Evolution equations for Markov processes: Application to the white-noise theory of filtering. (English) Zbl 0824.60075
Let \(X\) be a Markov process taking values in a complete separable metric space \(E\) and characterized via a martingale problem for an operator \(A\). The authors obtain a criterion for invariant measures when \(A\) is just a subset of continuous functions on \(E\). The authors then also prove uniqueness of the solutions to a measure-valued evolution equation. For the case of locally compact state space, similar results were developed by others [cf. P. E. Echeverria, Z. Wahrscheinlichkeitstheorie Verw. Geb. 61, 1-16 (1982; Zbl 0476.60074); S. N. Ethier and T. G. Kurtz, “Markov processes: Characterization and convergence” (1986; Zbl 0592.60049), p. 252]. By applying authors’ results to measure- valued filtering equation, uniqueness of the solutions to the analogue of the Zakai equation is proved in the class of all positive finite measures rather than only in a restricted class of measures.
Reviewer: Wu Chengxun (Shanghai)
MSC:
| 60J25 | Continuous-time Markov processes on general state spaces |
| 60G35 | Signal detection and filtering (aspects of stochastic processes) |
| 60G44 | Martingales with continuous parameter |
| 60J35 | Transition functions, generators and resolvents |
| 60G05 | Foundations of stochastic processes |
| 93E11 | Filtering in stochastic control theory |
Keywords:
Markov process; invariant measures; uniqueness of the solutions to a measure-valued evolution equation; measure-valued filtering equation; Zakai equationReferences:
| [1] | Bhatt AG, Karandikar, RL (to appear) Invariant measures and evolution equations for Markov processes characterized via martingale problems. Ann Probab |
| [2] | Bhatt AG, Karandikar RL (1993) Weak convergence to a Markov process: the martingale approach. Probab Theory Related Fields 96:335-351 · Zbl 0794.60074 · doi:10.1007/BF01292676 |
| [3] | Echeverria PE (1982) A criterion for invariant measures of Markov processes. Z Wahrsch Verw Gebiete 61:1-16 · Zbl 0476.60074 · doi:10.1007/BF00537221 |
| [4] | Ethier SN and Kurtz TG (1986) Markov Processes: Characterization and Convergence. Wiley, New York |
| [5] | Jacod J (1979) Calcul Stochastique et problèmes de martingales. Lecture Notes in Mathematics, Vol 714, Springer-Verlag, Berlin · Zbl 0414.60053 |
| [6] | 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 · Zbl 0592.93060 · doi:10.1007/BF00532792 |
| [7] | Kallianpur G, Karandikar, RL (1985) White noise calculus and nonlinear filtering theory. Ann Probab 13:1033-1107 · Zbl 0584.60055 · doi:10.1214/aop/1176992798 |
| [8] | Kalianpur G, Karandikar RL (1988) White Noise Theory of Prediction Filtering and Smoothing. Gordon and Breach, New York |
| [9] | Karandikar RL (1987) On the Feynman-Kac formula and its applications to filtering theory. Appl Math Optim 16:263-276 · Zbl 0636.60042 · doi:10.1007/BF01442195 |
| [10] | Métivier M (1982) Semimartingales: A Course on Stochastic Processes, de Gruyter, Berlin |
| [11] | Parthasarathy KR (1967) Probability Measures on Metric Spaces. Academic Press, New York · Zbl 0153.19101 |
| [12] | Stockbridge, RH (1990). Time average control of martingale problems: existence of a stationary solution. Ann Probab 18:190-225 · Zbl 0699.49018 · doi:10.1214/aop/1176990944 |
| [13] | Stroock DW, Varadhan SRS (1979) Multidimensional Diffusion Processes. Springer-Verlag, Berlin |
| [14] | Weiss G (1956) A note on Orlicz spaces, Portugal Math 15:35-47 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
