Class Of Point Processes Research Articles

A stochastic model for evaluating the peaks of commodities' returns

AbstractThis paper proposes a probabilistic model for the evaluation of the peak components of the return of a commodity. The ground of the study lies in the evidence that the spikes in the returns are due to the shocks occurring in the external environment. We follow an approach based on a particular class of point processes—the Spatial Mixed Poisson Processes—by exploiting an invariance property for such a class. The theoretical framework is used for presenting an estimation the procedure of the returns based on the available information. An empirical instance based on different commodities' returns and the abnormal returns of the volatility index as external shocks are presented to motivate our theoretical approach.

Fractional non-homogeneous Poisson and Pólya-Aeppli processes of order k and beyond

We introduce two classes of point processes: a fractional non-homogeneous Poisson process of order k and a fractional non-homogeneous Pólya-Aeppli process of order k. We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property.

Efficient simulation of Lévy-driven point processes

AbstractIn this paper, we introduce a new large family of Lévy-driven point processes with (and without) contagion, by generalising the classical self-exciting Hawkes process and doubly stochastic Poisson processes with non-Gaussian Lévy-driven Ornstein–Uhlenbeck-type intensities. The resulting framework may possess many desirable features such as skewness, leptokurtosis, mean-reverting dynamics, and more importantly, the ‘contagion’ or feedback effects, which could be very useful for modelling event arrivals in finance, economics, insurance, and many other fields. We characterise the distributional properties of this new class of point processes and develop an efficient sampling method for generating sample paths exactly. Our simulation scheme is mainly based on the distributional decomposition of the point process and its intensity process. Extensive numerical implementations and tests are reported to demonstrate the accuracy and effectiveness of our scheme. Moreover, we use portfolio risk management as an example to show the applicability and flexibility of our algorithms.

The endo–exo problem in high frequency financial price fluctuations and rejecting criticality

The endo–exo problem lies at the heart of statistical identification in many fields of science, and is often plagued by spurious strong-and-long memory due to improper treatment of trends, shocks and shifts in the data. A class of models that has shown to be useful in discerning exogenous and endogenous activity is the Hawkes process. This class of point processes has enjoyed great recent popularity and rapid development within the quantitative finance literature, with particular focus on the study of market microstructure and high frequency price fluctuations. We show that there are important lessons from older fields like time series and econometrics that should also be applied in financial point process modelling. In particular, we emphasize the importance of appropriately treating trends and shocks for the identification of the strength and length of memory in the system. We exploit the powerful Expectation Maximization algorithm and objective statistical criteria (BIC) to select the flexibility of the deterministic background intensity. With these methods, we strongly reject the hypothesis that the considered financial markets are critical at univariate and bivariate microstructural levels.

Stable random fields indexed by finitely generated free groups

In this work, we investigate the extremal behaviour of left-stationary symmetric $\alpha$-stable (S$\alpha$S) random fields indexed by finitely generated free groups. We begin by studying the rate of growth of a sequence of partial maxima obtained by varying the indexing parameter of the field over balls of increasing size. This leads to a phase-transition that depends on the ergodic properties of the underlying nonsingular action of the free group but is different from what happens in the case of S$\alpha$S random fields indexed by $\mathbb{Z}^{d}$. The presence of this new dichotomy is confirmed by the study of a stable random field induced by the canonical action of the free group on its Furstenberg–Poisson boundary with the measure being Patterson–Sullivan. This field is generated by a conservative action but its maxima sequence grows as fast as the i.i.d. case contrary to what happens in the case of $\mathbb{Z}^{d}$. When the action of the free group is dissipative, we also establish that the scaled extremal point process sequence converges weakly to a novel class of point processes that we have termed as randomly thinned cluster Poisson processes. This limit too is very different from that in the case of a lattice.

