Evolution equations for Markov processes: Application to the white-noise theory of filtering
Tóm tắt
LetX be a Markov process taking values in a complete, separable metric spaceE and characterized via a martingale problem for an operatorA. We develop a criterion for invariant measures when rangeA is a subset of continuous functions onE. Using this, uniqueness in the class of all positive finite measures of solutions to a (perturbed) measure-valued evolution equation is proved when the test functions are taken from the domain ofA. As a consequence, it is shown that in the characterization of the optimal filter (in the white-noise theory of filtering) as the unique solution to an analogue of Zakai (as well as Fujisaki-Kallianpur-Kunita) equation, it suffices to take domainA as the class of test functions where the signal process is the solution to the martingale problem forA.
Tài liệu tham khảo
Bhatt AG, Karandikar, RL (to appear) Invariant measures and evolution equations for Markov processes characterized via martingale problems. Ann Probab
Bhatt AG, Karandikar RL (1993) Weak convergence to a Markov process: the martingale approach. Probab Theory Related Fields 96:335–351
Echeverria PE (1982) A criterion for invariant measures of Markov processes. Z Wahrsch Verw Gebiete 61:1–16
Ethier SN and Kurtz TG (1986) Markov Processes: Characterization and Convergence. Wiley, New York
Jacod J (1979) Calcul Stochastique et problèmes de martingales. Lecture Notes in Mathematics, Vol 714, Springer-Verlag, Berlin
Kallianpur G, Karandikar RL (1984) Measure valued equations for the optimal filter in finitely additive nonlinear filtering theory. Z Wahrsch Verw Gebiete 66:1–17
Kallianpur G, Karandikar, RL (1985) White noise calculus and nonlinear filtering theory. Ann Probab 13:1033–1107
Kalianpur G, Karandikar RL (1988) White Noise Theory of Prediction Filtering and Smoothing. Gordon and Breach, New York
Karandikar RL (1987) On the Feynman-Kac formula and its applications to filtering theory. Appl Math Optim 16:263–276
Métivier M (1982) Semimartingales: A Course on Stochastic Processes, de Gruyter, Berlin
Parthasarathy KR (1967) Probability Measures on Metric Spaces. Academic Press, New York
Stockbridge, RH (1990). Time average control of martingale problems: existence of a stationary solution. Ann Probab 18:190–225
Stroock DW, Varadhan SRS (1979) Multidimensional Diffusion Processes. Springer-Verlag, Berlin
Weiss G (1956) A note on Orlicz spaces, Portugal Math 15:35–47