scholarly journals A GENERALIZED CONTAGION PROCESS WITH AN APPLICATION TO CREDIT RISK

2017 ◽  
Vol 20 (01) ◽  
pp. 1750003 ◽  
Author(s):  
ANGELOS DASSIOS ◽  
HONGBIAO ZHAO
Keyword(s):  
Credit Risk ◽  
Point Process ◽  
Point Processes ◽  
Probability Generating Function ◽  
Jump Processes ◽  
Hawkes Process ◽  
Cox Process ◽  
Model Framework ◽  
Special Cases ◽  
The Laplace Transform

We introduce a class of analytically tractable jump processes with contagion effects by generalizing the classical Hawkes process. This model framework combines the characteristics of three popular point processes in the literature: (1) Cox process with CIR intensity; (2) Cox process with Poisson shot-noise intensity; (3) Hawkes process with exponentially decaying intensity. Hence, it can be considered as a self-exciting and externally-exciting point process with mean-reverting stochastic intensity. Essential probabilistic properties such as moments, the Laplace transform of intensity process, and the probability generating function of point process as well as some important asymptotics have been derived. Some special cases and a method for change of measure are discussed. This point process may be applicable to modeling contagious arrivals of events for various circumstances (such as jumps, transactions, losses, defaults, catastrophes) in finance, insurance and economics with both endogenous and exogenous risk factors within one framework. More specifically, these exogenous factors could contain relatively short-lived shocks and long-lasting risk drivers. We make a simple application to calculate the default probability for credit risk and to price defaultable zero-coupon bonds.

scholarly journals Thinning spatial point processes into Poisson processes

2010 ◽  
Vol 42 (02) ◽  
pp. 347-358 ◽  
Author(s):  
Jesper Møller ◽  
Frederic Paik Schoenberg
Keyword(s):  
Point Process ◽  
Process Model ◽  
Goodness Of Fit ◽  
Point Processes ◽  
Death Process ◽  
Cox Process ◽  
Birth And Death Process ◽  
Conditional Intensity ◽  
Spatial Point Processes ◽  
Spatial Point

In this paper we describe methods for randomly thinning certain classes of spatial point processes. In the case of a Markov point process, the proposed method involves a dependent thinning of a spatial birth-and-death process, where clans of ancestors associated with the original points are identified, and where we simulate backwards and forwards in order to obtain the thinned process. In the case of a Cox process, a simple independent thinning technique is proposed. In both cases, the thinning results in a Poisson process if and only if the true Papangelou conditional intensity is used, and, thus, can be used as a graphical exploratory tool for inspecting the goodness-of-fit of a spatial point process model. Several examples, including clustered and inhibitive point processes, are considered.

A limit theorem for point processes by applications

1978 ◽  
Vol 15 (04) ◽  
pp. 726-747
Author(s):  
Prem S. Puri
Keyword(s):  
Limit Theorem ◽  
Point Process ◽  
Random Vector ◽  
Arbitrary Function ◽  
Point Processes ◽  
Probability Generating Function ◽  
Mild Conditions ◽  
Dimensional Distribution ◽  
Image Position

Let 0 ≦ T 1 ≦ T 2 ≦ ·· · represent the epochs in time of occurrences of events of a point process N(t) with N(t) = sup{k : Tk ≦ t}, t ≧ 0. Besides certain mild conditions on the process N(t) (see Conditions (A1)– (A3) in the text) we assume that for every k ≧ 1, as t →∞, the vector (t – TN (t), t – TN (t)–1, · ··, t – TN (t)–k+1) converges in law to a k-dimensional distribution which coincides with that of a random vector ξ k = (ξ 1, · ··, ξ k ) necessarily satisfying P(0 ≦ ξ 1 ≦ ξ 2 ≦ ·· ·≦ ξ k) = 1. Let R(t) be an arbitrary function defined for t ≧ 0, satisfying 0 ≦ R(t) ≦ 1, ∀0 ≦ t <∞, and certain mild conditions (see Conditions (B1)– (B4) in the text). Then among other results, it is shown that The paper also deals with conditions under which the limit (∗) will be positive. The results are applied to several point processes and to the situations where the role of R(t) is taken over by an appropriate transform such as a probability generating function, where conditions are given under which the limit (∗) itself will be a transform of an honest distribution. Finally the results are applied to the study of certain characteristics of the GI/G/∞ queue apparently not studied before.