The Endo-Exo Problem in High Frequency Financial Price Fluctuations and Rejecting Criticality

The endo-exo problem lies at the heart of statistical identi fication in many fields of science, and is often plagued by spurious strong-and-long memory due to improper treatment of trends, shocks and shifts in the data. A class of models that has shown to be useful in discerning exogenous and endogenous activity is the Hawkes process. This class of point processes has enjoyed great recent popularity and rapid development within the quantitative fi nance literature, with particular focus on the study of market microstructure and high frequency price fluctuations. We show that there are important lessons from older fi elds like time series and econometrics that should also be applied in fi nancial point process modelling. In particular, we emphasize the importance of appropriately treating trends and shocks for the identi fication of the strength and length of memory in the system. We exploit the powerful Expectation Maximization (EM) algorithm and objective statistical criteria (BIC) to select the flexibility of the deterministic background intensity. With these methods, we strongly reject the hypothesis that the considered financial markets are critical at univariate and bivariate microstructural levels.

Mixing properties and central limit theorem for associated point processes

Positively (resp. negatively) associated point processes are a class of point processes that induce attraction (resp. inhibition) between the points. As an important example, determinantal point processes (DPPs) are negatively associated. We prove $\alpha $-mixing properties for associated spatial point processes by controlling their $\alpha $-coefficients in terms of the first two intensity functions. A central limit theorem for functionals of associated point processes is deduced, using both the association and the $\alpha $-mixing properties. We discuss in detail the case of DPPs, for which we obtain the limiting distribution of sums, over subsets of close enough points of the process, of any bounded function of the DPP. As an application, we get the asymptotic properties of the parametric two-step estimator of some inhomogeneous DPPs.

A Gillespie Algorithm for Non-Markovian Stochastic Processes

The Gillespie algorithm provides statistically exact methods for simulating stochastic dynamics modelled as interacting sequences of discrete events including systems of biochemical reactions or earthquake occurrences, networks of queuing processes or spiking neurons, and epidemic and opinion formation processes on social networks. Empirically, the inter-event times of various phenomena obey long-tailed distributions. The Gillespie algorithm and its variants either assume Poisson processes (i.e., exponentially distributed inter-event times), use particular functions for time courses of the event rate, or work for non-Poissonian renewal processes, including the case of long-tailed distributions of inter-event times, but at a high computational cost. In the present study, we propose an innovative Gillespie algorithm for renewal processes on the basis of the Laplace transform. The algorithm makes use of the fact that a class of point processes is represented as a mixture of Poisson processes with different event rates. The method is applicable to multivariate renewal processes whose survival function of inter-event times is completely monotone. It is an exact algorithm and works faster than a recently proposed Gillespie algorithm for general renewal processes, which is exact only in the limit of infinitely many processes. We also propose a method to generate sequences of event times with a tunable amount of positive correlation between inter-event times. We demonstrate our algorithm with exact simulations of epidemic processes on networks, finding that a realistic amount of positive correlation in inter-event times only slightly affects the epidemic dynamics.

Capacity and Error Exponents of Stationary Point Processes under Random Additive Displacements

Consider a real-valued discrete-time stationary and ergodic stochastic process, called the noise process. For each dimension n, we can choose a stationary point process in ℝ n and a translation invariant tessellation of ℝ n . Each point is randomly displaced, with a displacement vector being a section of length n of the noise process, independent from point to point. The aim is to find a point process and a tessellation that minimizes the probability of decoding error, defined as the probability that the displaced version of the typical point does not belong to the cell of this point. We consider the Shannon regime, in which the dimension n tends to ∞, while the logarithm of the intensity of the point processes, normalized by dimension, tends to a constant. We first show that this problem exhibits a sharp threshold: if the sum of the asymptotic normalized logarithmic intensity and of the differential entropy rate of the noise process is positive, then the probability of error tends to 1 with n for all point processes and all tessellations. If it is negative then there exist point processes and tessellations for which this probability tends to 0. The error exponent function, which denotes how quickly the probability of error goes to 0 in n, is then derived using large deviations theory. If the entropy spectrum of the noise satisfies a large deviations principle, then, below the threshold, the error probability goes exponentially fast to 0 with an exponent that is given in closed form in terms of the rate function of the noise entropy spectrum. This is obtained for two classes of point processes: the Poisson process and a Matérn hard-core point process. New lower bounds on error exponents are derived from this for Shannon's additive noise channel in the high signal-to-noise ratio limit that hold for all stationary and ergodic noises with the above properties and that match the best known bounds in the white Gaussian noise case.

