Compound Poisson Process Research Articles

Fuzzy entropy complexity and multifractal behavior of statistical physics financial dynamics

The present paper introduces a novel stochastic statistical physics financial system by combining exclusion system with compound Poisson process. The financial system is a model concerning with price fluctuations derived from dissemination of trading attitudes of financial agents and random drastic jumps caused by exogenous economic factors, respectively. Multifractal detrended cross-correlation analysis and fuzzy entropy of simulation dataset are performed. Fractional fuzzy entropy based on fuzzy entropy method and fractional calculus, is also developed to explore nonlinear complexity of the proposed model in the paper. Empirical results display that there are significant multifractality and fractal asymmetry in the proposed model, and the complexity of return series increases when λ or γ increases, correspondingly. And same analyses for actual stock market dataset are comparatively investigated. The conclusions reveal that the novel compound dynamics can reproduce various nonlinear behaviors of actual stock markets to a certain extent.

Uniform asymptotics for compound Poisson processes with regularly varying jumps and vanishing drift

This paper addresses heavy-tailed large-deviation estimates for the distribution tail of functionals of a class of spectrally one-sided Lévy processes. Our contribution is to show that these estimates remain valid in a near-critical regime. This complements recent similar results that have been obtained for the all-time supremum of such processes. Specifically, we consider local asymptotics of the all-time supremum, the supremum of the process until exiting [0,∞), the maximum jump until that time, and the time it takes until exiting [0,∞). The proofs rely, among other things, on properties of scale functions.

Analysis of the stochastic cash balance problem using a level crossing technique

The simple cash management problem includes the following considerations: the opportunity cost of holding too much cash versus the penalty cost of not having enough cash to meet current needs; the cost incurred (or profit generated) when making changes to cash levels by increasing or decreasing them when necessary; the uncertainty in timing and magnitude of cash receipts and cash disbursements; and the type of control policy that should be used to minimize the required level of cash balances and related costs. In this paper, we study a version of this problem in which cash receipts and cash disbursements occur according to two independent compound Poisson processes. The cash balance is monitored continuously and an order-point, order-up-to-level, and keep-level $$ \left( {s, S, M} \right) $$ policy is used to monitor the content, where $$ s \le S \le M $$ . That is, (a) if, at any time, the cash level is below s, an order is immediately placed to raise the level to S; (b) if the cash level is between s and M, no action is taken; (c) if the cash level is greater than M, the amount in excess of M is placed into an earning asset. We seek to minimize the expected total costs per unit time of running the cash balance. We use a level-crossing approach to develop a solution procedure for finding the optimal policy parameters and costs. Several numerical examples are given to illustrate the tradeoffs.

Property Claim Services by Compound Poisson Process And Inhomogeneous Levy Process

In this paper, stochastic compound Poisson process is employed to value the catastrophic insurance options and model the claim arrival process for catastrophic events, which were written in the loss period , during which the catastrophe took place. Here, a time compound process gives the underlying loss index before and after whose losses are revaluated by inhomogeneous exponential Levy process factor. For this paper, an exponential Levy process is used to evaluate the well-known European call option in order to price Property Claim Services catastrophe insurance based on catastrophe index.

Transient Analysis of $$M^{[X_{1}]},M^{[X_{2}]}/G_{1},G_{2}/1$$ M [ X 1 ] , M [ X 2 ] / G 1 , G 2 / 1 Retrial Queueing System with Priority Services, Working Vacations and Vacation Interruption, Emergency Vacation, Negative Arrival and Delayed Repair

