Potential processes
HTML articles powered by AMS MathViewer
- by R. V. Chacon PDF
- Trans. Amer. Math. Soc. 226 (1977), 39-58 Request permission
Abstract:
The prototype of a potential process is a stochastic process which visits the same points in the same order as a Markov process, but at a rate obtained from a nonanticipating time change. The definition of a potential process may be given intrinsically and most generally without mention of a Markov process, in terms of potential theory. The definition may be given more directly and less generally in terms of potentials which arise from Markov processes, or more directly than this, as suitably time-changed Markov processes. The principal purpose of studying the class of potential processes, which may be shown to include martingales as well as Markov processes themselves, is to give a unified treatment to a wide class of processes which has potential theory at its core. That it is possible to do so suggests that potential rather than martingale results are central to the study of Markov processes. Furthermore, this also suggests that it is not the Markov property itself which makes Markov processes tractable, but rather the potential structure which can be constructed with the assistance of the Markov property. The general theory of potential processes is developed in a forthcoming paper. It will be shown there that a Markov process subject to an ordinary continuous nonanticipating time change is a local potential process. It may be seen, by examining examples, that it is necessary to consider randomized stopping times and randomized nonanticipating time changes in the general case. In the forthcoming paper a more general notion than randomized nonanticipating time changes is used to obtain a characterization of potential processes. It is an open problem whether randomization itself is sufficient in the general case, and whether ordinary nonanticipating time changes are sufficient for continuous parameter martingales and Brownian motion on the line. The emphasis in the present paper will be on developing the theory of discrete parameter martingales as a special case of the general theory.References
- D. G. Austin, G. A. Edgar, and A. Ionescu Tulcea, Pointwise convergence in terms of expectations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 30 (1974), 17–26. MR 358945, DOI 10.1007/BF00532860
- J. R. Baxter and R. V. Chacon, Potentials of stopped distributions, Illinois J. Math. 18 (1974), 649–656. MR 358960, DOI 10.1215/ijm/1256051015
- Leo Breiman, On the tail behavior of sums of independent random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9 (1967), 20–25. MR 226707, DOI 10.1007/BF00535464
- K. È. Dambis, On decomposition of continuous submartingales, Teor. Verojatnost. i Primenen. 10 (1965), 438–448 (Russian, with English summary). MR 0202179
- J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
- Lester E. Dubins and Gideon Schwarz, On continuous martingales, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 913–916. MR 178499, DOI 10.1073/pnas.53.5.913
- Lester E. Dubins, On a theorem of Skorohod, Ann. Math. Statist. 39 (1968), 2094–2097. MR 234520, DOI 10.1214/aoms/1177698036 W. Hall, On the Skorohod embedding theorem, J. Appl. Probability 7 (1970).
- J. Kiefer, Skorohod embedding of multivariate RV’s, and the sample DF, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 24 (1972), no. 1, 1–35. MR 1554013, DOI 10.1007/BF00532460
- Frank B. Knight, A reduction of continuous square-integrable martingales to Brownian motion, Martingales (Rep. Meeting, Oberwolfach, 1970) Lecture Notes in Math., Vol. 190, Springer, Berlin, 1971, pp. 19–31. MR 0370741
- Itrel Monroe, On embedding right continuous martingales in Brownian motion, Ann. Math. Statist. 43 (1972), 1293–1311. MR 343354, DOI 10.1214/aoms/1177692480
- V. A. Rohlin, On the fundamental ideas of measure theory, Mat. Sbornik N.S. 25(67) (1949), 107–150 (Russian). MR 0030584
- D. H. Root, The existence of certain stopping times on Brownian motion, Ann. Math. Statist. 40 (1969), 715–718. MR 238394, DOI 10.1214/aoms/1177697749
- Stanley Sawyer, A remark on the Skorohod representation, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 23 (1972), 67–74. MR 310939, DOI 10.1007/BF00536691
- Gordon Simons, A martingale decomposition theorem, Ann. Math. Statist. 41 (1970), 1102–1104. MR 261678, DOI 10.1214/aoms/1177696991
- Hiroshi Kunita and Shinzo Watanabe, On square integrable martingales, Nagoya Math. J. 30 (1967), 209–245. MR 217856, DOI 10.1017/S0027763000012484
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 226 (1977), 39-58
- MSC: Primary 60J45; Secondary 31D05
- DOI: https://doi.org/10.1090/S0002-9947-1977-0501374-5
- MathSciNet review: 0501374