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1
Content available remote Lévy processes, generalized moments and uniform integrability
Berger David, Kühn Franziska, Schilling René L
Probability and Mathematical Statistics
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2022
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Vol. 42, Fasc. 1
109--131
EN
We give new proofs of certain equivalent conditions for the existence of generalized moments of a Lévy process (Xt)t≥0; in particular, the existence of a generalized g-moment is equivalent to the uniform integrability of (g(Xt))t Є [0,1]. As a consequence, certain functions of a Lévy process which are integrable and local martingales are already true martingales. Our methods extend to moments of stochastically continuous additive processes, and we give new, short proofs for the characterization of lattice distributions and the transience of Leâvy processes.
2
Content available remote Ruin Probabilities for Two Collaborating Insurance Companies
Michna Zbigniew
Probability and Mathematical Statistics
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2020
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Vol. 40, Fasc. 2
360--386
EN
We find a formula for the supremum distribution of spectrally positive or negative Lévy processes with a broken linear drift. This gives formulas for ruin probabilities if two insurance companies (or two branches of the same company) divide between them both claims and premia in some specified proportions or if the premium rate for a given insurance portfolio is changed at a certain time. As an example we consider a gamma Lévy process, an -stable Lévy process and Brownian motion. Moreover we obtain identities for the Laplace transform of the distribution of the supremum of Lévy processes with a randomly broken drift (random time of the premium rate change) and on random intervals (random time when the insurance portfolio is closed).
3
Content available remote Stochastic complex integrals associated with homogeneous independently scattered random measures on the line
Yamamuro Kouji
Probability and Mathematical Statistics
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2019
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Vol. 39, Fasc. 1
219--236
EN
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.
4
Content available remote Series representation of time-stable stochastic processes
Kopp C., Molchanov I.
Probability and Mathematical Statistics
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2018
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Vol. 38, Fasc. 2
299--315
EN
A stochastically continuous process ξ(t), t ≥ 0, is said to be time-stable if the sum of n i.i.d. copies of ξ equals in distribution the time-scaled stochastic process ξ(nt), t ≥ 0. The paper advances the understanding of time-stable processes by means of their LePage series representations as the sum of i.i.d. processes with the arguments scaled by the sequence of successive points of the unit intensity Poisson process on [0,∞). These series yield numerous examples of stochastic processes that share one-dimensional distributions with a Lévy process.
5
Content available remote Fractional negative binomial and Pólya processes
Vellaisamy P., Maheshwari A.
Probability and Mathematical Statistics
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2018
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Vol. 38, Fasc. 1
77--101
EN
In this paper, we define a fractional negative binomial process (FNBP) by replacing the Poisson process by a fractional Poisson process (FPP) in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process (SFPP) is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.
6
Content available remote On extremal index of max-stable stationary processes
Dębicki K., Hashorva E.
Probability and Mathematical Statistics
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2017
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Vol. 37, Fasc. 2
299--317
EN
In this contribution we discuss the relation between Pickands-type constants defined for certain Brown-Resnick stationary proces W(t), t ϵ R, as [wzór] (set 0Z = R if δ = 0) and the extremal index of the associated max-stable stationary process ξW. We derive several new formulas and obtain lower bounds for ΉδW if W is a Gaussian or a Lévy process. As a by-product we show an interesting relation between Pickands constants and lower tail probabilities for fractional Brownian motions.
7
Content available remote Extremal particles in branching processes
Meller R.
Mathematica Applicanda
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2017
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Vol. 45, No. 1
3--20
EN
The purpose of this study is to investigate the behavior of extremal particles in a spatial branching process on R with the heavy-tailed compound Poisson process motion and inhomogeneous potential.
PL
W pracy rozważane są dwa modele procesów gałązkowych, pierwszy bazujący na procesie Galtona-Watsona, drugi na złożonym procesie Poissona. Oba procesy posiadają nieograniczony potencjał rozmnażania. Badana jest najwyższa cząstka w modelu poprzez określenie jej asymptotyki prawie na pewno z dokładnością do stałej. Oszacowanie z dołu polega na odgadnięciu prawie optymalnej strategii poruszania się cząstek. Oszacowanie z góry polega na badaniu innych modeli, prostszych w analizie, które stochastycznie szybciej się przemieszczają.
