Electronic Journal of Probability

Martingale Problems for Conditional Distributions of Markov Processes

Thomas Kurtz

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Abstract

Let $X$ be a Markov process with generator $A$ and let $Y(t)=\gamma (X(t))$. The conditional distribution $\pi_t$ of $X(t)$ given $\sigma (Y(s):s\leq t)$ is characterized as a solution of a filtered martingale problem. As a consequence, we obtain a generator/martingale problem version of a result of Rogers and Pitman on Markov functions. Applications include uniqueness of filtering equations, exchangeability of the state distribution of vector-valued processes, verification of quasireversibility, and uniqueness for martingale problems for measure-valued processes. New results on the uniqueness of forward equations, needed in the proof of uniqueness for the filtered martingale problem are also presented.

Article information

Source
Electron. J. Probab., Volume 3 (1998), paper no. 9, 29 pp.

Dates
Accepted: 6 July 1998
First available in Project Euclid: 29 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1454101769

Digital Object Identifier
doi:10.1214/EJP.v3-31

Mathematical Reviews number (MathSciNet)
MR1637085

Zentralblatt MATH identifier
0907.60065

Subjects
Primary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]
Secondary: 69J25 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 93E11: Filtering [See also 60G35] 60G09: Exchangeability 60G44: Martingales with continuous parameter

Keywords
partial observation conditional distribution filtering forward equation martingale problem Markov process Markov function quasireversibility measure-valued process

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Kurtz, Thomas. Martingale Problems for Conditional Distributions of Markov Processes. Electron. J. Probab. 3 (1998), paper no. 9, 29 pp. doi:10.1214/EJP.v3-31. https://projecteuclid.org/euclid.ejp/1454101769


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