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Abstract
The Poisson process is the most elementary continuous-time stochastic process that models a stream of repeating events. It is uniquely characterised by a single parameter called the rate. Instead of a single value for this rate, we here consider a rate interval and let it characterise two nested sets of stochastic processes. We call these two sets of stochastic process imprecise Poisson processes, explain why this is justified, and study the corresponding lower and upper (conditional) expectations. Besides a general theoretical framework, we also provide practical methods to compute lower and upper (conditional) expectations of functions that depend on the number of events at a single point in time.
Keywords
Poisson process, counting process, continuous-time Markov chain, imprecision

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MLA
Erreygers, Alexander, and Jasper De Bock. “First Steps towards an Imprecise Poisson Process.” Proceedings of the Eleventh International Symposium on Imprecise Probabilities : Theories and Applications, ISIPTA 2019, edited by Jasper De Bock et al., vol. 103, PMLR, 2019, pp. 175–84.
APA
Erreygers, A., & De Bock, J. (2019). First steps towards an imprecise Poisson process. In J. De Bock, C. P. de Campos, G. De Cooman, E. Quaeghebeur, & G. Wheeler (Eds.), Proceedings of the Eleventh International Symposium on Imprecise Probabilities : Theories and Applications, ISIPTA 2019 (Vol. 103, pp. 175–184). Thagaste, Ghent, Belgium: PMLR.
Chicago author-date
Erreygers, Alexander, and Jasper De Bock. 2019. “First Steps towards an Imprecise Poisson Process.” In Proceedings of the Eleventh International Symposium on Imprecise Probabilities : Theories and Applications, ISIPTA 2019, edited by Jasper De Bock, Cassio P. de Campos, Gert De Cooman, Erik Quaeghebeur, and Gregory Wheeler, 103:175–84. PMLR.
Chicago author-date (all authors)
Erreygers, Alexander, and Jasper De Bock. 2019. “First Steps towards an Imprecise Poisson Process.” In Proceedings of the Eleventh International Symposium on Imprecise Probabilities : Theories and Applications, ISIPTA 2019, ed by. Jasper De Bock, Cassio P. de Campos, Gert De Cooman, Erik Quaeghebeur, and Gregory Wheeler, 103:175–184. PMLR.
Vancouver
1.
Erreygers A, De Bock J. First steps towards an imprecise Poisson process. In: De Bock J, de Campos CP, De Cooman G, Quaeghebeur E, Wheeler G, editors. Proceedings of the Eleventh International Symposium on Imprecise Probabilities : Theories and Applications, ISIPTA 2019. PMLR; 2019. p. 175–84.
IEEE
[1]
A. Erreygers and J. De Bock, “First steps towards an imprecise Poisson process,” in Proceedings of the Eleventh International Symposium on Imprecise Probabilities : Theories and Applications, ISIPTA 2019, Thagaste, Ghent, Belgium, 2019, vol. 103, pp. 175–184.
@inproceedings{8623800,
  abstract     = {{The Poisson process is the most elementary continuous-time stochastic process that models a stream of repeating events. It is uniquely characterised by a single parameter called the rate. Instead of a single value for this rate, we here consider a rate interval and let it characterise two nested sets of stochastic processes. We call these two sets of stochastic process imprecise Poisson processes, explain why this is justified, and study the corresponding lower and upper (conditional) expectations. Besides a general theoretical framework, we also provide practical methods to compute lower and upper (conditional) expectations of functions that depend on the number of events at a single point in time.}},
  author       = {{Erreygers, Alexander and De Bock, Jasper}},
  booktitle    = {{Proceedings of the Eleventh International Symposium on Imprecise Probabilities : Theories and Applications, ISIPTA 2019}},
  editor       = {{De Bock, Jasper and de Campos, Cassio P. and De Cooman, Gert and Quaeghebeur, Erik and Wheeler, Gregory}},
  issn         = {{2640-3498}},
  keywords     = {{Poisson process,counting process,continuous-time Markov chain,imprecision}},
  language     = {{eng}},
  location     = {{Thagaste, Ghent, Belgium}},
  pages        = {{175--184}},
  publisher    = {{PMLR}},
  title        = {{First steps towards an imprecise Poisson process}},
  url          = {{http://proceedings.mlr.press/v103/}},
  volume       = {{103}},
  year         = {{2019}},
}