Summary
As a first step in the development of a general theory of set-indexed martingales, we define predictability on a general space with respect to a filtration indexed by a lattice of sets. We prove a characterization of the predictable σ-algebra in terms of adapted and “left-continuous” processes without any form of topology for the index set. We then define a stopping set and show that it is a natural generalization of the stopping time; in particular, the predictable σ-algebra can be characterized by various stochastic intervals generated by stopping sets.
References
Adler R.: The geometry of random fields. New York: Wiley 1981
Alexander, K.S., Pyke, R.: A uniform central limit theorem for set-indexed partial sum processes with finite variance. Ann. Probab.14, 582–597 (1986)
Allain, M.F.: Tribus prévisibles et espaces de processus à trajectoires continues indexés par un espace localement compact et métrisable. Sém. Rennes. Université de Rennes, 1979
Bass, R.F., Pyke, R.: The space D(A) and weak convergence for set-indexed processes. Ann. Probab.13, 860–884 (1985)
Dozzi, M., Ivanoff, B.G., Merzbach, E.: Doob-Meyer decomposition for set-indexed martingales. (Preprint 1993)
Dudley, R.M.: A course on empirical processes. In: Kim, A. C., Neumann, B.H. (eds.) Groups-Korea 1983. (Lect. Notes Math., vol. 1097, pp. 1–142) Berlin Heidelberg New York: Springer 1984
Dynkin, E.B.: Markov processes and random fields. Bull. Am. Math. Soc.3, 975–1000 (1980)
Evstigneev, I.V.: Markov times for random fields. Theory Probab. Appl.22, 563–569 (1977)
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.: A compendium of continuous lattices. Berlin Heidelberg New York: Springer 1980
Hajek, B., Wong, E.: Set parametered martingales and multiple stochastic integration. In: Williams, D. (ed.) Stochastic integrals. (Lect. Notes Math., vol. 851, pp. 119–151) Berlin Heidelberg New York: Springer 1981
Hurzeler, H.: The optional sampling theorem for processes indexed by a partially ordered set. Ann. Probab.13, 1224–1235 (1985)
Jagers, P.: Aspects of random measures and point processes. In: Ney, P., Port, S. (eds.) (Adv. probab. relat. top., vol. 3, pp. 179–239. New York Basel: Dekker 1974
Kallenberg, O.: Random measures, 4th ed. New York London: Academic Press 1986
Kallianpur, G., Mandrekar, V.: The Markov property for generalized Gaussian random fields. Ann. Inst. Fourier24, 143–167 (1974)
Krengel, U., Sucheston, L.: Stopping rules and tactics for processes indexed by a directed set. J. Multivariate Anal.11, 199–229 (1981)
Kunita, H.: Stochastic flows and stochastic differential equations. Cambridge: Cambridge University Press 1990
Kurtz, T.E.: The optional sampling theorem for martingales indexed by directed sets. Ann. Probab.8, 675–681 (1980)
Lawler, G.F., Vanderbei, R.J.: Markov strategies for optimal control problems indexed by a partially ordered set. Ann. Probab.11, 642–647 (1983)
Mandelbaum, A., Vanderbei, R.J.: Optional stopping and supermartingales over partially ordered sets. Z. Wahrscheinlichkeitstheor Verw. Geb.57, 253–264 (1981)
Mandrekar, V.: Markov properties for random fields. In: Bharucha-Reid, A.J. (ed.) (Probab. anal. relat. top. vol 3 pp. 161–193). New York London: Academic Press 1983
Matheron, G.: Random sets and integral geometry, New York: Wiley 1975
Métivier, M., Pellaumail, J.: Mesures stochastiques à valeurs dans les espaces Lo. Z. Wahrscheinlichkeitstheor. Verw. Geb.40, 101–114 (1977)
Meyer, P.A.: Théorie élémentaire des processus à deux indices. Korezlioglu, H. et al. (eds.) Processus aléatories á deux indices. (Lect. Notes Math., vol. 863, pp. 1–39) Berlin Heidelberg New York: Springer 1981
Norberg, T.: Stochastic integration on lattices (Tech. Rep. Chalmers Univ. Tech.) The University of Gőteberg 1989
Rosen, J.: Self intersections of random fields. Ann. Probab.12, 108–119 (1984)
Rozanov, Y.A.: Markov random fields, Berlin Heidelberg New York: Springer 1982
Walsh, J.B.: An introduction to stochastic partial differential equations. In: Hennequin, P.L. (ed.) École d'été de Probability de St. Flour. (Lect. Notes Math., vol. 1180, pp. 266–439) Berlin Heidelberg New York: Springer 1986
Washburn, R.B., Willsky, A.S.: Optional sampling of submartingales indexed by partially ordered sets. Ann. Probab.9, 957–970 (1981)
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Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada
Research partially done while the second author was visiting the University of Ottawa. He wishes to thank the Department of Mathematics for its hospitality
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Ivanoff, B.G., Merzbach, E. & Şchiopu-Kratina, I. Predictability and stopping on lattices of sets. Probab. Th. Rel. Fields 97, 433–446 (1993). https://doi.org/10.1007/BF01192958
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DOI: https://doi.org/10.1007/BF01192958
Mathematics Classification (1985)
- 60G07
- 60G60