Electronic Journal of Probability

Martingale Problems for Conditional Distributions of Markov Processes

Thomas Kurtz

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

This work is licensed under aCreative Commons Attribution 3.0 License.


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

Export citation


  • Bhatt, Abhay G. and Karandikar, Rajeeva L. (1993). Invariant measures and evolution equations for Markov processes characterized via martingale problems. Ann. Probab. 21, 2246-2268.
  • Bhatt, Abhay G. and Borkar, V. S. (1996). Occupation measures for controlled Markov processes: Characterization and optimality. Ann. Probab. 24, 1531-1562.
  • Bhatt, Abhay G., Kallianpur, G., Karandikar, Rajeeva L. (1995). Uniqueness and robustness of solution of measure-valued equations of nonlinear filtering. Ann. Probab. 23, 1895-1938.
  • Cameron, Murray A. (1973). A note on functions of Markov processes with an application to a sequence of statistics. J. Appl. Probab. 10, 895-900.
  • Dawson, Donald A. (1993). Measure-valued Markov processes. École d'Été de Probabilités de Saint-Flour XXI, 1991, 1-260, Lecture Notes in Math., 1541, Springer, Berlin.
  • Dellacherie, Claude (1972). Capacités et processus stochastiques. Springe, Berlin.
  • Donnelly, Peter E. and Kurtz, Thomas G. (1996). A countable representation of the Fleming-Viot measure-valued diffusion. Ann. Probab. 24, 698-742.
  • Donnelly, Peter E. and Kurtz, Thomas G. (1997). Particle representations for measure-valued population models. Ann. Probab. (to appear)
  • Ethier, Stewart N. and Kurtz, Thomas G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Ethier, Stewart N. and Kurtz, Thomas G. (1993). Fleming-Viot processes in population genetics. SIAM J. Control Optim. 31 (1993), 345-386.
  • Fan, Kaisheng, (1996). On a new approach to the solution of the nonlinear filtering equation of jump processes. Probab. Engrg. Inform. Sci. 10, 153-163.
  • Glover, Joseph (1991). Markov functions. Ann. Inst. H. Poincaré Probab. Statist. 27, 221-238.
  • Harrison, J. M. and Williams, R. J. (1990). On the quasireversibility of a multiclass Brownian service station. Ann. Probab. 18, 1249-1268.
  • Harrison, J. M. and Williams, R. J. (1992). Brownian models of feedforward queueing networks: Quasireversibility and product form solutions. Ann. Appl. Probab. 2, 263-293.
  • Karni, Shaul and Merzbach, Ely (1990). On the extension of bimeasures. J. Analyse Math. 55, 1-16.
  • Kelly, Frank P. (1982). Markovian functions of a Markov chain. Sankya Ser A 44, 372-379.
  • Kliemann, Wolfgang H., Koch, Georgio, and Marchetti, Federico (1990). On the unnormalized solution of the filtering problem with counting process observations. IEEE Trans. Inform. Theory 36, 1415-1425.
  • Kurtz, Thomas G. and Ocone, Daniel L. (1988). Unique characterization of conditional distributions in nonlinear filtering. Ann. Probab. 16, 80-107.
  • Kurtz, Thomas G. and Stockbridge, Richard H. (1998). Existence of Markov controls and characterization of optimal Markov controls. SIAM J. Cont. Optim. 36, 609-653.
  • Lenglart, E. Désintégration régulière de mesure sans conditions habituelles. Seminar on probability, XVII, 321-345, Lecture Notes in Math. 986, Springer-Verlag, Berlin-New York.
  • Rogers, L. C. G. and Pitman, J. W. (1981). Markov functions. Ann. Probab. 9, 573-582.
  • Rosenblatt, Murray (1966). Functions of Markov processes. Z. Wahrscheinlichkeitstheorie 5, 232-243.
  • Serfozo, Richard F. (1989). Poisson functionals of Markov processes and queueing networks. Adv. in Appl. Probab. 21, 595-611.
  • Yor, Marc (1977). Sur les théories du filtrage et de la prédiction. Séminaire de Probabilités XI. Lecture Notes in Math. 581, 257-297. Springer, Berlin.