicz,

Omey

et al. 2023

Results Math

Self Cite

View full text Add to dashboard Cite

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Cramér–Lundberg model for some classes of extremal Markov sequences

Jasiulis-Gołdyn,

Lechańska,

KrystynaMisiewicz

2023

Lith Math J

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The classical Cramér–Lundberg model was the first attempt to describe the financial condition of the insurance company. The incomes were approximated by a steady stream of money, and insurance payments were not limited and could take any value from zero to infinity. The society did not invest any part of its money and does not have any employees, shareholders, or enterprise maintenance costs. There exist many modifications of the Cramér–Lundberg model that cover at least some of the problems described here but usually require insight into the internal financial policy of the insurance company. We propose another modification based on Markov processes defined by generalized convolutions. Thanks to the generalized convolutions, we can stochastically approximate the internal financial policy of the company based on publicly available data. In this paper, we focus on computing the ruin probability in the Cramér–Lundberg model for an infinite time horizon for the Markov processes where the transition probabilities are defined by generalized convolutions, in particular, by the α-convolution, maximal convolution, or Kendall convolution.

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Asymptotic Properties of Extremal Markov Processes Driven by Kendall Convolution

Cited by 2 publications

References 32 publications

How Exceptional is the Extremal Kendall and Kendall-Type Convolution

How Exceptional is the Extremal Kendall and Kendall-Type Convolution

Cramér–Lundberg model for some classes of extremal Markov sequences

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