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Hypothesis testing and Skorokhod stochastic integration

Part of: Special processes Markov processes Stochastic analysis Parametric inference

Published online by Cambridge University Press:  14 July 2016

Nicolas Privault
Nicolas Privault*
Affiliation:
Université de La Rochelle
*
∗Postal address: Département de Mathématiques, Université de La Rochelle, Avenue Marillac, 17042 La Rochelle Cedex 1, France. Email address: [email protected]
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Abstract

We define a class of anticipative flows on Poisson space and compute its Radon-Nikodym derivative. This result is applied to statistical testing in an anticipative queueing problem.

Keywords

Hypothesis testingqueueing theorySkorokhod integralpoint processes

MSC classification

Primary: 62F03: Hypothesis testing 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60K25: Queueing theory 60J75: Jump processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 560 - 574
Copyright
Copyright © by the Applied Probability Trust 2000 

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References

Brémaud, P. (1981). Point Processes and Queues, Martingale Dynamics. Springer, Berlin.CrossRefGoogle Scholar
Buckdahn, R. (1991). Linear Skorohod stochastic differential equations. Prob. Theory Rel. Fields 90, 223–240.CrossRefGoogle Scholar
Buckdahn, R. (1992). Anticipative Girsanov transformations. Prob. Theory Rel. Fields 91, 211–238.Google Scholar
Carlen, E., and Pardoux, E. (1990). Differential calculus and integration by parts on Poisson space. In Stochastics, Algebra and Analysis in Classical and Quantum Dynamics. Kluwer, Dordrecht, pp. 63–73.Google Scholar
Dunford, N., and Schwartz, J. T. (1957). Linear Operators, Vol II. Interscience, New York.Google Scholar
Holden, H., Oksendal, B., Uböe, J., and Zhang, T. S. (1996). Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach. Birkhäuser, Basel.Google Scholar
Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales (Lecture Notes in Math. 714). Springer, Berlin.Google Scholar
Karr, A. F. (1986). Point Processes and their Statistical Inference. Marcel Dekker, New York.Google Scholar
Kusuoka, S. (1982). The nonlinear transformation of Wiener measure on Banach space and its absolute continuity. J. Fac. Sci. Tokyo Univ. Section IA, Mathematics 29, 567–598.Google Scholar
León, J. A., Ruiz de Chavez, J., and Tudor, C. (1996). Skorohod solutions of a stochastic differential equation on the Poisson space. Boletín de la Sociedad Matemática Mexicana 2, 55–63.Google Scholar
Nualart, D., and Pardoux, E. (1988). Stochastic calculus with anticipative integrands. Prob. Theory Rel. Fields 78, 535–582.Google Scholar
Nualart, D., and Vives, J. (1990). Anticipative calculus for the Poisson process based on the Fock space. In Séminaire de Probabilités XXIV, eds Azéma, J., Meyer, P. A. and Yor, M. (Lecture Notes in Math. 1426). Springer, Berlin, pp. 154–165.Google Scholar
Picard, J. (1996). Transformations et équations anticipantes pour les processus de Poisson. Ann. Math. Blaise Pascal 3, 111–124.CrossRefGoogle Scholar
Privault, N. (1994). Chaotic and variational calculus in discrete and continuous time for the Poisson process. Stoch. and Stoch. Repts 51, 83–109.CrossRefGoogle Scholar
Privault, N. (1996). Girsanov theorem for anticipative shifts on Poisson space. Prob. Theory Rel. Fields 104, 61–76.Google Scholar
Privault, N. (1996). Linear Skorohod stochastic differential equations on Poisson space. In Stochastic Analysis and Related Topics V: The Silivri Workshop, eds Körezliovglu, H., Oksendal, B. and Üstünel, A. S. Birkhäuser, Basel, pp. 237–253.Google Scholar
Russo, F., and Vallois, P. (1993). Forward, backward and symmetric stochastic integration. Prob. Theory Rel. Fields 97, 403–421.Google Scholar
Skorokhod, A. V. (1975). On a generalization of a stochastic integral. Theory Prob. Appl. XX, 219–233.Google Scholar
Üstünel, A. S., and Zakai, M. (1994). Transformation of the Wiener measure under non-invertible shifts. Prob. Theory Rel. Fields 99, 485–500.Google Scholar