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Cox process; filtering; Ornstein–Uhlenbeck process
Doubly stochastic point processes driven by non-Gaussian Ornstein–Uhlenbeck type processes are studied. The problem of nonlinear filtering is investigated. For temporal point processes the characteristic form for the differential generator of the driving process is used to obtain a stochastic differential equation for the conditional distribution. The main result in the spatio-temporal case leads to the filtering equation for the conditional mean.
[1] Barndorff-Nielsen O., Shephard N.: Non-Gaussian Ornstein–Uhlenbeck based models and some of their uses in financial economics. J. Roy. Statist. Soc. B 63 (2001), 167–241 DOI 10.1111/1467-9868.00282 | MR 1841412 | Zbl 0983.60028
[2] Beneš V., Prokešová M.: Nonlinear filtration in doubly stochastic point processes. In: Proc. 4th International Conference Aplimat 2005 (M. Kováčová, ed.), FME, Slovak Technical University, Bratislava 2005, pp. 415–420
[3] Brémaud P.: Point Processes and Queues: Martingale Dynamics. Springer–Verlag, Berlin 1981 MR 0636252 | Zbl 0478.60004
[4] Cont R., Tankov P.: Financial Modelling with Jump Processes. Chapman and Hall/CRC, Boca Raton 2004 MR 2042661
[5] Daley D. J., Vere-Jones D.: An Introduction to the Theory of Point Processes. Springer–Verlag, New York 1988 MR 0950166 | Zbl 1159.60003
[6] Daley D. J., Vere-Jones D.: An Introduction to the Theory of Point Processes, Vol. I: Elementary Theory and Methods. Second edition. Springer–Verlag, New York 2003 MR 1950431 | Zbl 1159.60003
[7] Fishman P. M., Snyder D.: The statistical analysis of space-time point processes. IEEE Trans. Inform. Theory 22 (1976), 257–274 DOI 10.1109/TIT.1976.1055558 | MR 0418216 | Zbl 0345.60033
[8] Gihman I., Dorogovcev A.: On stability of solutions of stochastic differential equations. Ukrain. Mat. Z. 6 (1965), 229–250 MR 0281278
[9] Jurek Z. J., Mason J. D.: Operator-limit Distributions in Probability Theory. Wiley, New York 1993 MR 1243181 | Zbl 0850.60003
[10] Karr A. F.: Point Processes and Their Statistical Inference. Marcel Dekker, New York 1986 MR 0851982 | Zbl 0733.62088
[11] Liptser R. S., Shiryayev A. N.: Statistics in Random Processes, Vol. II: Applications. Springer–Verlag, New York 2000 MR 0488267
[12] Sato K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge 1999 MR 1739520 | Zbl 0973.60001
[13] Sato K.: Basic results on Lévy processes. In: Lévy Processes – Theory and Applications (O. Barndorff-Nielsen, T. Mikosch, and S. Resnick, eds.), Birkäuser, Boston 2001 MR 1833689 | Zbl 0974.60036
[14] Snyder D. L.: Filtering and detection for doubly stochastic Poisson processes. IEEE Trans. Inform. Theory 18 (1972), 91–102 DOI 10.1109/TIT.1972.1054756 | MR 0408953 | Zbl 0227.62055
[15] Snyder D., Miller M.: Random Point Processes in Time and Space. Springer–Verlag, New York 1991 Zbl 0744.60050
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