Compound Poisson Process Research Articles

Bayesian analysis of compound Poisson process with change-point

The compound Poisson process is considered to model the frequency and the magnitude of the earthquake occurrences concurrently. Nevertheless, there are many debates on whether climate change influences the frequency of the natural disasters. In this study, we propose a compound Poisson process with change-point (CPPCP) model to fit the data with two-phase pattern. The hierarchical Bayesian method is employed via assigning a common distribution for the unit-specific parameters. For comparison purpose, we also develop the maximum-likelihood method. The simulation study illustrates the applicability of our proposed model and the validity of the hierarchical Bayesian method. In the analysis of the earthquake data, CPPCP model outperforms the quadratic linear regression model and the hierarchical Bayesian method is superior to the maximum-likelihood method in terms of the model fitting and prediction.

Application of compound Poisson process for modelling of ore flow in a belt conveyor system with cyclic loading

This paper deals with the analysis of the random process of cyclic loading of a mining belt conveyor with portions of ore discharged by loaders or trucks. Such transfer of transported ore from a cyclic to a continuous transport is typical for the specific mining operations implemented in the underground copper ore mines with room and pillar mining. The conveyors in such systems are usually significantly oversized in order to match the peak loads of cumulated discharges of ore hauled from mining fields by loaders. Therefore, the actual loadings that occur in the mining transportation systems need to be analysed to provide the data for more accurate design and control of belt conveyors. The large data-set of actual loadings of the belt conveyors has been used for the stochastic modelling of the analysed process. The compound Poisson process has been proposed as mathematical tool to analyse/describe properties of ore flow. The discussion of the chosen distribution functions and results of the fitted model simulations compared with the examined measurement data are presented. In this paper, the case of a belt conveyor loaded only by a single feeding point where loaders are randomly discharged have been analysed. More complex cases (several loading points, mixed ore supply from cyclic and continuous ore mass flow from preceding conveyors or ore bunkers) are under investigation and will be presented in the future.

Compound option pricing under a double exponential Jump-diffusion model

A compound option, an option on another option, plays an important role in financial field since it can be used to price American option and corporate debt with discrete coupons. In the real options literature, compound options are most suitable to be employed to investment problems involving sequential decision making. Most compound option and real options formulae are based on log-normal distribution while the empirical evidence shows that the return distribution in real market exhibits asymmetric leptokurtic feature, higher peak and two heavier tails. This paper introduces the jump-diffusion process into pricing compound options and derives the related valuation formulas. We assume that the dynamic of the underlying asset return process consists of a drift component, a continuous Wiener process and discontinuous jump-diffusion processes which have jump times that follow the compound Poisson process and the logarithm of jump size follows the double exponential distribution proposed by Kou (2002). Numerical results indicate that the advantage of combining the double exponential distribution and normal distribution is that it can capture the phenomena of both the asymmetric leptokurtic features and the volatility smile. Furthermore, the compound options under the double exponential jump diffusion model which we derived are more generalized than those proposed by Gukhal (2004) and Geske (1979), and thus have wider application.

Optimal reinsurance in a compound Poisson risk model with dependence

This paper considers the problem of optimal reinsurance in a compound Poisson risk model with dependent classes of insurance business. It is assumed that the risk process in each class follows a compound Poisson process, and that all classes are correlated due to the so-called thinning-dependence structure. Under the criterion of maximizing the adjustment coefficient, methods for finding the optimal reinsurance strategies are discussed for both the expected value premium principle and the variance premium principle. Numerical examples are also provided to illustrate the impact of the model parameters on the optimal reinsurance strategies.

Atomic clocks and the continuous-time random-walk

Atomic clocks play a fundamental role in many fields, most notably they generate Universal Coordinated Time and are at the heart of all global navigation satellite systems. Notwithstanding their excellent timekeeping performance, their output frequency does vary: it can display deterministic frequency drift; diverse continuous noise processes result in nonstationary clock noise (e.g., random-walk frequency noise, modelled as a Wiener process), and the clock frequency may display sudden changes (i.e., “jumps”). Typically, the clock’s frequency instability is evaluated by the Allan or Hadamard variances, whose functional forms can identify the different operative noise processes. Here, we show that the Allan and Hadamard variances of a particular continuous-time random-walk, the compound Poisson process, have the same functional form as for a Wiener process with drift. The compound Poisson process, introduced as a model for observed frequency jumps, is an alternative to the Wiener process for modelling random walk frequency noise. This alternate model fits well the behavior of the rubidium clocks flying on GPS Block-IIR satellites. Further, starting from jump statistics, the model can be improved by considering a more general form of continuous-time random-walk, and this could bring new insights into the physics of atomic clocks.

