The paper studies the filtering problem for a non-classical frame- work: we assume that the observation equation is driven by a signal dependent noise. We show that the support of the conditional distri- bution of the signal is on the corresponding level set of the derivative of the quadratic variation process. Depending on the intrinsic dimension of the noise, we distinguish two cases: In the first case, the conditional distribution has discrete support and we deduce an explicit represen- tation for the conditional distribution. In the second case, the filtering problem is equivalent to a classical one defined on a manifold and we deduce the evolution equation of the conditional distribution. The re- sults are applied to the filtering problem where the observation noise is an Ornstein-Uhlenbeck process.
References
Bhatt, Abhay G.; Karandikar, Rajeeva L. On filtering with Ornstein-Uhlenbeck process as noise. J. Indian Statist. Assoc. 41 (2003), no. 2, 205–220. 1331.93198 10.31390/cosa.4.1.08Bhatt, Abhay G.; Karandikar, Rajeeva L. On filtering with Ornstein-Uhlenbeck process as noise. J. Indian Statist. Assoc. 41 (2003), no. 2, 205–220. 1331.93198 10.31390/cosa.4.1.08
Evans, Lawrence C.; Gariepy, Ronald F. Measure theory and fine properties of functions.Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. viii+268 pp. ISBN: 0-8493-7157-0 0804.28001Evans, Lawrence C.; Gariepy, Ronald F. Measure theory and fine properties of functions.Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. viii+268 pp. ISBN: 0-8493-7157-0 0804.28001
Friedman, Avner. Stochastic differential equations and applications. Vol. 1.Probability and Mathematical Statistics, Vol. 28.Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. xiii+231 pp. (58 #13350a) 0323.60056Friedman, Avner. Stochastic differential equations and applications. Vol. 1.Probability and Mathematical Statistics, Vol. 28.Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. xiii+231 pp. (58 #13350a) 0323.60056
Gawarecki, L.; Mandrekar, V. On the Zakai equation of filtering with Gaussian noise. Stochastics in finite and infinite dimensions, 145–151, Trends Math., Birkhäuser Boston, Boston, MA, 2001. 0973.93055Gawarecki, L.; Mandrekar, V. On the Zakai equation of filtering with Gaussian noise. Stochastics in finite and infinite dimensions, 145–151, Trends Math., Birkhäuser Boston, Boston, MA, 2001. 0973.93055
Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes.Second edition.North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. xvi+555 pp. ISBN: 0-444-87378-3 MR1011252Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes.Second edition.North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. xvi+555 pp. ISBN: 0-444-87378-3 MR1011252
Joannides. M and LeGland. F (1997). Nonlinear Filtering with Perfect Discrete Time Observations, Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, December 13-15, 1995, pp. 4012-4017.Joannides. M and LeGland. F (1997). Nonlinear Filtering with Perfect Discrete Time Observations, Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, December 13-15, 1995, pp. 4012-4017.
Joannides. M and LeGland. F (1997). Nonlinear Filtering with Continuous Time Perfect Observations and Noninformative Quadratic Variation. Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, December 10-12, 1997, pp. 1645-1650.Joannides. M and LeGland. F (1997). Nonlinear Filtering with Continuous Time Perfect Observations and Noninformative Quadratic Variation. Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, December 10-12, 1997, pp. 1645-1650.
Kallianpur, G.; Karandikar, R. L. White noise theory of prediction, filtering and smoothing.Stochastics Monographs, 3. Gordon & Breach Science Publishers, New York, 1988. xiv+599 pp. ISBN: 2-88124-685-0 MR966884Kallianpur, G.; Karandikar, R. L. White noise theory of prediction, filtering and smoothing.Stochastics Monographs, 3. Gordon & Breach Science Publishers, New York, 1988. xiv+599 pp. ISBN: 2-88124-685-0 MR966884
Kunita, Hiroshi. Representation and stability of nonlinear filters associated with Gaussian noises. Stochastic processes, 201–210, Springer, New York, 1993. 0794.62070Kunita, Hiroshi. Representation and stability of nonlinear filters associated with Gaussian noises. Stochastic processes, 201–210, Springer, New York, 1993. 0794.62070
bibitem Z. Liu and J. Xiong (2009). Some computable examples for singular filtering. em Technical Report at http://www.math.utk.edu/textasciitilde jxiong/publist.html.bibitem Z. Liu and J. Xiong (2009). Some computable examples for singular filtering. em Technical Report at http://www.math.utk.edu/textasciitilde jxiong/publist.html.
Mandal, Pranab K.; Mandrekar, V. A Bayes formula for Gaussian noise processes and its applications. SIAM J. Control Optim. 39 (2000), no. 3, 852–871 (electronic). 0970.60046 10.1137/S0363012998343380Mandal, Pranab K.; Mandrekar, V. A Bayes formula for Gaussian noise processes and its applications. SIAM J. Control Optim. 39 (2000), no. 3, 852–871 (electronic). 0970.60046 10.1137/S0363012998343380
Rudin, Walter. Principles of mathematical analysis.Third edition.International Series in Pure and Applied Mathematics.McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. x+342 pp. MR0385023Rudin, Walter. Principles of mathematical analysis.Third edition.International Series in Pure and Applied Mathematics.McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. x+342 pp. MR0385023