8
Content available remote GI/GI/1 queues with infinite means of service time and interarrival time
Szczotka W.
Probability and Mathematical Statistics
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2016
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Vol. 36, Fasc. 2
279--293
EN
The main results deal with the GI/GI/1 queues with Infinite means of the service times and interarrival times. Theorem 3.1 gives an asymptotic, in a heavy traffic situation, of the sequence of waiting times of the consecutive customers. Theorem 4.1 gives an asymptotic of stationary waiting times in a heavy traffic situation. In a special case, the asymptotic stationary waiting times have an exponential distribution (Corollary 4.1).
9
Content available remote J1 convergence of partial sum processes with a reduced number of jumps
Krizmanić D.
Probability and Mathematical Statistics
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2015
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Vol. 35, Fasc. 1
107--128
EN
Various functional limit theorems for partial sum processes of strictly stationary sequences of regularly varying random variables in the space of càdlàg functions D[0, 1] with one of the Skorokhod topologies have already been obtained. The mostly used Skorokhod J1 topology is inappropriate when clustering of large values of the partial sum processes occurs. When all extremes within each cluster of high-threshold excesses do not have the same sign, Skorokhod M1 topology also becomes inappropriate. In this paper we alter the definition of the partial sum process in order to shrink all extremes within each cluster to a single one, which allows us to obtain the functional J1 convergence. We also show that this result can be applied to some standard time series models, including the GARCH(1, 1) process and its squares, the stochastic volatility models and m-dependent sequences.
10
Content available remote Coalescing stochastic processes in retrival from semantic memory
Queau P-Y., Woyczyński W. A., Lerner A. J.
Mathematica Applicanda
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2015
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Vol. 43, No. 2
189--224
EN
Semantic memory retrieval is one of the most fundamental cognitive functions in humans. It is not fully understood and researchers from various fields of science struggle to find a model that would correlate well with experimental results and help understanding the complex background processes involved. To study such a phenomenon we need a relevant experimental protocol which can isolate the basic cognitive functions of interest from other perturbations. A variety of existing medical tests can provide such information, and the one we analyze is the Category Fluency Test (CFT). It was originally designed to measure frontal brain lobe damages in injured patients, and it tests directly the semantic memory retrieval, which is affected in cases of injury but can be also influenced by dementia, Alzheimer syndrome, or just aging. This paper introduces a new paradigm in analysis of the temporal structure of CFT responses by utilizing coalescent stochastic process model. We believe that this particular model is relevant to how this cognitive function operates and can lead to a better understanding of the background processes. The method turns out to be better at separating the two cognitively different groups studied here than the Weibull model from our previous paper Meyer et al.(2012), and could potentially be used for early diagnostics of dementia or Alzheimer's disease. Two other models, one based on the concept of Levy processes, and one related to the fractional Poisson model, are also explored.
PL
Praca proponuje model procesów koalescencyjnych w celu wyjaśnienia mechanizmów odzysku pojęć i nazw z pamięci semantycznej. Model jest pretestowany używając dobrze znanego eksperymentalnego Testu Biegłości Kategorycznej, który jest standardowym narzędziem neurologów badających pacjentów z objawami demencji. Możliwości modelowania opartego na procesach Lévy’ego i ułamkowych procesach Poissona są również zbadane.
11
Content available A fuzzy approach to option pricing in a Levy process setting
Nowak P., Romaniuk M.
International Journal of Applied Mathematics and Computer Science
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2013
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Vol. 23, no. 3
613--622
EN
In this paper the problem of European option valuation in a Levy process setting is analysed. In our model the underlying asset follows a geometric Levy process. The jump part of the log-price process, which is a linear combination of Poisson processes, describes upward and downward jumps in price. The proposed pricing method is based on stochastic analysis and the theory of fuzzy sets.We assume that some parameters of the financial instrument cannot be precisely described and therefore they are introduced to the model as fuzzy numbers. Application of fuzzy arithmetic enables us to consider various sources of uncertainty, not only the stochastic one. To obtain the European call option pricing formula we use the minimal entropy martingale measure and Levy characteristics.
12
Content available Response of linear system to α-stable Levy input
Walczak J., Mazurkiewicz S.
