Nonhomogeneous Compound Poisson Process Research Articles

Multiple-Trigger Catastrophe Bond Pricing Model and Its Simulation Using Numerical Methods

Investor interest in single-trigger catastrophe bonds (STCB) has the potential to decline in the future. It is triggered by the increasing trend of global catastrophe loss and intensity every year, which increases the probability that a claim of STCB will occur. To increase investor interest again, the issuance of multiple-trigger catastrophe bonds (MTCB) can be one solution. However, to issue MTCB, its pricing is more complex because it involves more factors than STCB. Therefore, this study aims to design a simple MTCB pricing model. The claim trigger indices used are actual loss and fatality. Then, a nonhomogeneous compound Poisson process is used to model actual losses and fatalities aggregate to consider catastrophe intensity. In addition, this study proposes numerical methods, namely the continuous distribution approximation method and the Nuel recursive method, to facilitate the application of the model. Finally, an analysis of the effect of catastrophe intensity and other factors on MTCB prices is also presented. This study is expected to help special-purpose vehicles as MTCB issuers in MTCB pricing.

Optimal Time-Consistent Investment and Premium Control Strategies for Insurers with Constraint under the Heston Model

In this work, we study the optimal investment and premium control problem with the short-selling constraint under the mean-variance criterion. The claim process is assumed to follow the non-homogeneous compound Poisson process. The insurer invests the surplus in one risk-free asset and one risky asset described by the Heston model. Under these, we consider an optimization objective that maximizes the return (the expectation of terminal wealth) and minimizes the risk (the variance of terminal wealth). By constructing the extended Hamilton–Jacobi–Bellman (HJB) system with the dynamic programming method, the time-consistent strategies and the corresponding value function are obtained. Furthermore, we provide numerical examples to illustrate the effects of the model parameters on the optimal policies.

Modeling and Monitoring Erosion of the Leading Edge of Wind Turbine Blades

Leading edge surface erosion is an emerging issue in wind turbine blade reliability, causing a reduction in power performance, aerodynamic loads imbalance, increased noise emission, and, ultimately, additional maintenance costs, and, if left untreated, it leads to the compromise of the functionality of the blade. In this work, we first propose an empirical spatio-temporal stochastic model for simulating leading edge erosion, to be used in conjunction with aeroelastic simulations, and subsequently present a deep learning model to be trained on simulated data, which aims to monitor leading edge erosion by detecting and classifying the degradation severity. This could help wind farm operators to reduce maintenance costs by planning cleaning and repair activities more efficiently. The main ingredients of the model include a damage process that progresses at random times, across multiple discrete states characterized by a non-homogeneous compound Poisson process, which is used to describe the random and time-dependent degradation of the blade surface, thus implicitly affecting its aerodynamic properties. The model allows for one, or more, zones along the span of the blades to be independently affected by erosion. The proposed model accounts for uncertainties in the local airfoil aerodynamics via parameterization of the lift and drag coefficients’ curves. The proposed model was used to generate a stochastic ensemble of degrading airfoil aerodynamic polars, for use in forward aero-servo-elastic simulations, where we computed the effect of leading edge erosion degradation on the dynamic response of a wind turbine under varying turbulent input inflow conditions. The dynamic response was chosen as a defining output as this relates to the output variable that is most commonly monitored under a structural health monitoring (SHM) regime. In this context, we further proposed an approach for spatio-temporal dependent diagnostics of leading erosion, namely, a deep learning attention-based Transformer, which we modified for classification tasks on slow degradation processes with long sequence multivariate time-series as inputs. We performed multiple sets of numerical experiments, aiming to evaluate the Transformer for diagnostics and assess its limitations. The results revealed Transformers as a potent method for diagnosis of such degradation processes. The attention-based mechanism allows the network to focus on different features at different time intervals for better prediction accuracy, especially for long time-series sequences representing a slow degradation process.

An Adaptive Prognostic Approach for Partially Observable Degrading Products With Random Shocks

This paper proposes an adaptive remaining useful life (RUL) estimation method for partially observable degrading products with time-varying random shocks. In the modeling aspect of the proposed method, a shock degradation model is proposed to characterize the degradation process of the product, in which the continuous degradation process is described by the Wiener process, and the random shock process with a time-varying intensity is described by the non-homogeneous compound Poisson process (NHCPP). In the proposed model, to characterize the effect of random shocks on the degradation process, the degradation rate is related to the number of shocks and the magnitude of cumulative shocks. In the estimating aspect, we utilize a strong tracking filtering (STF) algorithm to estimate partially observable degradation states and a two-step expectation conditional maximization (ECM) algorithm to estimate the model parameters. In the prognostic aspect, the approximated analytical RUL distribution under the concept of the first hitting time (FHT) is obtained by incorporating the unit-to-unit variability and the uncertainty of the partially observable degradation state estimation from observations into the RUL estimation. As such, the RUL distribution can be updated according to the latest available degradation observation, thereby realizing the adaptive RUL estimation. Finally, the accuracy and effectiveness of the proposed approach is verified by a numerical example and a practical case study for the furnace wall, which provides higher estimation accuracy and better online capability than existing approaches.

