Summary
Suppose a measure μ dominates a measure ŋ in the ordering induced by the excessive functions of a transient Markov process. Rost shows than ŋ can be represented as the distribution of the process stopped at a randomized optional time and started with initial distribution μ. In this paper we introduce the shift operator to the class of randomized optional times, inducing the class of randomized quasi-terminal times and that of randomized terminal times. We analyze the algebraic properties of these classes and obtain some compactness results for the class of randomized quasi-terminal times. Some applications, including remplissage by hitting times, are presented.
References
Azéma, J.: Representation multiplicative d'une surmartingale bornée. Z. Wahrscheinlichkeitstheor. Verw. Geb. 45, 191–211 (1978)
Azéma, J., Meyer, P.-A.: Une nouvelle representation du type de Skorokhod. Séminaire de Probabilités VIII, Lecture Notes in Math., vol. 381, 1–10. Berlin Heidelberg New York: Springer 1974
Baxter, J.R., Chacon, R.V.: Compactness of stopping times. Z. Wahrscheinlichkeitstheor. Verw. Geb. 40, 169–181 (1977)
Çinlar, E., Jacod, J., Protter, P. Sharpe, M.J.: Semi-martingales and Markov processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 54, 161–219 (1980)
Falkner, N.: Embedding in Brownian motion. Ph.D. thesis, University of British Columbia, September 1978
Falkner, N.: On Skorokhod embedding in n-dimensional Brownian motion by means of natural stopping times. Strasbourg Seminaire de Probabilités XIV, Lecture Notes in Math., vol. 784 Berlin Heidelberg New York: Springer 1980
Falkner, N.: Hmbedding processes in Brownian motion in IR n. Trans. AMS 267 335–363 (1981)
Heath, D.: Skorokhod stopping in discrete time. Strasbourg Seminaire de Probabilités IX, Lecture Notes in Math., vol. 465, 515–517. Berlin Heidelberg New York: Springer 1975
Karr, A.F., Pittenger, A.O.: The inverse balayage problem for Markov chains. Stoch. Processes Appl. 7, 165–178 (1978)
Karr, A.F., Pittenger, A.O.: The inverse balayage problem for Markov chains, part II. Stochastic Processes Appl. 9, 35–53 (1979)
Karr, A.F., Pittenger, A.O.: An inverse balayage problem for Brownian motion. Ann. Probab. 7, 186–191 (1979)
Meyer, P.A.: Travaux de H. Rost en théorie du balayage. Strasbourg Seminaire de Probabilités VI, Lecture Notes in Math., vol. 191, 237–250. Berlin Heidelberg New York: Springer 1971
Meyer, P.A.: Cohvergence faible et compacité des temps d'arret d'apres Baxter et Chacon. Strasbourg Seminaire de Probabilités XII, Lecture Notes in Math., vol. 649, 411–423. Berlin Heidelberg New York: Springer 1977
Rost, H.: Die Stoppverteilungen eines Markoff-Prozesses mit lokalendlichem Potential. Manuscripta Math. 3, 321–330 (1970)
Rost, H.: Markov-Ketten bei sich füllenden Löchern im Zustandsraum. Ann. Inst. Fourier 21, 253–270 (1971)
Rost, H.: The stopping distribution of a Markov process. Invent. Math. 14, 1–16 (1971)
Walsh, J.B.: Some topologies connected with Lebesgue measure, Strasbourg Seminaire de Probabilités V, Lecture Notes in Math., vol. 191 290–310. Berlin Heidelberg New York: Springer 1971
Watanabe, T.: Balayées of excessive measures and functions with respect to resolvents, Strasbourg Seminaire de Probabilités V, Lecture Notes in Math., vol. 191 319–341. Berlin Heidelberg New York: Springer 1971
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Research supported in part by Air Force Office of Scientific Research grant AFOSR 82-0029B
Research supported in part by NSF grant MCS-8404539
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Karr, A.F., Pittenger, A.O. Structural properties of randomized times. Probab. Th. Rel. Fields 72, 395–415 (1986). https://doi.org/10.1007/BF00334193
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DOI: https://doi.org/10.1007/BF00334193
Keywords
- Stochastic Process
- Structural Property
- Probability Theory
- Statistical Theory
- Markov Process