Poisson process compound

3 Markov CTRW Models 3.3.1 Compound Poisson Process 

If the counting process N(t) is a Poisson process with the transition rate k, then the particle position 

Note that the Poisson process plays a very important role in random walk theory. It can be defined in two ways (1) as a continuous-time Markov chain with constant intensity, i.e., as a pure birth process with constant birth rate k (2) as a renewal process. In the latter case, it can be represented as (3.25) with T = Here 

Since the waiting time PDF (/ (f) is exponential, the random walk X(t) is a Markov process. The jump PDF w(z) is defined in (3.9). 

The mesoscopic particle density p(x, t) obeys the integro-differential equation 

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A process with independent increments can be generated by compounding Poisson processes in the following way. Take a random set of dots on the time axis forming shot noise as in (II.3.14) the density fx will now be called p. Define a process Z(t) by stipulating that, at each dot, Z jumps by an amount z (positive or negative), which is random with probability density w(z). Clearly the increment of Z between t and t + T is independent of previous history and its probability distribution has the form (IV.4.7). It is easy to compute. 

On comparing this expression with (6.4) one sees that the compound Poisson process defined here is the same as Z(t) in (6.3) if... 

Exercise. Argue that our compound Poisson process may be regarded as a superposition of simple Poisson processes as defined in IV.2. 

Exercise. Show that the random walk is a compound Poisson process. 

Exercise. Show that the Wiener process can be obtained as the limiting case of a compound Poisson process when the jumps become infinitely small, but infinitely dense on the time axis. Compare the Remark in 5. 

We have only shown that a compound Poisson process is at the same time a process with independent increments. The converse is proved in feller ii, p. 204. 

Now the probability p that fibril failure has occurred somewhere within the film square may be constructed using arguments appropriate to the compound Poisson process and is given by ... 

The integro-differential equation (3.74) can be derived in several ways. The following is probably the most instructive in the context of transport theory. Since a compound Poisson process is Markovian, the derivation of (3.74) is based on the idea that the particle density at time i -i- can be expressed in terms of the density... 

For illustrative purposes we begin with the transport of particles that follow the path of the compound Poisson process (3.71), X t) = Z,. The corresponding... 

In the previous two sections we gave a brief account of the compound Poisson process and the symmetric a-stable Levy process. This section is an introduction to general one-dimensional Ldvy processes. The compound Poisson process and symmetric a-stable process are simply examples of Markov processes of Ldvy type. Readers who are interested in this topic in greater detail are referred to the books by Applebaum [15] and Sato [378]. 

The compound Poisson process X (t), defined by (3.71), is an example of a pure jump process which has only a finite number of jumps on the finite time interval [0, t]. The Levy measure v(dz) = A,u (z)dz is finite on R, that is, v(dz) = X < 00. Note that v is not a probability measure, because /g v(dz) = X. The characteristic function for the compound Poisson process is... 

UX = const, w(z) does not depend on x, and v(x) = 0, we obtain the Kolmogorov-Feller equation (3.74), for which the underlying microscopic random movement is a compound Poisson process. 

Consider particles that follow a CTRW, such that the random time T between jumps is exponentially distributed with rate X, f(T > t) = exp(—A.t). The mean-field equation for the particle density is the Master equation for the compound Poisson process with logistic growth (5.2), Hyperbolic scaling yields... 

ABSTRACT In an earlier paper, we studied several approaches for modeling the accumulated severity of railway accidents in a certain time interval with the help of a compound Poisson process. The number of accidents can be modeled with a Homogeneous Poisson Process. The distribution of accident severity, however, has not been studied in detail until now. 

In an earlier paper, we modeled the accident statistics of railways with the help of a Compound Poisson Process (CPP), see Braband Schabe (2011), Braband Schabe (2013b). 

In this paper, we have given a theoretically well founded derivation of the distribution for the jump height of the compound Poisson process which can be used to describe counts of railway accidents. The distribution has been derived from the well-known F-N curve and turns out to be the Pareto distribution. We have used standard statistical textbooks to provide estimation and testing methods. In a practical example, we have derived the shape parameter that falls into an interval for parameters usually used for F-N curves. The distribution type, however, may in fact be more complex due to discretisation effects due the definition of FWI. This needs further research in order to come up with a distribution which fits the data and allows the derivation of optimal tests. 

Braband, I, Schabe, H., 2013. Comparison of compound Poisson processes as a general approach towards efficient evaluation of railway safety. Proceedings of the ESREL (2013), Safety, Reliability and Risk Analysis Beyond the Horizon—Steenbergen et al. (Eds) 2014 Taylor Francis Group, London, ISBN 978-1-138-00123-7), pp. 1297-1302. 

The parameters of the compound Poisson process are expressed in terms of the physics or failure of the polymer. 

Insurance models in actuarial sciences and financial engineering have a long history of development and wide applications in assessing underwriting risks. Popular approaches for characterizing catastrophic risk processes include a compound Poisson process (e.g., Rolski... 

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