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

2021 ◽  
Vol 2131 (2) ◽  
pp. 022107
Author(s):  
O Rusakov ◽  
Yu Yakubovich
Keyword(s):  
Poisson Process ◽  
Continuous Time ◽  
Random Variables ◽  
Compound Poisson Process ◽  
Poisson Processes ◽  
Finite Variance ◽  
Uhlenbeck Process ◽  
Skorokhod Space ◽  
Actuarial Mathematics ◽  
Ornstein Uhlenbeck Process

Abstract Weconsider a PSI-process, that is a sequence of random variables (&), i = 0.1,…, which is subordinated by a continuous-time non-decreasing integer-valued process N(t): <K0 = ÇN(ty Our main example is when /V(t) itself is obtained as a subordination of the standard Poisson process 77(s) by a non-decreasing Lévy process S(t): N(t) = 77(S(t)).The values (&)one interprets as random claims, while their accumulated intensity S(t) is itself random. We show that in this case the process 7V(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 (&) 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.

On some characterizations of the poisson process

1989 ◽  
Vol 26 (01) ◽  
pp. 176-181
Author(s):  
Wen-Jang Huang
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Arrival Time ◽  
Random Variables ◽  
Poisson Processes

In this article we give some characterizations of Poisson processes, the model which we consider is inspired by Kimeldorf and Thall (1983) and we generalize the results of Chandramohan and Liang (1985). More precisely, we consider an arbitrarily delayed renewal process, at each arrival time we allow the number of arrivals to be i.i.d. random variables, also the mass of each unit atom can be split into k new atoms with the ith new atom assigned to the process Di, i = 1, ···, k. We shall show that the existence of a pair of uncorrelated processes Di, Dj, i ≠ j, implies the renewal process is Poisson. Some other related characterization results are also obtained.

scholarly journals Berry–Esséen bound for the parameter estimation of fractional Ornstein–Uhlenbeck processes

2019 ◽  
Vol 20 (04) ◽  
pp. 2050023 ◽  
Author(s):  
Yong Chen ◽  
Nenghui Kuang ◽  
Ying Li
Keyword(s):  
Parameter Estimation ◽  
Brownian Motion ◽  
Continuous Time ◽  
Index Formula ◽  
Hurst Index ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Time Observation ◽  
Statistical Estimations ◽  
Drift Parameter

For an Ornstein–Uhlenbeck process driven by fractional Brownian motion with Hurst index [Formula: see text], we show the Berry–Esséen bound of the least squares estimator of the drift parameter based on the continuous-time observation. We use an approach based on Malliavin calculus given by Kim and Park [Optimal Berry–Esséen bound for statistical estimations and its application to SPDE, J. Multivariate Anal. 155 (2017) 284–304].

On the probability that a random point is the jth nearest neighbour to its own kth nearest neighbour

1986 ◽  
Vol 23 (01) ◽  
pp. 221-226 ◽  
Author(s):  
Norbert Henze
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Arbitrary Point ◽  
Limiting Distribution ◽  
Poisson Processes ◽  
Random Point ◽  
Independent Random Variables ◽  
Nearest Neighbour ◽  
Homogeneous Poisson Process

In a homogeneous Poisson process in R d , consider an arbitrary point X and let Y be its kth nearest neighbour. Denote by Rk the rank of X in the proximity order defined by Y, i.e., Rk = j if X is the jth nearest neighbour to Y. A representation for Rk in terms of a sum of independent random variables is obtained, and the limiting distribution of Rk, as k →∞, is shown to be normal. This result generalizes to mixtures of Poisson processes.

On a class of reflected AR(1) processes

2016 ◽  
Vol 53 (3) ◽  
pp. 818-832
Author(s):  
Onno Boxma ◽  
Michel Mandjes ◽  
Josh Reed
Keyword(s):  
Heavy Traffic ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Steady State Distribution ◽  
State Distribution ◽  
Uhlenbeck Process ◽  
Stationary Analysis ◽  
Ornstein Uhlenbeck Process ◽  
Phase Type

AbstractIn this paper we study a reflected AR(1) process, i.e. a process (Zn)n obeying the recursion Zn+1= max{aZn+Xn,0}, with (Xn)n a sequence of independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Zn (in terms of transforms) in case Xn can be written as Yn−Bn, with (Bn)n being a sequence of independent random variables which are all Exp(λ) distributed, and (Yn)n i.i.d.; when |a|<1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (Bn)n are of phase-type. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein–Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a normal random variable conditioned on being positive.