Capacity and Error Exponents of Stationary Point Processes under Random Additive Displacements

Consider a real-valued discrete-time stationary and ergodic stochastic process, called the noise process. For each dimensionn, we can choose a stationary point process in ℝnand a translation invariant tessellation of ℝn. Each point is randomly displaced, with a displacement vector being a section of lengthnof the noise process, independent from point to point. The aim is to find a point process and a tessellation that minimizes the probability of decoding error, defined as the probability that the displaced version of the typical point does not belong to the cell of this point. We consider the Shannon regime, in which the dimensionntends to ∞, while the logarithm of the intensity of the point processes, normalized by dimension, tends to a constant. We first show that this problem exhibits a sharp threshold: if the sum of the asymptotic normalized logarithmic intensity and of the differential entropy rate of the noise process is positive, then the probability of error tends to 1 withnfor all point processes and all tessellations. If it is negative then there exist point processes and tessellations for which this probability tends to 0. The error exponent function, which denotes how quickly the probability of error goes to 0 inn, is then derived using large deviations theory. If the entropy spectrum of the noise satisfies a large deviations principle, then, below the threshold, the error probability goes exponentially fast to 0 with an exponent that is given in closed form in terms of the rate function of the noise entropy spectrum. This is obtained for two classes of point processes: the Poisson process and a Matérn hard-core point process. New lower bounds on error exponents are derived from this for Shannon's additive noise channel in the high signal-to-noise ratio limit that hold for all stationary and ergodic noises with the above properties and that match the best known bounds in the white Gaussian noise case.

Some Classes of Point Processes

We dene four classes of point processes which we call A, B, *A, *B. Although we study point processes on R; these classes are suitable for generalizations for point processes on Rm and other measure metric spaces. The main result is the equivalence of classes *A and *B for point processes on R: As a matter of fact, we prove that A B A = B S; where S is the class of simple processes. We also relate these classes and the class of Poisson processes.

A model of point configurations given by a semiparametric interaction

A model of a point process specified by a density with respect to the Poisson measure is developed. The model proposed in the paper is rather flexible because of the choice of the interaction function in a semiparametric form. This allows us to simulate point patterns with various structural properties. Standard analytical properties of the process and its relations to known classes of point processes are discussed.

Extremal versus additive Matérn point processes

In the simplest Matern point processes one retains certain points of a Poisson point process in such a way that no pairs of points are at distance less than a threshold. This condition can be reinterpreted as a threshold condition on an extremal shot---noise field associated with the Poisson point process. This paper is focused on the case where one retains points that satisfy a threshold condition based on an additive shot---noise field of the Poisson point process. We provide an analytical characterization of the intensity of this class of point processes and we compare the packing obtained by the extremal and additive schemes and certain combinations thereof.

Stochastic order for alpha-permanental point processes

We establish relations of stochastic comparison among point processes elements of the set of alpha-permanental point processes. This set contains in particular, the determinantal point processes, the Poisson point processes and the permanental point processes. We show that these three classes of point processes can be ordered according to the increasing stochastic order. Elementary particles provide illustrations of some of the obtained relations of stochastic comparison.

Bound analysis of closed queueing networks with workload burstiness

Burstiness and temporal dependence in service processes are often found in multi-tier architectures and storage devices and must be captured accurately in capacity planning models as these features are responsible of significant performance degradations. However, existing models and approximations for networks of first-come first-served (FCFS) queues with general independent (GI) service are unable to predict performance of systems with temporal dependence in workloads. To overcome this difficulty, we define and study a class of closed queueing networks where service times are represented by Markovian Arrival Processes (MAPs), a class of point processes that can model general distributions, but also temporal dependent features such as burstiness in service times. We call these models MAP queueing networks. We introduce provable upper and lower bounds for arbitrary performance indexes (e.g., throughput, response time, utilization) that we call Linear Reduction (LR) bounds. Numerical experiments indicate that LR bounds achieve a mean accuracy error of 2 percent. The result promotes LR bounds as a versatile and reliable bounding methodology of the performance of modern computer systems.

