Abstract
In the previous chapters we described observable random processes X = (ξ t ), t ≥ 0, which possessed continuous trajectories and had properties analogous, to a certain extent, to those of a Wiener process. Chapters 18 and 19 will deal with the case of an observable process that is a point process whose trajectories are pure jump functions (a Poisson process with constant or variable intensity is a typical example).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Notes and References
Brémaud, P. (1972): Martingale Approach to Point Processes. ERL Memo M 345, University of California, Berkeley
Boel, R., Varaiya, P. and Wong E. (1975): Martingales on jump processes, III. Representation results. SIAM J. Control Optimization, 13, 5, 999–1060
Van Schuppen, I.L. (1973): Filtering for counting processes. Martingale approach. In: Proc. 4th Symposium on Nonlinear Estimation and Applications ( San Diego, CA )
Jacod, J. (1976): A General Theorem of Representation for Martingales. Dept. Math. Inform., Université de Rennes
Segal, A. (1973): A martingale approach to modeling, estimation and detection of jump processes. Tech. Report. No. 7050–21, MIT
Davis, M.H.A. (1974): The Representation of Martingales of Jump Processes. Research Report 74/78, Imperial College, London
Kabanov, Yu.M., Liptser, R.S. and Shiryaev, A.N. (1975): Martingale methods in the point process theory. Tr. Shcoly-Seminara po Teorii Sluchainykh Protsessov, II. ( Druskeninkai, 1974 ), 296–353
Segal, A. and Kailath, T. (1975): The modeling of random modulated jump processes. IEEE Trans. Inf. Theory, IT-21, 2, 135–42
Chou, C.S. and Meyer, P.A. (1974): La représentation des martingales relatives à un processus ponctuel discret. C. R. Acad. Sci., Paris, Sér. A-B, 278, 1561–3
Orey, S. (1974): Radon—Nikodym Derivative of Probability Measures: Martingale Methods. Dept. Found. Sci., Tokyo University of Education
Doléans-Dade, C. (1970): Quelques applications de la formule de changement de variables pour les semimartingales locales. Z. Wahrsch. Verw. Gebiete, 16, 181–94
Jacod, J. (1976): Un théorème de représentation pour les martingales discontinues. Z. Wahrsch. Verw. Gebiete, 34, 225–44
Brémaud, P. (1975): An extension of Watanabe’s theorem of characterization of a Poisson process over the positive real half time. J. Appl. Probab., 12, 2, 369–99
Jacod, J. (1979): Calcul Stochastique et Problèmes des Martingales, Lecture Notes in Mathematics 714, Springer-Verlag, Berlin Heidelberg New York
Jacod, J. and Shiryaev, A.N. (1987): Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin Heidelberg New York
Liptser, R.S. and Shiryaev, A.N. (1989): Theory of Martingales. Kluwer, Dordrecht (Russian edition 1986 )
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Liptser, R.S., Shiryaev, A.N. (2001). Random Point Processes: Stieltjes Stochastic Integrals. In: Statistics of Random Processes. Stochastic Modelling and Applied Probability, vol 6. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-10028-8_8
Download citation
DOI: https://doi.org/10.1007/978-3-662-10028-8_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08365-5
Online ISBN: 978-3-662-10028-8
eBook Packages: Springer Book Archive