A simple proof of the multivariate random time change theorem for point processes

1988 ◽  
Vol 25 (01) ◽  
pp. 210-214 ◽  
Author(s):  
Timothy C. Brown ◽  
M. Gopalan Nair
Keyword(s):  
Poisson Process ◽  
Simple Proof ◽  
Point Processes ◽  
Time Change ◽  
Random Time ◽  
Random Time Change ◽  
Special Case

A simple proof of the multivariate random time change theorem of Meyer (1971) is given. This result includes Watanabe's (1964) characterization of the Poisson process; even in this special case the present proof is simpler than existing proofs.

A simple proof of the multivariate random time change theorem for point processes

1988 ◽  
Vol 25 (1) ◽  
pp. 210-214 ◽  
Author(s):  
Timothy C. Brown ◽  
M. Gopalan Nair
Keyword(s):  
Poisson Process ◽  
Simple Proof ◽  
Point Processes ◽  
Time Change ◽  
Random Time ◽  
Random Time Change ◽  
Special Case

A simple proof of the multivariate random time change theorem of Meyer (1971) is given. This result includes Watanabe's (1964) characterization of the Poisson process; even in this special case the present proof is simpler than existing proofs.

The coincidence approach to stochastic point processes

1975 ◽  
Vol 7 (1) ◽  
pp. 83-122 ◽  
Author(s):  
Odile Macchi
Keyword(s):  
Poisson Process ◽  
Probability Space ◽  
Point Processes ◽  
Counting Process ◽  
Regular Point ◽  
General Point ◽  
Completely Regular ◽  
Photon Process ◽  
Stochastic Point Processes

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

Time-changing and truncating K-capacity queues from one K to another

1991 ◽  
Vol 28 (3) ◽  
pp. 647-655 ◽  
Author(s):  
Paul Glasserman ◽  
Wei-Bo Gong
Keyword(s):  
Queue Length ◽  
Sample Path ◽  
Time Change ◽  
Random Time ◽  
Random Time Change ◽  
Vacation Model

For , we obtain a K′- capacity queue from a K- capacity queue through a random time change and a truncation, provided arrivals are Poisson or service is exponential. In the case of an M/G/1/K queue, the time change erases service intervals that begin with more than K′ customers in the systems. This construction yields a straightforward sample path proof of Keilson's result on the proportionality of the ergodic queue length probabilities in M/G/1/K queues. The same approach proves a strengthened result for ‘detailed' state probabilities. It also reproduces a proportionality result for a vacation model, due to Keilson and Servi. A ‘dual' construction yields a different kind of proportionality for the G/M/1/K queue.

Well-posedness of Stratonovich multi-valued SDEs driven by semimartingales

2014 ◽  
Vol 14 (04) ◽  
pp. 1450005
Author(s):  
Jing Wu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Uniqueness Result ◽  
Time Change ◽  
Random Time ◽  
Approximation Technique ◽  
Well Posedness ◽  
Uniqueness Of Solutions ◽  
Random Time Change

In this paper we consider Stratonovich type multi-valued stochastic differential equations (MSDEs) driven by general semimartingales. Based on an existence and uniqueness result for MSDEs with respect to continuous semimartingales, we apply the random time change and approximation technique to prove existence and uniqueness of solutions to Stratonovich type multi-valued SDEs driven by general semimartingales with summable jumps.

An order statistic characterization of the poisson renewal process

1985 ◽  
Vol 22 (3) ◽  
pp. 717-722 ◽  
Author(s):  
Uri Liberman
Keyword(s):  
Poisson Process ◽  
Order Statistic ◽  
Renewal Process ◽  
Point Processes

Using the characterization of point processes having the order statistic property we prove that the only renewal process that has the order statistic property is the Poisson process.

Dynamical systems defined on point processes

1998 ◽  
Vol 35 (03) ◽  
pp. 581-588
Author(s):  
Laurence A. Baxter
Keyword(s):  
Stochastic Process ◽  
Poisson Process ◽  
Point Processes ◽  
Linear Boltzmann Equation ◽  
Semidynamical System ◽  
The Mean ◽  
Mean Function ◽  
Necessary And Sufficient ◽  
The Relationship

This paper introduces a new stochastic process in which the iterates of a dynamical system evolving in discrete time coincide with the events of a Poisson process. The autocovariance function of the stochastic process is studied and a necessary and sufficient condition for it to vanish is deduced. It is shown that the mean function of this process comprises a continuous-time semidynamical system if the underlying dynamical map is linear. The flow of probability density functions generated by the stochastic process is analysed in detail, and the relationship between the flow and the solutions of the linear Boltzmann equation is investigated. It is shown that the flow is a semigroup if and only if the point process defining the stochastic process is Poisson, thereby providing a new characterization of the Poisson process.

A framework of BSDEs with stochastic Lipschitz coefficients

2020 ◽  
Vol 24 ◽  
pp. 739-769
Author(s):  
Hun O ◽  
Mun-Chol Kim ◽  
Chol-Kyu Pak
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Large Class ◽  
Lipschitz Condition ◽  
Time Change ◽  
Random Time ◽  
Effective Technique ◽  
Random Time Change ◽  
Monotone Coefficients ◽  
Terminal Time

In this paper, we suggest an effective technique based on random time-change for dealing with a large class of backward stochastic differential equations (BSDEs for short) with stochastic Lipschitz coefficients. By means of random time-change, we show the relation between the BSDEs with stochastic Lipschitz coefficients and the ones with bounded Lipschitz coefficients and stopping terminal time, so they are possible to be exchanged with each other from one type to another. In other words, the stochastic Lipschitz condition is not essential in the context of BSDEs with random terminal time. Using this technique, we obtain a couple of new results of BSDEs with stochastic Lipschitz (or monotone) coefficients.

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