Convergence rates of Laplace-transform based estimators

This paper considers the problem of estimating probabilities of the form $\mathbb{P}(Y\leq w)$, for a given value of $w$, in the situation that a sample of i.i.d. observations $X_{1},\ldots,X_{n}$ of $X$ is available, and where we explicitly know a functional relation between the Laplace transforms of the non-negative random variables $X$ and $Y$. A plug-in estimator is constructed by calculating the Laplace transform of the empirical distribution of the sample $X_{1},\ldots,X_{n}$, applying the functional relation to it, and then (if possible) inverting the resulting Laplace transform and evaluating it in $w$. We show, under mild regularity conditions, that the resulting estimator is weakly consistent and has expected absolute estimation error $O(n^{-1/2}\log(n+1))$. We illustrate our results by two examples: in the first we estimate the distribution of the workload in an M/G/1 queue from observations of the input in fixed time intervals, and in the second we identify the distribution of the increments when observing a compound Poisson process at equidistant points in time (usually referred to as “decompounding”).

Compound Poisson Law for Hitting Times to Periodic Orbits in Two-Dimensional Hyperbolic Systems

We show that a compound Poisson distribution holds for scaled exceedances of observables $\phi$ uniquely maximized at a periodic point $\zeta$ in a variety of two-dimensional hyperbolic dynamical systems with singularities $(M,T,\mu)$, including the billiard maps of Sinai dispersing billiards in both the finite and infinite horizon case. The observable we consider is of form $\phi (z)=-\ln d(z,\zeta)$ where $d$ is a metric defined in terms of the stable and unstable foliation. The compound Poisson process we obtain is a P\'olya-Aeppli distibution of index $\theta$. We calculate $\theta$ in terms of the derivative of the map $T$. Furthermore if we define $M_n=\max\{\phi,\ldots,\phi\circ T^n\}$ and $u_n (\tau)$ by $\lim_{n\to \infty} n\mu (\phi >u_n (\tau) )=\tau$ the maximal process satisfies an extreme value law of form $\mu (M_n \le u_n)=e^{-\theta \tau}$. These results generalize to a broader class of functions maximized at $\zeta$, though the formulas regarding the parameters in the distribution need to be modified.

Optimal investment and premium control in a nonlinear diffusion model

This paper considers the optimal investment and premium control problem in a diffusion approximation to a non-homogeneous compound Poisson process. In the nonlinear diffusion model, it is assumed that there is an unspecified monotone function describing the relationship between the safety loading of premium and the time-varying claim arrival rate. Hence, in addition to the investment control, the premium rate can be served as a control variable in the optimization problem. Specifically, the problem is investigated in two cases: (i) maximizing the expected utility of terminal wealth, and (ii) minimizing the probability of ruin respectively. In both cases, some properties of the value functions are derived, and closed-form expressions for the optimal policies and the value functions are obtained. The results show that the optimal investment policy and the optimal premium control policy are dependent on each other. Most interestingly, as an example, we show that the nonlinear diffusion model reduces to a diffusion model with a quadratic drift coefficient when the function associated with the premium rate and the claim arrival rate takes a special form. This example shows that the model of study represents a class of nonlinear stochastic control risk model.

Complex and Entropy of Fluctuations of Agent-Based Interacting Financial Dynamics with Random Jump

This paper investigates the complex behaviors and entropy properties for a novel random complex interacting stock price dynamics, which is established by the combination of stochastic contact process and compound Poisson process, concerning with stock return fluctuations caused by the spread of investors’ attitudes and random jump fluctuations caused by the macroeconomic environment, respectively. To better understand the fluctuation complex behaviors of the proposed price dynamics, the entropy analyses of random logarithmic price returns and corresponding absolute returns of simulation dataset with different parameter set are preformed, including permutation entropy, fractional permutation entropy, sample entropy and fractional sample entropy. We found that a larger λ or γ leads to more complex dynamics, and the absolute return series exhibit lower complex dynamics than the return series. To verify the rationality of the proposed compound price model, the corresponding analyses of actual market datasets are also comparatively preformed. The empirical results verify that the proposed price model can reproduce some important complex dynamics of actual stock markets to some extent.

