Coterminal families and the strong Markov property
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- by A. O. Pittenger and C. T. Shih PDF
- Trans. Amer. Math. Soc. 182 (1973), 1-42 Request permission
Abstract:
Let ${E_\Delta }$ be a compact metric space and assume that a strong Markov process X is defined on ${E_\Delta }$. Under the assumption that X has right continuous paths with left limits, it is shown that a version of the strong Markov property extends to coterminal families, a class of random times which can be visualized as last exit times before t from a fixed subset of ${E_\Delta }$. Since the random times are not Markov times, the conditioning $\sigma$-field and the new conditional probabilities must be defined. If X is also assumed to be nearly quasileft continuous, i.e. branching points are permitted, two different conditionings are possible—one on the “past” of the random time and one on the “past plus present"—and two different conditional probabilities must be defined.References
- R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR 0264757
- R. M. Blumenthal and R. K. Getoor, A theorem on stopping times, Ann. Math. Statist. 35 (1964), 1348–1350. MR 169288, DOI 10.1214/aoms/1177703293
- K. L. Chung and J. L. Doob, Fields, optionality and measurability, Amer. J. Math. 87 (1965), 397–424. MR 214121, DOI 10.2307/2373011
- J. L. Doob, Compactification of the discrete state spaces of a Markov process, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 10 (1968), 236–251. MR 234525, DOI 10.1007/BF00536277
- Paul-A. Meyer, Probability and potentials, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. MR 0205288 —, Séminaire de probabilités. V, Springer-Verlag, Berlin, 1971, pp. 270-274.
- P. A. Meyer, R. T. Smythe, and J. B. Walsh, Birth and death of Markov processes, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 295–305. MR 0405600
- A. O. Pittenger, Last exit times and the $Q$-matrices of Markov chains, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 20 (1971/72), 143–162. MR 322966, DOI 10.1007/BF00536292
- A. O. Pittenger and C. T. Shih, Coterminal families and the strong Markov property, Bull. Amer. Math. Soc. 78 (1972), 439–443. MR 297019, DOI 10.1090/S0002-9904-1972-12935-5
- C. T. Shih, On extending potential theory to all strong Markov processes, Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 1, 303–315 (English, with French summary). MR 288845, DOI 10.5802/aif.343
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 182 (1973), 1-42
- MSC: Primary 60J40
- DOI: https://doi.org/10.1090/S0002-9947-1973-0336827-2
- MathSciNet review: 0336827