Compound Poisson Process Research Articles

Managing inventory with proportional transaction costs

We solve the problem of optimal inventory management for a CARA market-maker who faces proportional transaction costs and marking to market. Our model accommodates inventory shocks following an arbitrary compound Poisson process. We show that the no-trading region is always wider in the presence of inventory shocks.

A Multidimensional Problem of Optimal Dividends with Irreversible Switching: A Convergent Numerical Scheme

In this paper we study the problem of optimal dividend payment strategy which maximizes the expected discounted sum of dividends to a multidimensional set up of n associated insurance companies where the surplus process follows an n-dimensional compound Poisson process. The general manager of the companies has the possibility at any time to exercise an irreversible switch into another regime; we also take into account an expected discounted value at ruin. This multidimensional dividend problem is a mixed singular control/optimal problem. We prove that the optimal value function is a viscosity solution of the associated HJB equation and that it can be characterized as the smallest viscosity supersolution. The main contribution of the paper is to provide a numerical method to approximate (locally uniformly) the optimal value function by an increasing sequence of sub-optimal value functions of admissible strategies defined in an n-dimensional grid. As a numerical example, we present the optimal time of merger for two insurance companies.

Valuation of contingent convertible catastrophe bonds — The case for equity conversion

Within the context of banking-related literature on contingent convertible bonds, we comprehensively formalise the design and features of a relatively new type of insurance-linked security, called a contingent convertible catastrophe bond (CocoCat). We begin with a discussion on its design and compare its relative merits to catastrophe bonds and catastrophe-equity puts. Subsequently, we derive analytical valuation formulae for index-linked CocoCats under the assumption of independence between natural catastrophe and financial market risks. We model natural catastrophe losses by a time-inhomogeneous compound Poisson process, with the interest-rate process governed by the Longstaff model. By using an exponential change of measure on the loss process, as well as a Girsanov-like transformation to synthetically remove the correlation between the share and interest-rate processes, we obtain these analytical formulae. Using selected parameter values in line with earlier research, we numerically analyse our valuation formulae for index-linked CocoCats. An analysis of the results reveals that the CocoCat prices are most sensitive to changing interest-rates, conversion fractions and the threshold levels defining the trigger times.

Valuation of new-designed contracts for catastrophe risk management

In this study, we design and price a new kind of catastrophe equity put options, whose payoff depends on the ratio of accumulated losses and the expected level over the life of the option. We adopt a compound Poisson process to describe accumulated catastrophe losses and assume catastrophe losses affect the prices of the underlying stock. In the proposed framework, we obtain an explicit pricing formula of the new kind of catastrophe equity put options. Finally, numerical results are presented to investigate the values of this new class of options and illustrate the differences between the prices of vanilla catastrophe equity put options and the contracts designed in this paper. Interestingly, the prices of the new-designed contracts with different power exponents change oppositely as the intensity rises.

Empirical likelihood ratio for two-sample compound Poisson process problems

ABSTRACTIn this article, the author focus on the simple and yet very important case of making inference on the difference of two population means using the empirical likelihood approach of two-sample compound Poisson process. It is shown that the log empirical likelihood ratio statistic for the difference of two population means converges in distribution to as . The simulation results show that the empirical likelihood ratio method is applicable.

Steady-State Analysis of Buffer Occupancy for Different Forwarding Strategies in Mobile Opportunistic Network