A Cointegrated Regime-Switching Model Approach with Jumps Applied to Natural Gas Futures Prices

Energy commodities and their futures naturally show cointegrated price movements. However, there is empirical evidence that the prices of futures with different maturities might have, e.g., different jump behaviours in different market situations. Observing commodity futures over time, there is also evidence for different states of the underlying volatility of the futures. In this paper, we therefore allow for cointegration of the term structure within a multi-factor model, which includes seasonality, as well as joint and individual jumps in the price processes of futures with different maturities. The seasonality in this model is realized via a deterministic function, and the jumps are represented with thinned-out compound Poisson processes. The model also includes a regime-switching approach that is modelled through a Markov chain and extends the class of geometric models. We show how the model can be calibrated to empirical data and give some practical applications.

Point processes subordinated to compound Poisson processes

Point processes subordinated to compound Poisson processes

MODELLING THE ENERGY RELEASE PROCESS OF AFTERSHOCKS

A stochastic model for the study of Benioff strain release during aftershock sequences is suggested. The stochastic model is elaborated after a compound Poisson process and is applied on data of the M7.1 Ocober 18, 1989 Loma Prieta aftershock sequence in northern California, USA. The temporal evolution of the number of events is first modelled by the Restricted Epidemic Type Aftershock Sequence (RETAS) model and then the identified best fit model is incorporated in the energy release analysis. The suggested model is based on the assumptions that there is no relation between the magnitude and the occurrence time of an event first and second, that there is no relation between the magnitude of a certain event and magnitudes of previous events. The obtained results from the examination of the energy release reveal that the suggested model makes a good fit of the aftershock Benioff strain release and enables a more detailed study by identifying possible deviations between data and model. The real cumulative energy release values surpass the expected model ones, which proves that aftershocks, stronger than forecasted by the model, are clustered at the beginning of the Loma Prieta sequence.

Optimal control of a large dam using time-inhomogeneous Markov chains with an application to flood control

We consider a decision making framework for a dam modelled by a controllable time-inhomogeneous Markov chain with a compound Poisson process as an input, which approximates the inflow pattern of real dams with seasonal rainfall. The controls are responsible for output flow explicitly and implicitly via a price on water consumption. We consider the problem with system balance and demand shortfall performance criteria under constraints. The state space of the dam is discretized while the time domain remains continuous. We also add a controllable water release to reduce the probability of the dam overflowing and give the procedure for determining if an optimal solution exists by showing the consistency of constraints.

The large deviation principle for a compound Poisson process

For a compound Poisson process, under the moment Cramer condition, the extended large deviation principle is established in the space of functions of bounded variation with the Borovkov metric.

Optimal reinsurance and investment in a jump-diffusion financial market with common shock dependence

In this paper, we study the optimal reinsurance and investment problem in a financial market with jump-diffusion risky asset. It is assumed that the insurance risk model is modulated by a compound Poisson process, and that the jumps in both the risky asset and insurance risk process are correlated through a common shock. Under the criterion of maximizing the expected exponential utility, we adopt a nonstandard approach to examine the existence and uniqueness of the optimal strategy. Using the technique of stochastic control theory, closed-form expressions for the optimal strategy and the value function are derived not only for the expected value principle but also for the variance premium principle. Also, we investigate the effect of the common shock parameter as well as some other important parameters on the optimal strategies. In particular, a numerical example shows that the optimal investment strategy decreases as the degree of common shock dependence increases but the optimal insurance retention level does not behave the same.

Beating the omega clock: An optimal stopping problem with random time-horizon under spectrally negative Lévy models

We study the optimal stopping of an American call option in a random time-horizon under exponential spectrally negative Lévy models. The random time-horizon is modeled as the so-called Omega default clock in insurance, which is the first time when the occupation time of the underlying Lévy process below a level $y$, exceeds an independent exponential random variable with mean $1/q>0$. We show that the shape of the value function varies qualitatively with different values of $q$ and $y$. In particular, we show that for certain values of $q$ and $y$, some quantitatively different but traditional up-crossing strategies are still optimal, while for other values we may have two disconnected continuation regions, resulting in the optimality of two-sided exit strategies. By deriving the joint distribution of the discounting factor and the underlying process under a random discount rate, we give a complete characterization of all optimal exercising thresholds. Finally, we present an example with a compound Poisson process plus a drifted Brownian motion.

Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein–Uhlenbeck process with Lévy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein–Uhlenbeck process, where in particular we must restrict to operator-valued Lévy processes with “non-decreasing paths”. It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this class of stochastic volatility dynamics. Under a strong commutativity condition between the covariance operator of the Wiener process and the stochastic volatility, we can derive an analytical expression for the characteristic functional of the Ornstein–Uhlenbeck process perturbed by stochastic volatility if the noises are independent. The case of operator-valued compound Poisson processes as driving noise in the volatility is discussed as a particular example of interest. We apply our results to futures prices in commodity markets, where we discuss our proposed stochastic volatility model in light of ambit fields.

Analytic Loss Distributional Approach Models for Operational Risk from the -Stable Doubly Stochastic Compound Processes and Implications for Capital Allocation

Under the Basel II standards, the Operational Risk (OpRisk) advanced measurement approach is not prescriptive regarding the class of statistical model utilised to undertake capital estimation. It has however become well accepted to utlise a Loss Distributional Approach (LDA) paradigm to model the individual OpRisk loss process corresponding to the Basel II Business line/event type. In this paper we derive a novel class of doubly stochastic α-stable family LDA models. These models provide the ability to capture the heavy tailed loss process typical of OpRisk whilst also providing analytic expressions for the compound process annual loss density and distributions as well as the aggregated compound process annual loss models. In particular we develop models of the annual loss process in two scenarios. The first scenario considers the loss process with a stochastic intensity parameter, resulting in an inhomogeneous compound Poisson processes annually. The resulting arrival process of losses under such a model will have independent counts over increments within the year. The second scenario considers discretization of the annual loss process into monthly increments with dependent time increments as captured by a Binomial process with a stochastic probability of success changing annually. Each of these models will be coupled under an LDA framework with heavy-tailed severity models comprised of α-stable severities for the loss amounts per loss event. In this paper we will derive analytic results for the annual loss distribution density and distribution under each of these models and study their properties.

Impact of Insurance for Operational Risk: Is It Worthwhile to Insure or Be Insured for Severe Losses?

Under the Basel II standards, the Operational Risk (OpRisk) advanced measurement approach allows a provision for reduction of capital as a result of insurance mitigation of up to 20%. This paper studies different insurance policies in the context of capital reduction for a range of extreme loss models and insurance policy scenarios in a multi-period, multiple risk settings. A Loss Distributional Approach (LDA) for modelling of the annual loss process, involving homogeneous compound Poisson processes for the annual losses, with heavy-tailed severity models comprised of α-stable severities is considered. There has been little analysis of such models to date and it is believed, insurance models will play more of a role in OpRisk mitigation and capital reduction in future. The first question of interest is when would it be equitable for a bank or financial institution to purchase insurance for heavy-tailed OpRisk losses under different insurance policy scenarios? The second question pertains to Solvency II and addresses quantification of insurer capital for such operational risk scenarios. Considering fundamental insurance policies available, in several two risk scenarios, we can provide both analytic results and extensive simulation studies of insurance mitigation for important basic policies. The intention being to address questions related to VaR reduction under Basel II, SCR under Solvency II and fair insurance premiums in OpRisk for different extreme loss scenarios. In the process we provide closed-form solutions for the distribution of loss process and claims process in an LDA structure as well as closed-form analytic solutions for the Expected Shortfall, SCR and MCR under Basel II and Solvency II. We also provide closed-form analytic solutions for the annual loss distribution of multiple risks including insurance mitigation.

Lifetime prognostics for deteriorating systems with time-varying random jumps

In this paper, we propose a jump diffusion process with non-homogeneous compound Poisson process to model the degradation process with randomly occurring jumps, which combines two stochastic processes, i.e., traditional diffusion process to describe the continuous degradation and non-homogeneous compound Poisson process to depict random jumps with a time-varying intensity. The approximated analytical lifetime under the concept of the first passage time (FPT) is obtained by a time–space transformation technique. To identify the model parameters, we first present a general method based on Maximum Likelihood Estimation (MLE) for the proposed model, and then specifically provide a two-step approach for linear jump diffusion process via combining MLE and Expectation Conditional Maximization (ECM) algorithm. Finally, a numerical example and a study on the furnace wall are provided to illustrate the effectiveness of the proposed method.