Nonhomogeneus Generalisations of Poisson Process in the Modeling of Random Processes Related to Road Accidents

Abstract The stochastic processes theory provides concepts, and theorems, which allow to build the probabilistic models concerning accidents. “Counting process” can be applied for modelling the number of road, sea, and railway accidents in the given time intervals. A crucial role in construction of the models plays a Poisson process and its generalizations. The nonhomogeneous Poisson process, and the corresponding nonhomogeneous compound Poisson process are applied for modelling the road accidents number, and number of people injured and killed in Polish roads. To estimate model parameters were used data coming from the annual reports of the Polish police.

Nonhomogeneous Poisson Process and Compound Poisson Process in the Modelling of Random Processes Related to Road Accidents

Abstract The stochastic processes theory provides concepts and theorems that allow building probabilistic models concerning accidents. So called counting process can be applied for modelling the number of the road, sea and railway accidents in the given time intervals. A crucial role in construction of the models plays a Poisson process and its generalizations. The new theoretical results regarding compound Poisson process are presented in the paper. A nonhomogeneous Poisson process and the corresponding nonhomogeneous compound Poisson process are applied for modelling the road accidents number and number of injured and killed people in the Polish road. To estimate model parameters were used data coming from the annual reports of the Polish police [9, 10]. Constructed models allowed anticipating number of accidents at any time interval with a length of h and the accident consequences. We obtained the expected value of fatalities or injured and the corresponding standard deviation in the given time interval. The statistical distribution of fatalities number in a single accident and statistical distribution of injured people number and also probability distribution of fatalities or injured number in a single accident are computed. It seems that the presented examples explain basic concepts and results discussed in the paper.

Nonhomogeneous Stochastic Processes Connected to Poisson Process

Abstract Some generalizations of the Poisson process and their properties are presented in the paper. The non-homogeneous Poisson process allows to construct a probabilistic model describing the different kinds of accidents number. The nonhomogeneous compound Poisson process enables to describe mathematically the various types of accidents consequences. Theoretical results give possibility to anticipate the accidents number and their consequences.

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.

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.

Queueing model with non-homogeneous bulk arrivals having state-dependent service rates

This paper considers a single-server queueing model with non-homogeneous bulk arrivals and state-dependent service rates. The bulk arrival queueing model is important in developing optimal operating policies for queueing in transportation systems and communication systems. Arrivals are assumed to be characterised by a non-homogeneous compound Poisson process, and the service rate depends on the content of the buffer connecting to the service station. Using difference-differential equations and a probability generating function, the system’s behaviour is analysed. Explicit expressions are derived for performance measures including average number of customers in the queue and probability of queue emptiness, etc. Sensitivity analysis of the model revealed that assuming non-homogeneous arrival significantly influences the system performance measures. The state-dependent service rate strategy can control queue congestion and mean delay in transmission. This model is effective in evaluating the performance of time-dependent communication systems.

Beam Noise Characteristics

Beam-noise spectrum levels were measured with high-resolution horizontal arrays of 22 to 100 wavelengths long at frequencies between 50 and 320 Hz in the Mediterranean Sea. Results indicate that the low-frequency noise field has two major components one of which is due to the dynamic and temporally variable distant shipping. Beam noise level differences of 10 to 20 dB were observed between the resolved distant shipping and the uncorrelated background noise. The measured distributions of beam-noise-intensity fluctuations were found to be non-Gaussian. The beam-noise-intensity distribution was found to depend on aperture length, frequency, bandwidth, integration time, and steering angle. The shapes of the beam-noise-intensity cumulative distribution functions were variable and qualitatively described by nonhomogeneous compound Poisson process as shown with a simulation of the shipping-induced noise field. Temporal characteristics of beam-noise-intensity fluctuations were found to have noise level fades at 30% probability below the median noise level with durations of 1 h for observations over 10 h.

The infinite server queue with arrivals generated by a non-homogeneous compound poisson process and heterogeneous customers

We consider the infinite server queue where customers arrive in groups of various sizes according to a non-homogeneous Poisson process. The customers in each group may be of different types and also may have dependent service times. Assuming that the system is initially empty, we obtain the probability generating function of the transient distribution of the numbers of customers of various types present in the system, thus extending recent relevant results. At the same time the probability generating function of the corresponding departure counting process is derived as well.