Integrated Volatility and UHF-GARCH Models: A Comparison Using High Frequency Financial Data

A very promising literature has been recently devoted to the modeling of ultra-high-frequency (UHF) data. The main problem encountered to measure volatility dynamics of high-frequency intra-daily data lies in the irregularity at which observations arrive. Engle and Russell (1998) have proposed a new class of point processes with dependent arrival rates: autoregressive conditional duration (ACD) point processes. Engle (2000) has generalized this approach and developed a Ultra-High-Frequency GARCH model. At the same time, Andersen and Bollerslev (1998) and Bollerslev and Wright (2001) have studied the volatility of UHF financial data by introducing the concept of integrated volatility. This integrated volatility is measured by the squared value of intra-daily returns. Our first aim in this paper is to develop an empirical application of ACD GARCH models in forecasting future volatilities. Then we propose another contribution in comparing the performances of ACD GARCH models with the integrated volatility. Therefore we propose a procedure to take into account the calendar effect or the time of day effect. We introduce an adjusted duration to calculate volatilities before comparing volatility forecasts. We show that the models induced by the two approaches lead to poor performances. Nevertheless, our results show that the integrated volatility method tends to outperform UHF-GARCH models.

Cluster processes: a natural language for network traffic

We introduce a new approach to the modeling of network traffic, consisting of a semi-experimental methodology combining models with data and a class of point processes (cluster models) to represent the process of packet arrivals in a physically meaningful way. Wavelets are used to examine second-order statistics, and particular attention is paid to the modeling of long-range dependence and to the question of scale invariance at small scales. We analyze in depth the properties of several large traces of packet data and determine unambiguously the influence of network variables such as arrival patterns, durations, and volumes of transport control protocol (TCP) flows and internal flow structure. We show that session-level modeling is not relevant at the packet level. Our findings naturally suggest the use of cluster models. We define a class where TCP flows are directly modeled, and each model parameter has a direct meaning in network terms, allowing the model to be used to predict traffic properties as networks and traffic evolve. The class has the key advantage of being mathematically tractable, in particular, its spectrum is known and can be readily calculated, its wavelet spectrum deduced, interarrival distributions can be obtained, and it can be simulated in a straightforward way. The model reproduces the main second-order features, and results are compared against a simple black box point process alternative. Discrepancies with the model are discussed and explained, and enhancements are outlined. The elephant and mice view of traffic flows is revisited in the light of our findings.

Point processes with finite-dimensional conditional probabilities

We study the structure of point processes N with the property that the P (θ t N∈· | F t ) vary in a finite-dimensional space where θ t is the shift and F t the σ -field generated by the counting process up to time t . This class of point processes is strictly larger than Neuts’ class of Markovian arrival processes. On the one hand, it allows for more general features like interarrival distributions which are matrix-exponential rather than phase type, on the other the probabilistic interpretation is a priori less clear. Nevertheless, the properties are very similar. In particular, finite-dimensional distributions of interarrival times, moments, Laplace transforms, Palm distributions, etc., are shown to be given by two fundamental matrices C , D just as for the Markovian arrival process. We also give a probabilistic interpretation in terms of a piecewise deterministic Markov process on a compact convex subset of R p , whose jump times are identical to the epochs of N .

On connected component Markov point processes

We note some interesting properties of the class of point processes which are Markov with respect to the ‘connected component’ relation. Results in the literature imply that this class is closed under random translation and independent cluster generation with almost surely non-empty clusters. We further prove that it is closed under superposition. A wide range of examples is also given.

On connected component Markov point processes

We note some interesting properties of the class of point processes which are Markov with respect to the ‘connected component’ relation. Results in the literature imply that this class is closed under random translation and independent cluster generation with almost surely non-empty clusters. We further prove that it is closed under superposition. A wide range of examples is also given.