Computer Applications in Electrical Engineering
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2012
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Vol. 10
50--60
EN
This article suggests the method of analysis stochastic processes in deterministic linear SISO system of first order. Using the definition of α-stable random variable, the definition of α-stable Levy process has been introduced and presented their properties. Transformation of α-stable white noise process in the system has been investigated as well. Obtained results has been illustrated by an example.
13
Content available A note on several meteorological topics related to polar regions
Sienicki K.
Problemy Klimatologii Polarnej
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2011
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nr 21
39-76
EN
Analysis of the meteorology of Polar Regions is fundamental to the process of understanding the global climatology of the Earth and Earth-like planets. The nature of air circulation in a polar vortex is of preliminary importance. I have show the local and continental spatiotemporal relationship between near surface wind events in terms of self-organized criticality. In particular, the wind event size, wind event duration, and duration of quiescent wind event are well approximated by power-law distributions. On a continental scale, the wind events in the Antarctic tend to be self-organized criticality with ergodic properties. A similar self-organized criticality wind event was also found in Taylor Valley located at McMurdo Dry Valleys discovered by Captain Scott.s expedition. Captain Scott.s meteorological Terra Nova record was also examined. I have also revisited and re-analyzed wind events in Hornsund at Spitsbergen Island, in terms of marginal probabilities and marginal copulas which describe positive Lévy process.
PL
Badanie meteorologii regionów polarnych jest fundamentalne w procesie zrozumienia i opisania globalnej klimatologii Ziemi i jej podobnych planet. Natura cyrkulacji powietrza w Wirze Polarnym jest podstawowej wagi. Wykorzystując dane dotyczące prędkości wiatru na Antarktydzie z około 40 stacji meteorologicznych pokazano, że zdarzenia wiatrowe rozumiane jako wielkość siły wiatru, czasu wiania i czasu niewiania są samo-zorganizowaną krytycznością opisaną przez rozkład prawdopodo-bieństwa w postaci skalującej funkcji potęgowej, Pokazano, że zdarzenia wiatrowe na Antarktydzie tworzą kontynentalny system, który z punktu widzenia mechaniki statystycznej jest ergo-dyczny. Ta podstawowa charakterystyka wiatrów powinna leżeć w konstrukcji ułamkowego równania Fokkera-Plancka dla rozkładu gęstości prawdopodobieństwa. W artykule pokazano, że w związku z samo-zorganizowaną krytycznością wiatrów na Antarktydzie nie można obliczyć ich wartości średnich. Analiza katabatycznych wiatrów w Suchych Dolinach w rejonie McMurdo jest szczegółowo opisana w powiązaniu z ich orografią. Wykorzystując metodę sieci neuronowych oraz statystyki zdarzeń wiatrowych na Lodowej Barierze Rossa pokazano, że dwa ekstremalne zdarzenia pogodowe (czarne łabędzie): bardzo niska temperatura w okresie 27 luty – 19 marca 1912 oraz katabatyczny sztorm 21-29 marca 1912 roku, opisane przez Kapitana Scotta nie wydarzyły się. Pokazano, że samo-zorganizowana krytyczność wiatrów nie jest charakterystyczna tylko dla Antarktydy ale jest obserwowana również w innych obszarach polarnych tworzących lodowe plateau jak na przykład w Hornsundzie na Spitsbergenie. Wykorzystując twierdzenia dotyczące procesów o nieskończonej wariancji (proces Lévy.ego) pokazano, że uprzednio zaproponowany opis pogody przez jej podział na klasy pogodowe ma fundamentalne uzasadnienie matematyczne w postaci marginałów i kopuł tworzących skalującą funkcję potęgową nieposiadającą wartości średniej. Bazując na pomiarach meteorologicznych w Hornsundzie zilustrowano, że pogoda opisana przez kwartety (4 elementy): temperatura, wiatr, zachmurzenie i opady tworzy funkcję potęgową charakterystyczną dla samo-zorganizowanych krytycznie układów. Przedstawione wyniki udowodniają, że dotychczasowe paradygmaty meteorologiczne powinny być zrewidowane. W szczególności, powszechnie bez uzasadnienia przyjęte założenie o skończenie wymiarowym gaussowskim rozkładzie zmiennych meteorologicznych powinno być odrzucone i zastąpione przez stochastyczne alfa-stabilne procesy Lévy.ego dla których wariancja może przyjmować dowolną wartość w przedziale od zera do nieskończoności.