scholarly journals Multivariate General Compound Point Processes in Limit Order Books

Risks ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 98
Author(s):  
Qi Guo ◽  
Bruno Remillard ◽  
Anatoliy Swishchuk
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Hawkes Process ◽  
Limit Order ◽  
Large Numbers ◽  
Order Books ◽  
Limit Order Books ◽  
Functional Central Limit Theorems ◽  
Multivariate Point ◽  
Functional Central Limit

In this paper, we focus on a new generalization of multivariate general compound Hawkes process (MGCHP), which we referred to as the multivariate general compound point process (MGCPP). Namely, we applied a multivariate point process to model the order flow instead of the Hawkes process. The law of large numbers (LLN) and two functional central limit theorems (FCLTs) for the MGCPP were proved in this work. Applications of the MGCPP in the limit order market were also considered. We provided numerical simulations and comparisons for the MGCPP and MGCHP by applying Google, Apple, Microsoft, Amazon, and Intel trading data.

scholarly journals A dynamic contagion process

2011 ◽  
Vol 43 (3) ◽  
pp. 814-846 ◽  
Author(s):  
Angelos Dassios ◽  
Hongbiao Zhao
Keyword(s):  
Point Process ◽  
Probability Generating Function ◽  
Dependence Structure ◽  
Mathematical Framework ◽  
Exponential Distributions ◽  
Piecewise Deterministic Markov Process ◽  
The Laplace Transform ◽  
Analytic Expressions ◽  
Exogenous Factors ◽  
Contagion Process

We introduce a new point process, the dynamic contagion process, by generalising the Hawkes process and the Cox process with shot noise intensity. Our process includes both self-excited and externally excited jumps, which could be used to model the dynamic contagion impact from endogenous and exogenous factors of the underlying system. We have systematically analysed the theoretical distributional properties of this new process, based on the piecewise-deterministic Markov process theory developed in Davis (1984), and the extension of the martingale methodology used in Dassios and Jang (2003). The analytic expressions of the Laplace transform of the intensity process and the probability generating function of the point process have been derived. An explicit example of specified jumps with exponential distributions is also given. The object of this study is to produce a general mathematical framework for modelling the dependence structure of arriving events with dynamic contagion, which has the potential to be applicable to a variety of problems in economics, finance, and insurance. We provide an application of this process to credit risk, and a simulation algorithm for further industrial implementation and statistical analysis.

Thinning of Point Processes—Martingale Method

1990 ◽  
Vol 4 (1) ◽  
pp. 117-129 ◽  
Author(s):  
Shengwu He ◽  
Jiagang Wang
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Classical Problem ◽  
Arbitrary Point ◽  
Affirmative Answer ◽  
Poisson Processes ◽  
Jump Processes ◽  
Martingale Method ◽  
Distribution Law ◽  
Nonhomogeneous Poisson Processes

