Nonlinear filtering for Markov systems with delayed observations

• Autorzy: Antonella Calzolari , Patrick Florchinger , Giovanna Nappo
• Języki publikacji: EN

Abstrakty

EN

This paper deals with nonlinear filtering problems with delays, i.e., we consider a system (X,Y), which can be represented by means of a system (X,Ŷ), in the sense that Yt = Ŷa(t), where a(t) is a delayed time transformation. We start with X being a Markov process, and then study Markovian systems, not necessarily diffusive, with correlated noises. The interest is focused on the existence of explicit representations of the corresponding filters as functionals depending on the observed trajectory. Various assumptions on the function a(t) are considered.

Słowa kluczowe

EN

nonlinear filtering; jump processes; diffusion processes; Markov processes; stochastic delay differential equation

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2009
• Tom: 19
• Numer: 1
• Strony: 49-57

Daty

wydano

2009

otrzymano

2008-03-01

poprawiono

2008-06-13

Twórcy

autor

Antonella Calzolari

• Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica 1, I 00133 Roma, Italy

autor

Patrick Florchinger

• Département de Mathématiques, Université de Metz, 23 Allée des Oeillets, F 57160 Moulins les Metz, France

autor

Giovanna Nappo

• Dipartimento di Matematica, Università di Roma “La Sapienza”, piazzale A. Moro 2, I 00185 Roma, Italy

Bibliografia

• Baras, J. S., Blankenship, G. L. and Hopkins, Jr., W. E. (1983). Existence, uniqueness, and asymptotic behavior of solutions to a class of Zakai equations with unbounded coefficients, IEEE Transactions on Automatic Control 28(2): 203-214.
• Bhatt, A. G., Kallianpur, G. and Karandikar, R. L. (1995). Uniqueness and robustness of solution of measure-valued equations of nonlinear filtering, The Annals of Probability 23(4): 1895-1938.
• Brémaud, P. (1981). Point Processes and Queues. Martingale Dynamics, Springer-Verlag, New York, NY.
• Calzolari, A., Florchinger, P. and Nappo, G. (2003). Nonlinear filtering for Markov diffusion systems with delayed observations, Proceedings of the 42nd Conference on Decision and Control, Maui, HI, USA, pp. 1404-1405.
• Calzolari, A., Florchinger, P. and Nappo, G. (2006). Approximation of nonlinear filters for Markov systems with delayed observations, SIAM Journal on Control and Optimization 45(2): 599-633.
• Calzolari, A., Florchinger, P. and Nappo, G. (2007). Convergence in nonlinear filtering for stochastic delay systems, SIAM Journal on Control and Optimization 46(5): 1615-1636.
• Cannarsa, P. and Vespri, V. (1985). Existence and uniqueness of solutions to a class of stochastic partial differential equations, Stochastic Analysis and Applications 3(3): 315-339.
• Clark, J. M. C. (1978). The design of robust approximations to the stochastic differential equations of nonlinear filtering, Communication systems and random process theory (Proceedings of the 2nd NATO Advanced Study Institute, Darlington, 1977), Vol. 25 of NATO Advanced Study Institute Series E: Applied Sciences, Sijthoff & Noordhoff, Alphen aan den Rijn, pp. 721-734.
• Cvitanić, J., Liptser, R. and Rozovskii, B. (2006). A filtering approach to tracking volatility from prices observed at random times, The Annals of Applied Probability 16(3): 1633-1652.
• Davis, M. H. A. (1982). A pathwise solution of the equations of nonlinear filtering, Akademiya Nauk SSSR. Teoriya Veroyatnoste˘ı i ee Primeneniya 27(1): 160-167.
• Elliott, R. J. and Kohlmann, M. (1981). Robust filtering for correlated multidimensional observations, Mathematische Zeitschrift 178(4): 559-578.
• Florchinger, P. (1993). Zakai equation of nonlinear filtering with unbounded coefficients. The case of dependent noises, Systems & Control Letters 21(5): 413-422.
• Frey, R., Prosdocimi, C. and Runggaldier, W. J. (2007). Affine credit risk models under incomplete information, in J. Akahori, S. Ogawa, S. Watanabe (Eds), Stochastic Processes and Applications to Mathematical Finance. Proceedings of the 6th Ritsumeikan International Symposium, Ritsumeikan University, Japan, World Scientific Publishing Co., pp. 97-113.
• Hopkins, Jr., W. E. (1982). Nonlinear filtering of nondegenerate diffusions with unbounded coefficients, Ph.D. thesis, University of Maryland at College Park.
• Joannides, M. and Le Gland, F. (1995). Nonlinear filtering with perfect discrete time observations, Proceedings of the 34th Conference on Decision and Control, New Orleans, LA, USA, pp. 4012-4017.
• Kirch, M. and Runggaldier, W. J. (2004/05). Efficient hedging when asset prices follow a geometric Poisson process with unknown intensities, SIAM Journal on Control and Optimization 43(4): 1174-1195.
• Kliemann, W., Koch, G. and Marchetti, F. (1990). On the unnormalized solution of the filtering problem with counting observations, IEEE Transactions on Information Theory 316(6): 1415-1425.
• Kunita, H. (1984). Stochastic differential equations and stochastic flows of diffeomorphisms, École d'été de probabilités de Saint-Flour, XII-1982, Vol. 1097 of Lecture Notes in Mathematics, Springer, Berlin, pp. 143-303.
• Liptser, R. S. and Shiryaev, A. N. (1977). Statistics of random processes. I, Vol. 5 of Applications of Mathematics, Expanded Edn, Springer-Verlag, Berlin.
• Pardoux, E. (1991). Filtrage non linéaire et équations aux dérivées partielles stochastiques associées, Ecole d'Eté de Probabilités de Saint-Flour XIX-1989, Vol. 1464 of Lecture Notes in Mathematics, Springer, Berlin, pp. 67-163.
• Schweizer, M. (1994). Risk-minimizing hedging strategies under restricted information, Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics 4(4): 327-342.
• Sussmann, H. J. (1981). Rigorous results on the cubic sensor problem, Stochastic systems: The mathematics of filtering and identification and applications (Les Arcs, 1980), Vol. 78 of NATO Advanced Study Institute Series C: Mathematical and Physical Sciences, Reidel, Dordrecht, pp. 637-648.
• Zakai, M. (1969). On the optimal filtering of diffusion processes, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 11: 230-243.
• Zeng, Y. (2003). A partially observed model for micromovement of asset prices with Bayes estimation via filtering, Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics 13(3): 411-444.

Identyfikatory

DOI

10.2478/v10006-009-0004-8