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 Electronic Journal of Probability > Vol. 3 (1998) > Paper 9 open journal systems 


Martingale Problems for Conditional Distributions of Markov Processes

Thomas G. Kurtz, University of Wisconsin, Madison


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 $sigma (Y(s):sleq t)$ 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.


Full text: PDF

Pages: 1-29

Published on: July 6, 1998


Bibliography

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. Math. Review 95d:60120

Bhatt, Abhay G. and Borkar, V. S. (1996). Occupation measures for controlled Markov processes: Characterization and optimality. Ann. Probab. 24, 1531-1562. Math. Review 97i:90105

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. Math. Review 97a:60061

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. Math. Review 50:8703

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. Math. Review 94m:60101

Dellacherie, Claude (1972). Capacités et processus stochastiques. Springe, Berlin. Math. Review 56:6810

Donnelly, Peter E. and Kurtz, Thomas G. (1996). A countable representation of the Fleming-Viot measure-valued diffusion. Ann. Probab. 24, 698-742. Math. Review 98f:60162

Donnelly, Peter E. and Kurtz, Thomas G. (1997). Particle representations for measure-valued population models. Ann. Probab. (to appear) Math. Review

Ethier, Stewart N. and Kurtz, Thomas G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York. Math. Review 88a:60130

Ethier, Stewart N. and Kurtz, Thomas G. (1993). Fleming-Viot processes in population genetics. SIAM J. Control Optim. 31 (1993), 345-386. Math. Review 94d:60131

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. Math. Review 97c:60110

Glover, Joseph (1991). Markov functions. Ann. Inst. H. Poincaré Probab. Statist. 27, 221-238. Math. Review 93a:60106

Harrison, J. M. and Williams, R. J. (1990). On the quasireversibility of a multiclass Brownian service station. Ann. Probab. 18, 1249-1268. Math. Review 91i:60204

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. Math. Review 93d:60126

Karni, Shaul and Merzbach, Ely (1990). On the extension of bimeasures. J. Analyse Math. 55, 1-16. Math. Review 92f:28005

Kelly, Frank P. (1982). Markovian functions of a Markov chain. Sankya Ser A 44, 372-379. Math. Review 85d:60129

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. Math. Review

Kurtz, Thomas G. and Ocone, Daniel L. (1988). Unique characterization of conditional distributions in nonlinear filtering. Ann. Probab. 16, 80-107. Math. Review 88m:93146

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. Math. Review

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. Math. Review 86i:60096

Rogers, L. C. G. and Pitman, J. W. (1981). Markov functions. Ann. Probab. 9, 573-582. Math. Review 82j:60133

Rosenblatt, Murray (1966). Functions of Markov processes. Z. Wahrscheinlichkeitstheorie 5, 232-243. Math. Review 34:3659

Serfozo, Richard F. (1989). Poisson functionals of Markov processes and queueing networks. Adv. in Appl. Probab. 21, 595-611. Math. Review 91h:60106

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. Math. Review 57:10801















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Electronic Journal of Probability. ISSN: 1083-6489