By using the martingale method, we show that thinning of an arbitrary point process produces independent thinned processes if and only if the original point process is (nonhomogeneous) Poisson. Thinning is a classical problem for point processes. It is well-known that independent homogeneous Poisson processes result from constant Bernoulli thinnings of homogeneous Poisson processes. In fact, the conclusion remains true for nonconstant Bernoulli thinnings of nonhomogeneous Poisson processes. processes. In fact, the conclusion remains true for nonconstant Bernoulli thinnings of nonhomogeneous Poisson processes. It is easy to see that the converse is also true, i.e., if the thinned processes are independent nonhomogeneous Poisson processes, so are the original processes. But if we only suppose the thinned processes are independent, nothing is concerned with their distribution law, the problem of whether or not the original processes are (nonhomogeneous) Poisson becomes interesting and challenging, which is the objective of this paper. It is considerably surprising for us to arrive at the affirmative answer. So far as this problem is concerned, the most work was done under the renewal assumption. For example, Bremaud [1] showed that for arbitrary delayed renewal processes, the existence of a pair of uncorrelated thinned processes is sufficient to guarantee that the original process is Poisson. It is natural that the mathematical tools to solve the problem in this case be typical ones for renewal theory, such as renewal equations and Laplace-Stieltjes transformations. Obviously, they are not available for nonrenewal processes. We find out that the martingale method is the most efficient one to solve this problem in the general case. More precisely, we use mainly the dual predictable projections of point processes. In fact, the distribution law of a point process is determined uniquely by its dual predictable projection (see [5]), and Ma [6] offered us a very useful criterion of independence of jump processes having no common jump time through their dual predictable projections. Based on these results, it is not a long way to arrive at the destination.

Point processes generated by transitions of Markov chains

1973 ◽  
Vol 5 (2) ◽  
pp. 262-286 ◽  
Author(s):  
Mats Rudemo
Keyword(s):  
Markov Chain ◽  
Point Process ◽  
Point Processes ◽  
Queueing System ◽  
Initial Distribution ◽  
Recurrence Time ◽  
Asynchronous Sampling ◽  
Doubly Stochastic ◽  
Special Cases ◽  
Forward Recurrence Time

For a continuous time Markov chain the time points of transitions, belonging to a subset of the set of all transitions, are observed. Special cases include the point process generated by all transitions and doubly stochastic Poisson processes with a Markovian intensity. Equations are derived for the conditional distribution of the state of the Markov chain, given observations of the point process. This distribution may be used for prediction. For the forward recurrence time of the point process, distributions corresponding to synchronous and asynchronous sampling are also derived. The Palm distribution for the point process is specified in terms of the corresponding initial distribution for the Markov chain. In examples the point processes of arrivals and departures in a queueing system are studied. Two biological applications deal with estimation of population size and detection of epidemics.

Algorithmically deconstructing shot locations as a method for shot quality in hockey

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Devan G. Becker ◽  
Douglas G. Woolford ◽  
Charmaine B. Dean
Keyword(s):  
Point Process ◽  
Process Model ◽  
Point Processes ◽  
Learning Algorithm ◽  
Basis Functions ◽  
Cox Process ◽  
Spatial Point Processes ◽  
Professional Basketball ◽  
Spatial Point ◽  
Log Gaussian Cox Process

AbstractSpatial point processes have been successfully used to model the relative efficiency of shot locations for each player in professional basketball games. Those analyses were possible because each player makes enough baskets to reliably fit a point process model. Goals in hockey are rare enough that a point process cannot be fit to each player’s goal locations, so novel techniques are needed to obtain measures of shot efficiency for each player. A Log-Gaussian Cox Process (LGCP) is used to model all shot locations, including goals, of each NHL player who took at least 500 shots during the 2011–2018 seasons. Each player’s LGCP surface is treated as an image and these images are then used in an unsupervised statistical learning algorithm that decomposes the pictures into a linear combination of spatial basis functions. The coefficients of these basis functions are shown to be a very useful tool to compare players. To incorporate goals, the locations of all shots that resulted in a goal are treated as a “perfect player” and used in the same algorithm (goals are further split into perfect forwards, perfect centres and perfect defence). These perfect players are compared to other players as a measure of shot efficiency. This analysis provides a map of common shooting locations, identifies regions with the most goals relative to the number of shots and demonstrates how each player’s shot location differs from scoring locations.