This paper deals with the analysis of two classes of batch arrivals, one is high priority and other is low priority(retrial) customers with non-preemptive priority service, working vacations, negative arrival, emergency vacation for an unreliable server, which consists of a break-down by negative arrival and delay time to start repair. When there are no customers in the queue and in the orbit at the time of service completion for a positive customers, the server goes for a working vacation. The server works at a lower service rate during working vacation period. If there are no customers in the system at the end of vacation, the server becomes idle and ready for serving the new arrivals with probability p(single working vacation) or it remains on vacation with probability $$q=(1{-}p)$$ (multiple working vacations). Here we assume that customers arrive according to compound Poisson process in which high priority customers are assigned to class one and class two customers are of a low priority type. The priority customers that find the server busy are queued and then served in accordance with FCFS discipline. The arriving low priority customers on finding the server busy are queued in the orbit in accordance with FCFS constant retrial policy. The arrival of negative customer remove the customer being in service and makes the server breakdown and the server fail for a short interval of time. When the server fails it will send to repair. But the repair time do not start immediately, there is a delay time to start the repair process. We consider an emergency vacation which means that during the service time the server suddenly go for a vacation and the interrupted customer waits to getting the remaining service. The retrial time, service time, emergency vacation time, delay time and repair time are all follows general(arbitrary) distribution, working vacation time and negative arrival follows exponential distribution. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results are obtained explicitly. Also some performance measures of our model such as the system state probabilities, reliability indices, the average number of customer in the high priority queue and the low priority in the orbit and the average waiting time are derived. Numerical results are computed and graphical representations are presented.

Shipment consolidation with two demand classes: Rationing the dispatch capacity

We analyze the problem faced by a logistics provider that dispatches shipment orders (parcels or larger packages) of two order classes, viz. expedited and regular. Shipment orders arrive according to a compound Poisson process for each class. Upon an arrival, the logistics provider may continue consolidating arriving orders by paying a holding cost. Alternatively, the provider may dispatch, at a fixed cost, a vehicle containing (a portion of) the load consolidated so far. In addition, the provider must specify the composition of each dispatch by allocating (rationing) the volume of the vehicle between expedited and regular shipment orders. We model this problem as a continuous-time Markov Decision Process and minimize the expected discounted total cost. We prove the existence of quantity-based optimal threshold policies under particular conditions. We also structurally analyze the thresholds of these optimal policies. Based on these structural properties, we develop an efficient solution approach for large problem instances which are difficult to solve using the conventional policy-iteration method. For two real-life applications, we show that the quantity-based threshold policies derived using the proposed approach outperform the time policies used in practice.

A Markov-switching generalized additive model for compound Poisson processes, with applications to operational loss models

This paper is concerned with modelling the behaviour of random sums over time. Such models are particularly useful to describe the dynamics of operational losses, and to correctly estimate tail-related risk indicators. However, time-varying dependence structures make it a difficult task. To tackle these issues, we formulate a new Markov-switching generalized additive compound process combining Poisson and generalized Pareto distributions. This flexible model takes into account two important features: on the one hand, we allow all parameters of the compound loss distribution to depend on economic covariates in a flexible way. On the other hand, we allow this dependence to vary over time, via a hidden state process. A simulation study indicates that, even in the case of a short time series, this model is easily and well estimated with a standard maximum likelihood procedure. Relying on this approach, we analyse a novel data-set of 819 losses resulting from frauds at the Italian bank UniCredit. We show that our model improves the estimation of the total loss distribution over time, compared to standard alternatives. In particular, this model provides estimations of the 99.9% quantile that are never exceeded by the historical total losses, a feature particularly desirable for banking regulators.

Exchange option pricing in jump-diffusion models based on esscher transform

ABSTRACTIn the real world, we introduce a dynamic model about the risky asset which is governed by Brownian motion, stationary compound Poisson process and its compensation process. By choosing Esscher transform parameters, we obtain a risk-neural measure Q under which the discounted value of the risky underlying asset is a martingale. Then, we give the pricing formulas of Exchange option by change of numeraire. At last, we analyze the option pricing formula and provide numerical illustrations by introducing BBY stock and SBUX stock.

