Compound Cox Processes Research Articles

Evolution of statistical properties of microturbulence during transient process under electron cyclotron resonance heating of the L-2M stellarator plasma

The formalism of compound Cox processes was used for statistical analysis of turbulent density fluctuations of plasma during a transient process at electron cyclotron resonance heating of L-2M stellarator plasmas. Short-wavelength (k = 20–30 cm−1) density fluctuations in the central region of the plasma column were measured by the collective backscattering technique. The transient process was caused by sputtering of wall coating and subsequent pulsed impurity injection into the plasma after a step-like increase of the heating power. The evolution of the following parameters was studied: probability density function (PDF) and dynamic and diffusive components of variance of density fluctuations increments. It was found that density fluctuations increments increase simultaneously with average electron density increase and electron temperature decrease during the transient process. It was shown that the PDF of fluctuation increments is characterized by the existence of heavy tails and it can be represented as a composition of four normal laws. Deviation from normal law that happens in the form of random bursts was observed. Duration and frequency of the bursts increases with the level of fluctuation increments and this increase is accompanied by the increase of both components of the variance.

Forecasting counting and time statistics of compound Cox processes: a focus on intensity phase type process, deletions and simultaneous events

Compound Cox processes (CCP) are flexible marked point processes due to the stochastic nature of their intensity. This paper states closed-form expressions of their counting and time statistics in terms of the intensity and of the mean processes. They are forecast by means of principal components prediction models applied to the mean process in order to reach attainable results. A proposition proves that only weak restrictions are needed to estimate the probability of a new occurrence. Additionally, the phase type process is introduced, which important feature is that its marginal distributions are phase type with random parameters. Since any non-negative variable can be approximated by a phase-type distribution, the new stochastic process is proposed to model the intensity process of any point process. The CCP with this type of intensity provides an especially general model. Several simulations and the corresponding study of the estimation errors illustrate the results and their accuracy. Finally, an application to real data is performed; extreme temperatures in the South of Spain are modeled by a CPP and forecast.

A regime-switching model with jumps and its application to bond pricing and insurance

In this paper, we consider a Markovian, regime-switching model with jumps and its application to bond pricing and insurance. The jumps in the model are described by a compound Cox process, where the arrival intensity of the counting number-process follows a regime-switching shot noise process. Using a martingale method, we derive exponential-affine form expressions for the price of a zero-coupon bond and the joint Laplace transform of the aggregate accumulated claims and the arrival intensity.

A Note on Functional Limit Theorems for Compound Cox Processes*

An improved and corrected version of the functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes (compound Cox processes) to Levy processes in the Skorokhod space under more realistic moment conditions. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Levy processes with variance-mean mixed normal distributions, in particular, to stable Levy processes, generalized hyperbolic and generalized variance-gamma Levy processes.

On a Spectral Analysis and Modeling of Non-Gaussian Processes in the Structural Plasma Turbulence*

The paper presents an empirical approach to the analysis of broadband Fourier-spectra of the low-frequency structural plasma turbulence based on a priori assumptions concerning the number of processes and the Gaussian form of the spectral components. This type of turbulence in toroidal plasma devices is described by the mathematical model of non-stationary continuous-time random walk, namely compound doubly stochastic Poisson process or compound Cox process. Measurements of low-frequency (in ranges of frequencies to 5 MHz) long-wave (3..6 cm) fluctuations spectra were obtained at the plasma edge in the L-2M stellarator by the technique of Doppler reflectometry. Under some experimental conditions, the efficiency of the methodology was demonstrated. The spectra were successfully decomposed into a few components. There are more than two harmonics in broadband Fourier-spectra besides a harmonic connected with poloidal rotation of the plasma. Those additional harmonics correspond to fluctuations rotating in the directions of electron and ion drift in toroidal plasma devices. In all modes it was possible to reveal the components corresponding to the poloidal rotation of the plasma (which could be defined by a radial electric field) and phase velocity of the two types of structural turbulence (which could be defined by electron gradient and ion gradient instabilities). The suggested description of probabilistic and spectral characteristics of low-frequency plasma turbulence allow us to formulate the correct problem of characterization of structural turbulence by a system of stochastic differential equations. The equations of the system should involve stochastic processes with densities in the form of mixtures of probability distributions. Such a comprehensive approach assumes a correct comparison of different structural turbulence models (caused by drift dissipative, ion-sound, gradient instabilities, etc.) with characteristics of the obtained stochastic processes.