We perform a study on the buffer occupancy 1 1 The term buffer occupancy refers to the amount (or number) of data (or packets) present in the buffer. We shall be using the terms buffer occupancy and system size interchangeably as deemed fit. and its steady-state properties in a mobile opportunistic network (MON) for different forwarding strategies. Nodes in MON adhere to replication based strategy in order to ensure reliable delivery. Such replications occur in bulks during intermittent and very short contact events, which often lead to buffer congestion in the relay nodes, thereby affecting the network performance. The state-of-the-art on buffer management in MON relies on local information exchange to measure buffer occupancy, which leads to overhead. Therefore, to find the average buffer occupancy, we first study the aggregated bulk transfer size using real-life contact traces and find that it follows a log-normal distribution. Since the contact rate has been observed to follow a Poisson distribution, we model the packet arrival process in each node as a compound Poisson process. Furthermore, the departure process is modeled as packets dropped due to time-to-live expiry, which is shown to have an exponential distribution. Using the above model, we derive the steady-state probability distribution of buffer occupancy of an arbitrary node for three different classes of replication scheme, viz., epidemic routing, probabilistic routing, and time-varying probabilistic routing. The results of this paper help in assessing how fast a node buffer gets depleted under different routing algorithms, which will help in designing better buffer management techniques and routing algorithms.

Controlling distribution inventory systems with shipment consolidation and compound Poisson demand

We consider a one-warehouse-multiple-retailer inventory system where the retailers face stochastic customer demand, modelled as compound Poisson processes. Deliveries from the central warehouse to groups of retailers are consolidated using a time based shipment consolidation policy. This means that replenishment orders have to wait until a vehicle departures, which increases the lead time for the retailers and therefore also the safety stock. Thus, a trade-off exists between expected shipment costs and holding costs. Our aim is to determine the shipment intervals and the required amount of safety stock for each retailer and the warehouse to minimize total cost, both for backorder costs and fill rate constraints. Previous work has focused on exact solutions which are computationally demanding and not applicable for larger real world problems. The focus of our present work is on the development of computationally attractive heuristics that can be applied in practice. A numerical study shows that the proposed heuristics perform well compared to the exact cost minimizing solutions. We also illustrate that the approaches are appropriate for solving real world problems using data from a large European company.

Quasi-likelihood estimation of structure-changed threshold double autoregressive models

This paper investigates the quasi-maximum likelihood estimator (QMLE) of the structure-changed and two-regime threshold double autoregressive model. It is shown that both the estimated threshold and change-point are n-consistent, and they converge weakly to the smallest minimizer of a compound Poisson process and the location of minima of a two-sided random walk, respectively. Other estimated parameters are n−consistent and asymptotically normal. The performance of the QMLE is assessed via simulation studies and a real example is given to illustrate our procedure.

Fixed relative precision estimators of growth rate for compound Poisson and Lévy processes

We consider compound Poisson processes or, more generally, Lévy processes X(t) with positive bounded jumps. The problem is to estimate the “growth rate” μ=EX(t)∕t with fixed relative precision, i.e. to construct an estimator μˆ such that P(|μˆ−μ|<με)≥1−α, for a given precision parameter ε and confidence parameter α, given a trajectory X(t) for 0≤t≤T. Such an estimator must be sequential, i.e. the length T of the observed trajectory must be random and chosen adaptively. Assume that the upper bound on jumps is known (w.l.o.g. equal to 1). We consider the estimator μˆr=r∕Tr, where Tr=min{t:X(t)≥r}, with a suitably chosen r=r(ε,α). We show that this estimator is “nearly worst case optimal” in a certain asymptotic sense, for ε→0 and α→0. The “worst case” turns out to be the process with jumps 1, i.e. the Poisson process with intensity μ.

Convergence and stability of the backward Euler method for jump–diffusion SDEs with super-linearly growing diffusion and jump coefficients

This paper firstly investigates convergence of the backward Euler method for stochastic differential equations (SDEs) driven by Brownian motion and compound Poisson process. Different from existing works, we do not impose a globally Lipschitz condition on the jump coefficient but formulate a more relaxed condition to allow for its super-linear growth. It is shown that the mean square convergence order of this method can be arbitrarily close to 1 2 under mild assumptions imposed on SDEs, allowing for possibly super-linearly growing drift, diffusion and jump coefficients. An exact order 1 2 is recovered when further differentiability assumption is put on the coefficients. Furthermore, the considered method is able to inherit the mean square stability of a wider class of Lévy noise driven SDEs for all stepsizes. These results are finally supported by some numerical experiments.