Mechanisms of dispersion in a porous medium

This paper studies the mechanisms of dispersion in the laminar flow through the pore space of a three-dimensional porous medium. We focus on preasymptotic transport prior to the asymptotic hydrodynamic dispersion regime, in which solute motion may be described by the average flow velocity and a hydrodynamic dispersion coefficient. High-performance numerical flow and transport simulations of solute breakthrough at the outlet of a sand-like porous medium evidence marked deviations from the hydrodynamic dispersion paradigm and identify two distinct regimes. The first regime is characterised by a broad distribution of advective residence times in single pores. The second regime is characterised by diffusive mass transfer into low-velocity regions in the wake of solid grains. These mechanisms are quantified systematically in the framework of a time-domain random walk for the motion of marked elements (particles) of the transported material quantity. Particle transitions occur over the length scale imprinted in the pore structure at random times given by heterogeneous advection and diffusion. Under globally advection-dominated conditions, i.e., Péclet numbers larger than 1, particles sample the intrapore velocities by diffusion and the interpore velocities through advection. Thus, for a single transition, particle velocities are approximated by the mean pore velocity. In order to quantify this advection mechanism, we develop a model for the statistics of the Eulerian velocity magnitude based on Poiseuille’s law for flow through a single pore and for the distribution of mean pore velocities, both of which are linked to the distribution of pore diameters. Diffusion across streamlines through immobile zones in the wake of solid grains gives rise to exponentially distributed residence times that decay on the diffusion time over the pore length. The trapping rate is determined by the inverse diffusion time. This trapping mechanism is represented by a compound Poisson process conditioned on the advective residence time in the proposed time-domain random walk approach. The model is parameterised with the characteristics of the porous medium under consideration and captures both preasymptotic regimes. Macroscale transport is described by an integro-differential equation for solute concentration, whose memory kernels are given in terms of the distribution of mean pore velocities and trapping times. This approach quantifies the physical non-equilibrium caused by a broad distribution of mass transfer time scales, both advective and diffusive, on the representative elementary volume (REV). Thus, while the REV indicates the scale at which medium properties like porosity can be uniquely defined, this does not imply that transport can be characterised by hydrodynamic dispersion.

Pricing exchange options with correlated jump diffusion processes

We study the applicability to energy facilities of a model for correlated Poisson processes generated by self-decomposable jumps. In this context, the implementation of our approach, both to shape power or gas dynamics, and to evaluate transportation assets seen as spread or exchange options, is rather natural. In particular we first enhance the Merton market with two underlying assets making jumps at times ruled by correlated Poisson processes. Here however— at variance with the existing literature—the correlation is no longer provided only by a systemic common source of synchronous macroeconomic shocks, but also by a delayed synaptic propagation of the shocks themselves between the assets. In a second step, we consider a price dynamics driven by an exponential mean-reverting geometric Ornstein–Uhlenbeck process plus a compound Poisson process: a combination which is well suited for the energy markets. In our specific instance, for each underlying, we adopt a jumping price spot dynamics that has the advantage of being exactly treatable to find no-arbitrage conditions. As a result we are able to find closed form formulas for vanilla options, so that the price of the spread options can subsequently be calculated (again in closed form) using the Margrabe formula if the strike is zero (exchange options), or with some other suitable procedures available in the literature. The exchange option values obtained in our numerical examples show that, compared to the other Poisson models we analysed, the dependence introduced by the self-decomposition gives more relevance to the timing of the jumps and not only to their frequency.

An M/M/1 Queueing-Inventory System with Geometric Batch Demands and Lost Sales

This paper studies an M/M/1 queueing-inventory system with batch demands. Customers arrive in the system according to a compound Poisson process, where the size of the batch demands for each arrival is a random variable that follows a geometric distribution. The inventory is replenished according to the standard (s,S) policy. The replenishment time follows an exponential distribution. Two models are considered. In the first model, if the on-hand inventory is less than the size of the batch demands of an arrived customer, the customer takes away all the items in the inventory, and a part of the customer’s batch demands is lost. In the second model, if the on-hand inventory is less than the size of the batch demands of an arrived customer, the customer leaves without taking any item from the inventory, and all of the customer’s batch demands are lost. For these two models, the authors derive the stationary conditions of the system. Then, the authors derive the stationary distributions of the product-form of the joint queue length and the on-hand inventory process. Besides this, the authors obtain some important performance measures and the average cost functions by using these stationary distributions. The results are illustrated by numerical examples.