14
Content available remote Several forms of stochastic integral representations of gamma random variables and related topics
Aoyama T., Maejima M., Ueda Y.
Probability and Mathematical Statistics
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2011
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Vol. 31, Fasc. 1
99--118
EN
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.
15
Content available remote Chain dependent continuous time random walk
Szczotka W., Żebrowski P.
Probability and Mathematical Statistics
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2011
|
Vol. 31, Fasc. 2
239--261
EN
An asymptotic behavior of a continuous time random walk is investigated in the case when the sequence of pairs of jump vectors and times between jumps is chain dependent.
16
Content available remote On relations between Urbanik and Mehler semigroups
Jurek Z. J.
Probability and Mathematical Statistics
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2009
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Vol. 29, Fasc. 2
297--308
EN
It is shown that operator-selfdecomposable measures or, more precisely, their Urbanik decomposability semigroups induce generalized Mehler semigroups of bounded linear operators. Moreover, those semigroups can be represented as random integrals of operator valued functions with respect to stochastic Lévy processes. Our Banach space setting is in contrast with the Hilbert spaces on which so far and most often the generalized Mehler semigroups were studied. Furthermore, we give new proofs of the random integral representation.
17
Content available remote Nested subclasses of some subclass of the class of type G selfdecomposable distributions on Rd
Aoyama T.
Probability and Mathematical Statistics
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2009
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Vol. 29, Fasc. 1
135--154
EN
Nested subclasses, denoted by Mn(Rd); n = 1; 2,…,of the class M(Rd), a subclass of the class of type G and selfdecomposable distributions on Rd are studied. An analytic characterization in terms of Lévy measures and a probabilistic characterization by stochastic integral representations for M(Rd) are known. In this paper, analytic characterizations for Mn(Rd); n = 1; 2,…,are given in terms of Lévy measures as well as probabilistic characterizations by stochastic integral representations are shown. A relationship with stable distributions is given.
18
Content available remote Intrinsic ultracontractivity for Lévy processes
Grzywny T.
Probability and Mathematical Statistics
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2008
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Vol. 28, Fasc. 1
91--106
EN
We prove the intrinsic ultracontractivity for semigroups generated by a large class of symmetric Lévy processes killed on exiting a bounded and connected Lipschitz set under some conditions about the behavior of the Lévy measure in the neighborhood of the origin.
19
Content available remote A calculus on Lévy exponents and selfdecomposability on Banach spaces
Jurek Z. J.
Probability and Mathematical Statistics
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2008
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Vol. 28, Fasc. 2
271--280
EN
In infinite-dimensional Banach spaces there is no complete characterization of the Lévy exponents of infinitely divisible probability measures. Here we propose a calculus on Lévy exponents that is derived from some random integrals. As a consequence we prove that each selfdecomposable measure can by factorized as another selfdecomposable measure and its background driving measure that is s-selfdecomposable. This complements a result from the paper of Iksanov, Jurek and Schreiber in the Annals of Probability (2004).
20
Content available remote A proposal of portfolio choice for infinitely divisible distributions of asset returns
Kliber P.
Foundations of Computing and Decision Sciences
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2008
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Vol. 33, No. 1
43-52
EN
In the paper \ve present a proposal of augmenting portfolio analysis for the infinitely divisible distributions of returns - so that the prices of assets can follow Levy processes. In the classical portfolio analysis (by Markovitz or Sharp) the portfolio is evaluated according to two criteria: mean return and variance of returns. Such an approach is cumbersome second moments of assets' returns do not exist or if the interdependence between the returns of different assets can not be described only by covariation. In this article we propose a model in which asset prices follow multidimensional Levy process and the interdependence between assets are described by covariance (Gaussian part) and multidimensional jump measure (Poisson pan). Then we propose to choose the optimal portfolio based on three criteria: mean return, total variance of diffusion and a measure of jump risk. We also consider augmenting this multi-criteria choice setup for the costs of possible portfolio adjustments.
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