Isotropic Multiple Scattering Processes on Hyperspheres

This paper presents several results about isotropic random walks and multiple scattering processes on hyperspheres ${\mathbb S}^{p-1}$ . It allows one to derive the Fourier expansions on ${\mathbb S}^{p-1}$ of these processes. A result of unimodality for the multiconvolution of symmetrical probability density functions on ${\mathbb S}^{p-1}$ is also introduced. Such processes are then studied in the case where the scattering distribution is von Mises–Fisher (vMF). Asymptotic distributions for the multiconvolution of vMFs on ${\mathbb S}^{p-1}$ are obtained. Both Fourier expansion and asymptotic approximation allow us to compute estimation bounds for the parameters of compound cox processes on ${\mathbb S}^{p-1}$ .

Modeling High-Frequency Non-Homogeneous Order Flows by Compound Cox Processes*

A micro-scale model is proposed for the evolution of a limit order book in modern high-frequency trading applications. Within this model, order flows are described by doubly stochastic Poisson processes (also called Cox processes) taking account of the stochastic character of the intensities of order flows. The models for the number of order imbalance (NOI) process and order flow imbalance (OFI) process are introduced as two-sided risk processes that are special compound Cox processes. These processes are sensitive indicators of the current state of the limit order book since time intervals between events in a limit order book are usually so short that price changes are relatively infrequent events. Therefore price changes provide a very coarse and limited description of market dynamics at time micro-scales. NOI and OFI processes track best bid and ask queues and change much faster than prices. They incorporate information about build-ups and depletions of order queues and they can be used to interpolate market dynamics between price changes and to track the toxicity of order flows. The proposed multiplicative model of stochastic intensities makes it possible to analyze the characteristics of the order flows as well as the instantaneous proportion of the forces of buyers and sellers without modeling the external information background. The proposed model gives the opportunity to link the micro-scale high-frequency dynamics of the limit order book with the macroscale models of stock price processes of the form of subordinated Wiener processes by means of limit theorems for special random sums and hence, to give a deeper insight in the nature of popular models of statistical regularities of the evolution of characteristics of financial markets such as generalized hyperbolic distributions and other normal variance-mean mixtures.

Convergence of nonhomogeneous random walks generated by compound Cox processes to generalized variance-gamma Lévy processes

Functional limit theorems on the convergence of nonhomogeneous random walks generated by compound Cox processes to Levy processes with generalized one-dimensional variance-gamma distributions, in particular, to subordinate Wiener processes with subordinator being a Levy–Weibull process, are proved.

Application of Compound Cox Processes In Modeling Order Flows with Non-Homogeneous Intensities

A micro-scale model is proposed for the evolution of a limit order book in modern high-frequency trading applications. Within this model, order flows are described by doubly stochastic Poisson processes (also called Cox processes) taking account of the stochastic character of the intensities of order flows. The models for the number of orders imbalance (NOI) process and order flow imbalance (OFI) process are introduced as two-sided risk processes which are special compound Cox processes. These processes are sensitive indicators of the current state of the limit order book since time intervals between events in a limit order book are usually so short that price changes are relatively infrequent events. Therefore price changes provide a very coarse and limited description of market dynamics at time micro-scales. NOI and OFI processes track best bid and ask queues and change much faster than prices. They incorporate information about build-ups and depletions of order queues and they can be used to interpolate market dynamics between price changes and to track the toxicity of order flows. The proposed multiplicative model of stochastic intensities makes it possible to analyze the characteristics of the order flows as well as the instantaneous proportion of the forces of buyers and sellers without modeling the external information background. The proposed model gives the opportunity to link the micro-scale high-frequency dynamics of the limit order book with the macro-scale models of stock price processes of the form of subordinated Wiener processes by means of limit theorems for special random sums and hence, to give a deeper insight in the nature of popular models of statistical regularities of the evolution of characteristics of financial markets such as generalized hyperbolic distributions and other normal variance-mean mixtures.

On convergence of random walks generated by compound Cox processes to Lévy processes

A functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes to Lévy processes in the Skorokhod space. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Lévy processes with mixed normal distributions, and in particular, to stable Lévy processes.

Limit theorems for small deviation probabilities of some iterated Stochastic processes

We derive logarithmic asymptotics for probabilities of small deviations for some iterated processes. We show that under appropriate conditions, these asymptotics are the same as those of the processes which generate the iterated processes. When these conditions do not hold, the asymptotics of small deviations for iterated processes are quite different. We apply our results to the iterated processes generated by the compound Cox processes and compound renewal processes. Bibliography: 10 titles.