Out of the Antilles: Fossil phylogenies support reverse colonization of bats to South America

AbstractAimPrevious phylogenies of extant short‐faced bats (Chiroptera: Stenodermatina) supported either two colonization events from the mainland to the Antilles, or reverse colonization, but lacked both fossil data and statistical modelling of biogeography. Recent multi‐locus phylogenies of noctilionoid bats and likelihood modelling of ancestral ranges support a continental origin for the clade. We include all known extinct and extant stenodermatina species and apply statistical modelling to test competing biogeographical hypotheses.LocationThe Neotropics, including the Antilles.MethodsWe combined mitochondrial and nuclear sequences with 302 new morphological characters to infer phylogenies. Bayesian tip‐dating analyses applied codon models to protein‐coding genes, with relaxed molecular clocks fitting a compound Poisson process. The combined maximum clade credibility tree was used in comparisons of alternative biogeographical models.ResultsThe new phylogenies support the fossil Cubanycteris silvai as sister to all extant species of short‐faced bats. Among Artibeus (the sister group to short‐faced bats), the Antillean fossil A. anthonyi has distinctive characters and is nested within the subgenus Artibeus. The common ancestor of all short‐faced bats is inferred to be Antillean, as a mainland origin is unlikely. Founder‐event speciation is the most probable process explaining the distribution of these highly divergent fossil lineages.Main conclusionsDated, character‐based phylogenies of fossil species are indispensable for biogeographical inference: without fossils, biogeographical analyses find a mainland origin for short‐faced bats. The rate of founder speciation in this clade is twice as high as the estimate from noctilionoids in general, highlighting the role of founder events in the diversification of island taxa. Although rare, reverse colonization contributes key species to continental communities. Short‐faced bats, including Cubanycteris, share biomechanical adaptations for a strong bite conferring access to harder figs. We hypothesize these adaptations and characters related to roosting ecology enabled ancestral lineages to successfully establish and diversify on the mainland.

Stochastic interest model driven by compound Poisson process and Brownian motion with applications in life contingencies

In this paper, we introduce a class of stochastic interest model driven by a compoundPoisson process and a Brownian motion, in which the jumping times of force of interest obeyscompound Poisson process and the continuous tiny fluctuations are described by Brownian motion, andthe adjustment in each jump of interest force is assumed to be random. Based on the proposed interestmodel, we discuss the expected discounted function, the validity of the model and actuarial presentvalues of life annuities and life insurances under different parameters and distribution settings. Ournumerical results show actuarial values could be sensitive to the parameters and distribution settings,which shows the importance of introducing this kind interest model.

A Poisson-Gamma Model for Zero Inflated Rainfall Data

Rainfall modeling is significant for prediction and forecasting purposes in agriculture, weather derivatives, hydrology, and risk and disaster preparedness. Normally two models are used to model the rainfall process as a chain dependent process representing the occurrence and intensity of rainfall. Such two models help in understanding the physical features and dynamics of rainfall process. However rainfall data is zero inflated and exhibits overdispersion which is always underestimated by such models. In this study we have modeled the two processes simultaneously as a compound Poisson process. The rainfall events are modeled as a Poisson process while the intensity of each rainfall event is Gamma distributed. We minimize overdispersion by introducing the dispersion parameter in the model implemented through Tweedie distributions. Simulated rainfall data from the model shows a resemblance of the actual rainfall data in terms of seasonal variation, means, variance, and magnitude. The model also provides mechanisms for small but important properties of the rainfall process. The model developed can be used in forecasting and predicting rainfall amounts and occurrences which is important in weather derivatives, agriculture, hydrology, and prediction of drought and flood occurrences.