scholarly journals Thinning spatial point processes into Poisson processes

2010 ◽  
Vol 42 (2) ◽  
pp. 347-358 ◽  
Author(s):  
Jesper Møller ◽  
Frederic Paik Schoenberg
Keyword(s):  
Point Process ◽  
Process Model ◽  
Goodness Of Fit ◽  
Point Processes ◽  
Death Process ◽  
Cox Process ◽  
Birth And Death Process ◽  
Conditional Intensity ◽  
Spatial Point Processes ◽  
Spatial Point

In this paper we describe methods for randomly thinning certain classes of spatial point processes. In the case of a Markov point process, the proposed method involves a dependent thinning of a spatial birth-and-death process, where clans of ancestors associated with the original points are identified, and where we simulate backwards and forwards in order to obtain the thinned process. In the case of a Cox process, a simple independent thinning technique is proposed. In both cases, the thinning results in a Poisson process if and only if the true Papangelou conditional intensity is used, and, thus, can be used as a graphical exploratory tool for inspecting the goodness-of-fit of a spatial point process model. Several examples, including clustered and inhibitive point processes, are considered.

scholarly journals EXPLICIT COMPUTATIONS FOR A FILTERING PROBLEM WITH POINT PROCESS OBSERVATIONS WITH APPLICATIONS TO CREDIT RISK

2011 ◽  
Vol 25 (3) ◽  
pp. 393-418 ◽  
Author(s):  
Vincent Leijdekker ◽  
Peter Spreij
Keyword(s):  
Credit Risk ◽  
Point Process ◽  
Generating Function ◽  
Moment Generating Function ◽  
Cox Process ◽  
Risk Modeling ◽  
Conditional Moment ◽  
Credit Risk Modeling ◽  
Conditional Moment Generating Function ◽  
Filtering Problem

We consider the filtering problem for a doubly stochastic Poisson or Cox process, where the intensity follows the Cox–Ingersoll–Ross model. In this article we assume that the Brownian motion, which drives the intensity, is not observed. Using filtering theory for point process observations, we first derive the dynamics for the intensity and its moment-generating function, given the observations of the Cox process. A transformation of the dynamics of the conditional moment-generating function allows us to solve in closed form the filtering problem, between the jumps of the Cox process as well as at the jumps, which constitutes the main contribution of the article. Assuming that the initial distribution of the intensity is of the Gamma type, we obtain an explicit solution to the filtering problem for all t>0. We conclude the article with the observation that the resulting conditional moment-generating function at time t, after Nt jumps, corresponds to a mixture of Nt+1 Gamma distributions. Currently, the model that we analyze has become popular in credit risk modeling, where one uses the intensity-based approach for the modeling of default times of one or more companies. In this approach, the default times are defined as the jump times of a Cox process. In such a model, one only has access to observations of the Cox process, and thus filtering comes in as a natural technique in credit risk modeling.

Point processes generated by transitions of Markov chains

1973 ◽  
Vol 5 (02) ◽  
pp. 262-286 ◽  
Author(s):  
Mats Rudemo
Keyword(s):  
Markov Chain ◽  
Point Process ◽  
Point Processes ◽  
Queueing System ◽  
Initial Distribution ◽  
Recurrence Time ◽  
Asynchronous Sampling ◽  
Doubly Stochastic ◽  
Special Cases ◽  
Forward Recurrence Time

For a continuous time Markov chain the time points of transitions, belonging to a subset of the set of all transitions, are observed. Special cases include the point process generated by all transitions and doubly stochastic Poisson processes with a Markovian intensity. Equations are derived for the conditional distribution of the state of the Markov chain, given observations of the point process. This distribution may be used for prediction. For the forward recurrence time of the point process, distributions corresponding to synchronous and asynchronous sampling are also derived. The Palm distribution for the point process is specified in terms of the corresponding initial distribution for the Markov chain. In examples the point processes of arrivals and departures in a queueing system are studied. Two biological applications deal with estimation of population size and detection of epidemics.

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