On the non-homogeneous service system MX/G/∞

An infinite-server queueing system is studied in which units arrive in batches of variable size following a non-homogeneous compound Poisson process and each unit arriving at time t has its own arbitrary service time distribution. Using the standard properties of the Poisson process and elementary probability arguments, we derive the probability generating functions (PGF) of N(t), the number of units in the system at time t and D(t), the number of service completions up to time t or in the interval (0, t] in the transient state. Covariance functions between N(t), D(t); N(t), R(t), and R(t), D(t) are obtained separately where R(t) represents the number of arrivals in the time interval (0, t]. Expressions are obtained for the mean and variance of N(t) and D(t). Finally, some steady state results are obtained under certain conditions.

Public Works: Rational Choice of Design Period

Engineers have relied generally on judgment in selecting a design period for a project. Yet it can be argued that designers of public works owe a duty to future people to minimize constraints upon future actions to the extent that such constraints arise from present decisions. It is held that the public policy which most nearly satisfies this moral obligation to the future is to find an “economic life” over which the annual total project cost is minimized. Total cost is defined as capital cost plus all future cost throughout the design period. A mathematical model is developed, a nonhomogeneous compound Poisson process which gives optimum design period. One required parameter is the ratio between capital cost and initial annual maintenance and operation cost. The other parameter is an empirically determined exponent which accounts for the increase of annual costs with time. Parameters are estimated and examples given.

The Infinite Server Queue with Arrivals Generated by a Non-Homogeneous Compound Poisson Process

This paper considers the infinite server queue with arrivals generated by a non-homogeneous compound Poisson process. In such a system, customers arrive in groups of variable size, the arrival epochs of groups being points of a non-homogeneous Poisson process, and they are served without delay. The service times of customers who belong to the same group need not be independent nor identically distributed. Assuming that the system is initially empty, the transient distribution of the queue size and of the counting departure process is obtained. Also the limiting queue size distribution (when it exists) is determined and it is found to be insensitive to the form the service time distribution functions.

The Infinite Server Queue with Arrivals Generated by a Non-Homogeneous Compound Poisson Process

The Infinite Server Queue with Arrivals Generated by a Non-Homogeneous Compound Poisson Process

Array beam noise characterization

The formulation of the stochastic process for beam noise of a horizontal array operating at low frequency is analyzed based on the array's response to the distant ship-induced noise field. The general form of beam noise characteristic function obtained with nonhomogeneous compound Poisson process assumption for shipping distribution is related to three major operation environment parameters: the mean number of shipping density, the array aperture length and beam pattern, and random fluctuation and range dependence in the transmission loss function. This characteristic function can be used to derive the probability of observing low noise region in terms of frequency-azimuth-time representation of noise surface for some specific operation scenarios. Computer modeling for ship lane dominant noise field and uniform shipping distributed noise field are performed for investigating the sensitivity of the relevant parameters to the array's beam noise statistical property.

The occurrence of low beam noise level in ship induced noise field

The formulation of the stochastic process for beam noise of a horizontal array operating at low frequency is analyzed based on the array's response to the distant ship-induced noise field. The general form of beam noise characteristic function obtained with nonhomogeneous compound Poisson process assumption for shipping distribution is related to three major environmental parameters: the mean number of shipping density, the array aperture length and beam pattern, and random fluctuation and range dependence in the transmission loss function. This characteristic function can be used to derive the probability of observing low noise regions in terms of frequency-azimuth-time representation of noise surface for certain acoustic environments. Computer modeling for ship lane dominant noise field and uniform shipping distributed noise field were performed for investigating the relationship among the relevant parameters in the array's beamforming output. The results illustrate the sensitivity of beam noise statistical characteristics to the angular ship density.

A diffusion process model for the optimal investment strategies of an R & D project

In this article we examine an R & D project in which the project status changes according to a diffusion process. The decision variables include a resource expenditure strategy and a stopping policy which determines when the project should be terminated. The drift and the diffusion parameters of the project status process are assumed to be functions of the resource expenditure rate. The terminal reward from the project is a non-decreasing function of the project status. Our purpose is to select optimal investment strategies under the discounted return criterion. The value of the project is shown to be a solution of a second order, non-linear differential equation. Finally, we derive the optimal investment strategies for an R & D project in which the project status changes according to a non-homogeneous compound Poisson process by using diffusion approximation.

A diffusion process model for the optimal investment strategies of an R & D project

In this article we examine an R & D project in which the project status changes according to a diffusion process. The decision variables include a resource expenditure strategy and a stopping policy which determines when the project should be terminated. The drift and the diffusion parameters of the project status process are assumed to be functions of the resource expenditure rate.The terminal reward from the project is a non-decreasing function of the project status. Our purpose is to select optimal investment strategies under the discounted return criterion.The value of the project is shown to be a solution of a second order, non-linear differential equation. Finally, we derive the optimal investment strategies for an R & D project in which the project status changes according to a non-homogeneous compound Poisson process by using diffusion approximation.