A Poisson Gamma Model for Zero Inflated Rainfall Data

Rainfall modeling is significant for prediction and forecasting purposes in agriculture, weather derivatives, hydrology, risk and disaster preparedness. Normally two models are used to model the rainfall process as a chain dependent representing the occurrence and intensity of rainfall. Such two models helps to understand the physical features and dynamics of rainfall process. However rainfall data is zero inflated and exhibit overdispersion which is always underestimated by such models. In this study we have modeled the two processes simultaneously as a compound Poisson process. The rainfall events are modeled as a Poisson process while the intensity of each rainfall event is Gamma distributed. We minimize overdispersion by introducing the dispersion parameter in the model implemented through Tweedie distributions. Simulated rainfall data from the model shows a resemblance of the actual rainfall data in terms of seasonal variation, means, variance and magnitude. The model also provides mechanisms for small but important properties of the rainfall process. The model developed can be used in forecasting and predicting rainfall amounts and occurrences which is important in weather derivatives, agriculture, hydrology, and predicting of drought and flood occurrences.

Stochastic interest model driven by compound Poisson process and Brownian motion with applications in life contingencies

Stochastic interest model driven by compound Poisson process and Brownian motion with applications in life contingencies

Fractional compound Poisson processes with multiple internal states

For the particles undergoing the anomalous diffusion with different waiting time distributions for different internal states, we derive the Fokker-Planck and Feymann-Kac equations, respectively, describing positions of the particles and functional distributions of the trajectories of particles; in particular, the equations governing the functional distribution of internal states are also obtained. The dynamics of the stochastic processes are analyzed and the applications, calculating the distribution of the first passage time and the distribution of the fraction of the occupation time, of the equations are given. For the further application of the newly built models, we make very detailed discussions on the none-immediately-repeated stochastic process, e.g., the random walk of smart animals.

Probability distribution of the seismic damage cost over the life cycle of structures

In the life-cycle analysis, the total cost of damage caused by earthquakes is a significant but highly uncertain component. In the current literature, the seismic risk analysis is largely limited to the evaluation of the average cost of damage, which is not informative about the full extent of variability in the cost. The paper presents a systematic development of the stochastic modeling of seismic risk analysis problem and reformulates the damage cost analysis as a superposition of compound Poisson processes. An explicit analytical solution for the distribution of damage cost is derived in form of a recursive equation. The proposed approach extends the capability of the existing framework of seismic risk analysis, which can be used to optimize initial design and retrofitting of structures.

Optimal Cash Management Problem for Compound Poisson Processes with Two-Sided Jumps

This paper presents a continuous time cash management problem where the uncontrolled money stock follows a compound poisson process with two-sided jumps and negative drift. The main goal is to minimize the total expected discounted sum of the opportunity cost of holding cash and the adjustment cost coming from deposits and withdrawals. This adjustment cost can have a proportional and a fixed component. In this paper it is assumed that the controlled money stock cannot be negative. We show that the optimal value function is an a.e. strong solution of the corresponding Hamilton–Jacobi–Bellman equation and that it can be characterized as the largest a.e. strong supersolution. We prove that the optimal policy has an impulse multi-band structure with multiple trigger-target pairs. We present examples where the optimal value function is not differentiable and an example where the optimal impulse policy has more that one band.

Optimal dividend and investment problems under Sparre Andersen model

In this paper, we study a class of optimal dividend and investment problems assuming that the underlying reserve process follows the Sparre Andersen model, that is, the claim frequency is a “renewal” process, rather than a standard compound Poisson process. The main feature of such problems is that the underlying reserve dynamics, even in its simplest form, is no longer Markovian. By using the backward Markovization technique, we recast the problem in a Markovian framework with expanded dimension representing the time elapsed after the last claim, with which we investigate the regularity of the value function, and validate the dynamic programming principle. Furthermore, we show that the value function is the unique constrained viscosity solution to the associated HJB equation on a cylindrical domain on which the problem is well defined.