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Nonlinear filtering in spatio-temporal doubly stochastic point processes driven by OU processes. (English) Zbl 1249.60097
Summary: 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 of 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.
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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[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 · Zbl 0983.60028
[2] Beneš V., Prokešová M.: Nonlinear filtration in doubly stochastic point processes. Proc. 4th International Conference Aplimat 2005 (M. Kováčová, FME, Slovak Technical University, Bratislava 2005, pp. 415-420
[3] Brémaud P.: Point Processes and Queues: Martingale Dynamics. Springer-Verlag, Berlin 1981 · Zbl 0478.60004
[4] Cont R., Tankov P.: Financial Modelling with Jump Processes. Chapman and Hall/CRC, Boca Raton 2004 · Zbl 1052.91043
[5] Daley D. J., Vere-Jones D.: An Introduction to the Theory of Point Processes. Springer-Verlag, New York 1988 · 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 · 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 · Zbl 0345.60033
[8] Gihman I., Dorogovcev A.: On stability of solutions of stochastic differential equations. Ukrain. Mat. Z. 6 (1965), 229-250
[9] Jurek Z. J., Mason J. D.: Operator-limit Distributions in Probability Theory. Wiley, New York 1993 · Zbl 0850.60003
[10] Karr A. F.: Point Processes and Their Statistical Inference. Marcel Dekker, New York 1986 · Zbl 0733.62088
[11] Liptser R. S., Shiryayev A. N.: Statistics in Random Processes, Vol. II: Applications. Springer-Verlag, New York 2000 · Zbl 0364.60004
[12] Sato K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge 1999 · Zbl 0973.60001
[13] Sato K.: Basic results on Lévy processes. Lévy Processes - Theory and Applications (O. Barndorff-Nielsen, T. Mikosch, and S. Resnick, Birkäuser, Boston 2001 · Zbl 0974.60036
[14] Snyder D. L.: Filtering and detection for doubly stochastic Poisson processes. IEEE Trans. Inform. Theory 18 (1972), 91-102 · 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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