Equivalent constant-amplitude fatigue load method based on the energy equivalence principle

An equivalent constant-amplitude cyclic loading method for random vehicle load is proposed based on the concept of energy equivalence. The filtered compound Poisson process is adopted to describe random vehicle load, through which the vehicle load spectrum is gained. The total dissipated energy due to fatigue loads subjected to concrete structures is deduced by introducing a multi-scale model, in which the energy dissipation induced by the nano-cracks level is presented, and the energy transition from nano to macro scales is derived. By assuming that the total energy dissipation under random loading equals that under equivalent constant-amplitude loading, the equivalent load ratio and amplitude as well as the number of equivalent cycles for random vehicle load are obtained. To validate the effectiveness of the proposed method, a set of numerical simulations is presented. The fatigue damage accumulations for concrete structures under both the random load and the equivalent loads converted by the proposed method and the classic root mean square method are calculated. By comparing with the root mean square method, the accuracy and advantage of the proposed equivalent load model are verified.

Non-Parametric Threshold Estimation for the Wiener–Poisson Risk Model

In this paper, we consider the Wiener–Poisson risk model, which consists of a Wiener process and a compound Poisson process. Given the discrete record of observations, we use a threshold method and a regularized Laplace inversion technique to estimate the survival probability. In addition, we also construct an estimator for the distribution function of jump size and study its consistency and asymptotic normality. Finally, we give some simulations to verify our results.

Compound Poisson Noise Sources in Diffusion-Based Molecular Communication

Diffusion-based molecular communication (DMC) is one of the most promising approaches for realizing nano-scale communications for healthcare applications. The DMC systems in in-vivo environments may encounter biological entities that release molecules identical to the molecules used for signaling as part of their functionality. Such entities in the environment act as external noise sources from the DMC system’s perspective. In this paper, the release of molecules by external bio-inspired noise sources is particularly modeled as a compound Poisson process. The impact of compound Poisson noise sources (CPNSs) on the performance of a point-to-point DMC system is investigated. To this end, the noise from the CPNS observed at the receiver is characterized. Considering a simple on-off keying modulation and formulating symbol-by-symbol maximum likelihood (ML) detector, the performance of the DMC system in the presence of the CPNS is analyzed. For the special case of CPNS in a high-rate regime, the noise received from the CPNS is approximated as a Poisson process whose rate is normally distributed. In this case, it is proved that a simple single-threshold detector is an optimal ML detector. Our results reveal that in general, adopting the conventional simple homogeneous Poisson noise model may lead to overly optimistic performance predictions, if a CPNS is present.

Exponential Dispersion Process for Degradation Analysis

The methods for analyzing degradation data are usually based on a specific degradation model. However, this may result in a large bias to estimate the reliability of the product when the assumed model is wrong. In this paper, a new stochastic process, called exponential dispersion process, is developed to describe the degradation path of the product's physical or chemical characteristics. Exponential dispersion process includes the most used degradation models (Wiener process, gamma process, inverse Gaussian process, and compound Poisson process) as special cases. Thus, it is useful for suggesting an appropriate degradation model for a specific dataset. We investigate the exponential dispersion model with nonlinear degradation path, and use approximated maximum likelihood method and bootstrap approach to obtain the point and confidence interval estimates of the unknown model parameters. Furthermore, exponential dispersion process models with acceleration factors and random effects are also explored. The developed methodologies are then applied to real data analysis.

Optimal dividend of compound poisson process under a stochastic interest rate

In this paper we assume the insurance wealth process is driven by the compound Poisson process. The discounting factor is modelled as a geometric Brownian motion at first and then as an exponential function of an integrated Ornstein-Uhlenbeck process. The objective is to maximize the cumulated value of expected discounted dividends up to the time of ruin. We give an explicit expression of the value function and the optimal strategy in the case of interest rate following a geometric Brownian motion. For the case of the Vasicek model, we explore some properties of the value function. Since we can not find an explicit expression for the value function in the second case, we prove that the value function is the viscosity solution of the corresponding HJB equation.

