Abstract
The main purpose of this chapter is to provide a martingale characterization of the Poisson process obtained in Watanabe (1964). This will be aided by the development of a special stochastic calculus that exploits its non-decreasing, right-continuous, step-function sample path structure when viewed as a counting process; i.e., for which stochastic integrals can be defined in terms of standard Lebesgue integration theory.
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Notes
- 1.
A more comprehensive treatment of point processes from a martingale perspective is given in Brémaud (1981).
- 2.
- 3.
See BCPT p. 228, for Lebesgue–Stieltjes measure and integration.
- 4.
See Feller (1971), p. 430.
References
Brémaud P (1981) Point processes and queues: martingale dynamics. Springer, Berlin.
Le Cam L (1960b) A stochastic description of precipitation. In: Neyman J (ed)Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, vol III, pp165–186.
Sun Y, Stein ML (2015) A stochastic space-time model for intermittent precipitation occurrences. Ann Appl Statist 9(4):2110–2132.
Watanabe S (1964) On discontinuous additive functionals and Levy measures of a Markov process. Japanese J Math 34:53–70.
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Bhattacharya, R., Waymire, E.C. (2021). Stochastic Calculus for Point Processes and a Martingale Characterization of the Poisson Process. In: Random Walk, Brownian Motion, and Martingales. Graduate Texts in Mathematics, vol 292. Springer, Cham. https://doi.org/10.1007/978-3-030-78939-8_15
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