Some genetalizations of the poisson process and related processes

Abstract

The homogeneous Poisson process (HPP) is the most useful and popular counting process. HPP is used for example to model the arrivals/departures in queuing theory, arrivals of jumps in various jump-di usion models, number of customers at service stations and number of claims to an insurance company in a particular time period etc. However, the number of arrivals in many real-life phenomena may not be governed by HPP. In many real life scenario, there could be burst arrivals, the intensity may be time dependent, the inter-arrival times may be heavy tailed, the arrival rate may be stochastic in nature or the counting needs to be done over a space rather than over a time interval. To overcome these limitations the Poisson process is generalized in several directions for example nonhomogeneous poison process, time-fractional Poisson process (TFPP), space-fractional Poisson process (SFPP), Poisson process of order k, doubly stochastic Poisson process (or Cox process) and Poisson random elds etc. In this thesis work, we extend these models in several directions. We introduce the tempered space fractional Poisson process (TSFPP) by taking the tempered fractional shift operator in place of ordinary fractional shift operator in governing di erential-di erence equation of SFPP. Moreover, we also provide the subordination representation of this process. Further, using the same idea a large class of fractional Poisson processes is introduced, namely, tempered time-space fractional Poisson processes (TTSFPP) which may give more exibility in modeling of counting types real world data. Many insurance models generally use Poisson process to model the arrival of claims which has a limitation of not having more than one claim in small time intervals. However, in many extreme scenarios like natural disaster, terrorists attack etc, the claim arrivals may be in groups and which may contain arbitrary number of claims in small time intervals. To overcome this di culty a variant of the standard Poisson process known as Poisson process of order k (PPoK) is introduced in literature, where the number of arrivals in a small time interval is a discrete uniform random variable taking values in [0; k]. This process can model the claims arrival in groups of size k. We extend PPoK to time-changed PPoK in one dimensional space, which has stochastic intensity. We further extend PPoK in higher dimensional space called Poisson elds of order k which can be used to model random points in space representing distributions of mobile phones or TV units in a geographic location. In tick-by-tick high frequency trading data to model the number of positive and negative jumps (also called up-ticks and down-ticks) the Skellam process is used in literature which is the di erence of two independent Poisson processes. However, Skellam process has limitation, as it allows only -1, 0 and 1 jump sizes in small time period. To allow arbitrary jump size, we introduce Skellam process of order k (SPoK) where number of jumps can vary between 􀀀k and k in small time intervals. The arrival time between the positive and negative jumps are exponentially distributed in SP and SPoK. We also introduce fractional Poisson elds of order k in n-dimensional Euclidean space Rn +. We also work on time-fractional Poisson process of order k, space-fractional Poisson process of order k and tempered version of time-space fractional Poisson process of order k. These processes are de ned in terms of fractional compound Poisson processes. Timefractional Poisson process of order k naturally generalizes the Poisson process and Poisson process of order k to a heavy tailed waiting times counting process. The space-fractional Poisson process of order k, allows on average in nite number of arrivals in any interval. We derive the marginal probabilities, governing di erence-di erential equations of the introduced processes. We also provide Watanabe martingale characterization for some time-changed Poisson processes. Moreover, the stable subordinator, inverse stable subordinator, tempered stable subordinator, inverse tempered stable subordinator, compositions of tempered stable subordinators and compositions of inverse tempered stable subordinators are central elements in time-changed Poisson processes or generalized Poisson processes. Hence, we discuss the distributional properties of these subordinators and inverse subordinators. The in nite series form of the probability densities of tempered stable and inverse tempered stable subordinators are obtained using Mellin transform. Further, the densities of the products and quotients of stable and inverse stable subordinators are worked out. The asymptotic behaviours of these densities are obtained as x ! 0+. Similar results for tempered and inverse tempered stable subordinators are discussed. Our results provide alternative methods to nd the densities of these subordinators and complement the results available in literature. We further introduce mixtures of tempered stable subordinators. These mixtures de ne a class of subordinators which generalize tempered stable subordinators (TSS). The main properties like the probability density function (pdf), L evy density, moments, governing Fokker-Planck-Kolmogorov (FPK) type equations and the asymptotic form of potential density are derived. Further, the governing FPK type equation and the asymptotic form of the renewal function for the corresponding inverse subordinator are discussed. We generalize these results to n-th order mixtures of TSS. The governing fractional di erence and di erential equations of the time-changed Poisson process and Brownian motion are also discussed.