Efficient Rare-Event Simulation for Multiple Jump Events in Regularly Varying Random Walks and Compound Poisson Processes

We propose a class of strongly efficient rare-event simulation estimators for random walks and compound Poisson processes with a regularly varying increment/jump-size distribution in a general large deviations regime. Our estimator is based on an importance sampling strategy that hinges on a recently established heavy-tailed sample-path large deviations result. The new estimators are straightforward to implement and can be used to systematically evaluate the probability of a wide range of rare events with bounded relative error. They are “universal” in the sense that a single importance sampling scheme applies to a very general class of rare events that arise in heavy-tailed systems. In particular, our estimators can deal with rare events that are caused by multiple big jumps (therefore, beyond the usual principle of a single big jump) as well as multidimensional processes such as the buffer content process of a queueing network. We illustrate the versatility of our approach with several applications that arise in the context of mathematical finance, actuarial science, and queueing theory.

Infinite-server systems with Coxian arrivals

We consider a network of infinite-server queues where the input process is a Cox process of the following form: The arrival rate is a vector-valued linear transform of a multivariate generalized (i.e., being driven by a subordinator rather than a compound Poisson process) shot-noise process. We first derive some distributional properties of the multivariate generalized shot-noise process. Then these are exploited to obtain the joint transform of the numbers of customers, at various time epochs, in a single infinite-server queue fed by the above-mentioned Cox process. We also obtain transforms pertaining to the joint stationary arrival rate and queue length processes (thus facilitating the analysis of the corresponding departure process), as well as their means and covariance structure. Finally, we extend to the setting of a network of infinite-server queues.

Distribution of the time to buffer overflow in the M/G/1/N-type queueing model with batch arrivals and multiple vacation policy

A single-channel FIFO queueing model with finite buffer capacity and the multiple vacation policy is investigated, in which jobs arrive according to a compound Poisson process and are being processed individually with a general-type distribution function of the service time. A multiple vacation period, consisting of a number of independent generally-distributed server vacations, is being started each time when the system becomes empty. During this period, the processing of jobs is suspended. Successive server vacations are being initialised until at least one job is present in the buffer at the completion epoch of one of them. A compact formula for the Laplace transform of the distribution of the time to the first buffer overflow, conditioned by initial number of packets present in the buffer, is found. The analytical approach is based on the paradigm of embedded Markov chain, integral equations and Korolyuk’s potential idea. Numerical illustrating examples are attached as well.

Analysis of an M^([X])/G(a,b)/1 Unreliable G-queue with Loss, Instantaneous Bernoulli Feedback, Vacation and Two Delays of Verification

This paper deals with a batch arrival that customers arrive to the system according to a compound Poisson process. The customer’s behavior is incorporated according to which loss with a certain probability and the server begins to provide a service only when a queue size minimum say ‘a’ and maximum service capacity is ‘b’. Once the server completes the service, the unsatisfied customers may get the same service under Bernoulli schedule is termed as instantaneous Bernoulli feedback. The occurrence of negative customer cause the server to fail and removes a group of customers or an amount of work if present upon its arrival. As soon as the failure instant, the service channel send to the two delays of verification, the first verification delay starts before the repair process and the second verification delay begins after the repair process. We use the generating function method to derive the stationary queue size distribution. Some important performance measures such as different states of the system and the expected length of the queue explicitly. Some important special cases and numerical examples are determined.

Estimating the Expected Discounted Penalty Function in a Compound Poisson Insurance Risk Model with Mixed Premium Income

In this paper, we consider an insurance risk model with mixed premium income, in which both constant premium income and stochastic premium income are considered. We assume that the stochastic premium income process follows a compound Poisson process and the premium sizes are exponentially distributed. A new method for estimating the expected discounted penalty function by Fourier-cosine series expansion is proposed. We show that the estimation is easily computed, and it has a fast convergence rate. Some numerical examples are also provided to show the good properties of the estimation when the sample size is finite.