Measuring Tail Dependence for Aggregate Collateral Losses Using Bivariate Compound Cox Process with Shot Noise Intensity

A catastrophic event such as flood, storm, hail, bushfire and earthquake brings about damages in properties, motors and interruption of businesses collaterally. Also a couple of losses incurred collaterally from the World Trade Centre (WTC) catastrophe, Hurricane Katrina and Victorian Bushfire. However it has not been developed a suitable model for insurance companies either to measure tail dependence between these collateral losses or relevant risk measures that can be used as insurance risk premiums. The first aim of this paper is to measure tail dependence between collateral losses as insurance industry is more concerned with dependence between extreme losses. The second is to calculate conditional probabilities and conditional expectations as relevant risk measures. To achieve these aims, we use bivariate compound process where a Cox process with shot noise intensity is used to count collateral losses from catastrophic events. Homogeneous Poisson process is also examined as its counterpart for the case where the catastrophic loss frequency rate is deterministic. Using a member of Farlie-Gumbel-Morgenstern copula with exponential margins, we derive explicit expressions of joint Laplace transforms of aggregate collateral losses. Fast Fourier transform is used to obtain the joint distributions of aggregate collateral losses, with which we calculate relevant risk measures. The figures of the joint distributions of collateral losses, their contours and numerical calculations of risk measures are provided.

On asymptotic behavior for probabilities of large deviations of compound Cox processes

We derive logarithmic asymptotics for probabilities of large deviations for compound Cox processes. We show that under appropriate conditions, these asymptotics are the same as those for sums of independent random variables and processes with independent increments. When these conditions fail, the asymptotics of large deviations probabilities for compound Cox processes are quite different. Bibliography: 5 titles.

On probabilities of small deviations for compound cox processes

We derive logarithmic asymtotics for probabilities of small deviations for compound Cox processes. We show that under appropriate conditions, these asymptotics are the same as those for sums of independent random variables and processes with independent increments. When these conditions do not hold, the asymptotics of small deviations for compound Cox processes are quite different. Bibliography: 9 titles.

Functional approach to the random mean of a compound Cox process

The parametric process and counting statistics of a marked point process whose marks belong to a given subset of the mark space of a compound Cox process are derived in this paper by means of functional data analysis. They are illustrated by means of an example and simulation study with different intensity processes for the CCP.

Jump diffusion processes and their applications in insurance and finance

For insurance risks, jump processes such as homogeneous/non-homogeneous compound Poisson processes and compound Cox processes have been used to model aggregate losses. If we consider the economic assumption of a positive interest to aggregate losses, Lévy processes have proven to be useful. Also in financial modelling, it has been observed that diffusion models are not robust enough to capture the appearance of jumps in underlying asset prices and interest rates. As a result, jump diffusion processes, which are, simply speaking, combinations of compound Poisson processes with Brownian motion, have gained popularity for modelling in insurance and finance. In this paper, considering a jump diffusion process, we obtain the explicit expression of the joint Laplace transform of the distribution of a jump diffusion process and its integrated process, assuming that jump size follows the mixture of two exponential distributions, which is a special case of phase-type distributions. Based on this Laplace transform, we derive the moments of the aggregate accumulated claim amounts of insurance risk. For a financial application, we concern non-defaultable zero-coupon bond pricing. We also provide several numerical examples for the moments of aggregate accumulated claims and default-free zero-coupon bond prices.

Optimal investment for investors with state dependent income, and for insurers

Abstract. An optimal control problem is considered where a risky asset is used for investment, and this investment is financed by initial wealth as well as by a state dependent income. The objective function is accumulated discounted expected utility of wealth, where the utility function is nondecreasing and bounded. This problem is investigated for constant as well as for stochastic discount rate, where the stochastic model is a time homogeneous finite state Markov process. We prove that the Bellman equation to this optimization problem has a classical solution and give a verification argument. Based on this we deal with the problem of optimal investment for an insurer with an insurance business modelled by a compound Poisson or a compound Cox process, under the presence of constant as well as (finite state space Markov) stochastic interest rate.

Compound Cox Processes and Option Pricing

Compound Cox Processes and Option Pricing

Asymptotic Properties of Extrema of Compound Cox Processes and Their Applications to Some Problems of Financial Mathematics

Necessary and sufficient conditions are presented for the weak convergence of one-dimensional distributions of extrema of compound doubly stochastic Poisson processes whose jumps have zero expectation and finite variance. Convergence rate estimates are given. The obtained results are applied to the problem of prediction of stock prices.

The asymptotic behavior of extrema of compound Cox processes with nonzero means

We give necessary and sufficient conditions for the weak convergence of one-dimensional distributions of extrema of compound Cox processes with jumps having finite variances and nonzero expectations. No moment-type restrictions are imposed on the controlling process. As corollaries we obtain criteria of the normal convergence of extrema of compound Cox processes with nonzero means. Convergence rate estimates are also presented.