The research on the strong Markov property

Tang Rong, Huang Yonghui


Let X(t, ω) = {xt(ω): t ≥ 0} be a Markov process defined on a probability space (Ω,F, P) and valued in a measurable space (E; E ). In this paper, we give the definitions of σ-algebras prior to α and post-α and discuss their properties. At the same time, we prove that the strong Markov property holds for an arbitrary Markov process, that is, we prove that the Markov property is equivalent to the strong Markov property.


σ-algebra prior to α, σ-algebra post-α; Markov property; the strong Markov property; stopping time; homogeneity

Mathematics Subject Classification

60J25, 60J27


Ash, Robert B. and Catherine A. Doléans-Dade. "Porbability & measurable theory." Beijing: Post & telecom Press, 2007 (in Chinese).

Bauer, H. "Probability theory and elements of measure theory." Second english edition. London-New York: Academic Press, 1981

Chenggao, Wan. "The limit theory of martingale." Beijing: Science Press, 2002 (in Chinese).

Chung, Kai Lai. "Markov chains with stationary transition probabilities." New York: Springer-Verlang, 1967.

Doob, J.L. "Stochastic processes." New York: John Wiley&Sons, 1953.

Dynkin, E.B "Markov processes." Berlin-Göttingen-Heidelberg: Springer-Verlag, 1965.

Feller, W. "On the Integro-differential Equation of Purely Discontinuous Markoff Process." Trans. Amer. Math. Soc. 48 (1940): 488-515.

Gikhman, Iosif I. and Anatoli V. Skorokhol. "The theory of stochastic process (II)." Berlin: Springer-Verlag, 2004.

Jiaan, Yan "The theory of martingale and stochastic integral." Shanghai: Shanghai Science and Technology Press, 1981 (in Chinese).

Mingpin, Qian and Gong Guanglu. "The theory stocastic processes." Beijing: Peking University Press, 1997 (in Chinese).

Mukherjea, A. and K. Pothoven. "Real and functional analysis." New York-London: Plenum Press, 1978.

Ray, D. "Stationary Markov processes with continuous paths." Trans. Amer. Math. Soc. 82 (1956): 452-493.

Shijian, Yan et al. "The foundation of probability." Beijing: Science Press, 1982 (in Chinese).

Zhenting, Hou et al. "The Q-matrix Problem of Markov Process." Hunan: Hunan Scientific Press, 1994 (in Chinese).

Zikun, Wang. "The general theory of stochastic process." Beijing: Beijing Normal University Press, 1996 (in Chinese).

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