An invariance property of Poisson processes

1969 ◽  
Vol 6 (02) ◽  
pp. 453-458 ◽  
Author(s):  
Mark Brown
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Random Variables ◽  
Poisson Processes ◽  
Invariance Property ◽  
Homogeneous Poisson Process ◽  
Arrival Times ◽  
Random Functions

In this paper we shall investigate point processes generated by random variables of the form 〈gi (Ti ]), i=± 1, ± 2, … 〉, where 〈Ti, i= ± 1, … 〉 is the set of arrival times from a (not necessarily homogeneous) Poisson process or mixture of Poisson processes, and 〈gi, i = ± 1, … 〉 is an independently and identically distributed (i.i.d.) or interchangeable sequence of random functions, independent of 〈Ti 〉.

scholarly journals A Note on Embedding Certain Bernoulli Sequences in Marked Poisson Processes

2008 ◽  
Vol 45 (04) ◽  
pp. 1181-1185 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Poisson Processes ◽  
Bernoulli Random Variables ◽  
Conditional Poisson

A sequence of independent Bernoulli random variables with success probabilities a / (a + b + k − 1), k = 1, 2, 3, …, is embedded in a marked Poisson process with intensity 1. Using this, conditional Poisson limits follow for counts of failure strings.

scholarly journals On the evolution of topology in dynamic clique complexes

2016 ◽  
Vol 48 (4) ◽  
pp. 989-1014 ◽  
Author(s):  
Gugan C. Thoppe ◽  
D. Yogeshwaran ◽  
Robert J. Adler
Keyword(s):  
Markov Chains ◽  
Betti Number ◽  
Continuous Time ◽  
Time Varying ◽  
Inclusion Probability ◽  
Uhlenbeck Process ◽  
Continuous Time Markov Chains ◽  
Ornstein Uhlenbeck Process

AbstractWe consider a time varying analogue of the Erdős–Rényi graph and study the topological variations of its associated clique complex. The dynamics of the graph are stationary and are determined by the edges, which evolve independently as continuous-time Markov chains. Our main result is that when the edge inclusion probability is of the form p=nα, where n is the number of vertices and α∈(-1/k, -1/(k + 1)), then the process of the normalised kth Betti number of these dynamic clique complexes converges weakly to the Ornstein–Uhlenbeck process as n→∞.

UPPER FIRST-EXIT TIMES OF COMPOUND POISSON PROCESSES REVISITED

2003 ◽  
Vol 17 (4) ◽  
pp. 459-465 ◽  
Author(s):  
W. Stadje ◽  
S. Zacks
Keyword(s):  
Poisson Process ◽  
Compound Poisson Process ◽  
Hitting Time ◽  
Poisson Processes ◽  
Conditional Density ◽  
Exit Times ◽  
Compound Poisson ◽  
Straight Line ◽  
First Exit Times ◽  
First Time

For a compound Poisson process (CPP) with only positive jumps, an elegant formula connects the density of the hitting time for a lower straight line with that of the process itself at time t, h(x; t), considered as a function of time and position jointly. We prove an analogous (albeit more complicated) result for the first time the CPP crosses an upper straight line. We also consider the conditional density of the CPP at time t, given that the upper line has not been reached before t. Finally, it is shown how to compute certain moment integrals of h.

Asymptotic distributions of pontograms

1987 ◽  
Vol 101 (1) ◽  
pp. 131-139 ◽  
Author(s):  
Miklós Csörgő ◽  
Lajos Horváth
Keyword(s):  
Poisson Process ◽  
Covariance Function ◽  
Weighted Approximation ◽  
Asymptotic Distributions ◽  
Uhlenbeck Process ◽  
Intensity Parameter ◽  
Ornstein Uhlenbeck Process ◽  
Image Position

Let{N(x), x ≥ 0} be a Poisson process with intensity parameter λ > 0, and introduceWhen looking for the changepoint in the Land's End data, Kendall and Kendall [7] proved for all 0 < ε1 < 1 − ε2 < 1 thatwhere {V(s), − ∞ < s < ∞} is an Ornstein–Uhlenbeck process with covariance function exp (−|t − s|). D. G. Kendall has posed the problem of replacing ε1 and ε2 by zero or by sequences εi(n) → 0 (n → ∞) (i = 1, 2), in (1·1). In this paper we study the latter problem and also its L2 version. The proofs will be based on the following weighted approximation of Zn.

scholarly journals A Note on Embedding Certain Bernoulli Sequences in Marked Poisson Processes

2008 ◽  
Vol 45 (4) ◽  
pp. 1181-1185 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Poisson Processes ◽  
Bernoulli Random Variables ◽  
Conditional Poisson

A sequence of independent Bernoulli random variables with success probabilities a / (a + b + k − 1), k = 1, 2, 3, …, is embedded in a marked Poisson process with intensity 1. Using this, conditional Poisson limits follow for counts of failure strings.

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