Summary
Consider the set\(C\) of all possible distributions of triples (τ, κ, η), such that τ is a finite stopping time with associated mark κ in some fixed Polish space, while η is the compensator random measure of (τ, κ). We prove that\(C\) is convex, and that the extreme points of\(C\) are the distributions obtained when the underlying filtration is the one induced by (τ, κ). Moreover, every element of\(C\) has a corresponding unique integral representation. The proof is based on the peculiar fact that EV τ, κ=0 for every predictable processV which satisfies a certain moment condition. From this it also follows thatT τ, κ isU(0, 1) wheneverT is a predictable mapping into [0, 1] such that the image of ζ, a suitably discounted version of η, is a.s. bounded by Lebesgue measure. Iterating this, one gets a time change reduction of any simple point process to Poisson, without the usual condition of quasileftcontinuity. The paper also contains a very general version of the Knight-Meyer multivariate time change theorem.
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References
Aalen, O.O., Hoem J.M.: Random time changes for multivariate counting processes. Scand. Actuarial J., 81–101 (1978)
Brémaud, P.: Point processes and queues. Berlin Heidelberg New York: Springer 1981
Brown, T.C., Nair, M.G.: A simple proof of the multivariate random time change theorem for point processes. J. Appl. Probab.25, 210–214 (1988a)
Brown, T.C., Nair, M.G.: Poisson approximations for time-changed point processes. Stochastic Processes Appl.29, 247–256 (1988b)
Cocozza, C., Yor, M.: Démonstration d'un théorème de F. Knight à l'aide de martingales exponentielles. (Lect. Notes Math., vol. 784, pp. 496–499) Berlin Heidelberg New York: Springer 1980
Dambis, K.E.: On the decomposition of continuous submartingales. Theory Probab. Appl.10, 401–410 (1965)
Dellacherie, C., Meyer, P.A.: Probabilités et potentiel. Chap. I–VIII. Paris: Hermann 1975/1980
Dubins, L.E., Schwarz, G.: On continuous martingales. Proc. Natl. Acad. Sci. USA53, 913–916 (1965)
Elliott, R.J.: Stochastic calculus and applications. Berlin Heidelberg New York: Springer 1982
Grigelionis, B.: On representation of integer-valued random measures by means of stochastic integrals with respect to the Poisson measure (in Russian). Litov. Mat. Sb.11, 93–108 (1971)
Hall, P., Heyde, C.C.: Martingale limit theory and its application. New York: Academic Press 1980
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam, Tokyo: North Holland & Kodansha 1981
Jacod, J.: Calcul Stochastique et Problèmes de Martingales. (Lect. Notes Math., vol. 714) Berlin Heidelberg New York: Springer 1979
Jacod, J., Yor, M.: Étude des solutions extrémales et représentation intégrale des solutions pour certains problèmes de martingales. Z. Wahrscheinlichkeits. Verw. Geb.38, 83–125 (1977)
Kallenberg, O.: Random measures. 4th edn. Berlin London: Akademie-Verlag & Academic Press 1986
Kallenberg, O.: Spreading and predictable sampling in exchangeable sequences and proceses. Ann. Probab.16, 508–534 (1988)
Kallenberg, O.: General Wald-type identities for exchangeable sequences and processes. Probab. Th. Rel. Fields83, 447–487 (1989)
Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. Berlin Heidelberg New York: Springer 1988
Karoui, N., El, Lepeltier, J.P.: Représentation des processus ponctuels multivariés à l'aide d'un processus de Poisson. Z. Wahrscheinlichkeits. Verw. Geb.39, 111–133 (1977)
Knight, F.B.: An infinitesimal decomposition for a class of Markov processes. Ann. Math. Statist.41, 1510–1529 (1970)
Knight, F.B.: A reduction of continuous, square-integrable martingales to Brownian motion. (Lect. Notes Math., vol 190, pp. 19–31) Berlin Heidelberg New York: Springer 1971
Kurtz, T.G.: Representations of Markov processes as multiparameter time changes. Ann. Probab.8, 682–715 (1980)
Lépingle, D., Meyer, P.A., Yor, M.: Extrémalité et remplissage de tribus pour certaines martingales purement discontinue. (Lect. Notes Math., vol. 850, pp. 604–617) Berlin Heidelberg New York: Springer 1981
Liptser, R.S., Shiryayev, A.N.: Statistics of random processes, vol. 2. Berlin Heidelberg New York: Springer 1978
Merzbach, E., Nualart, D.: A characterization of the spatial Poisson process and changing time. Ann. Probab.14, 1380–1390 (1986)
Meyer, P.A.: Démonstration simplifiée d'un théorème de Knight. (Lect. Notes Math., vol. 191, pp. 191–195) Berlin Heidelberg New York: Springer 1971
Neveu, J.: Martingales à Temps Discret. Paris: Masson et Cie 1972
Papangelou, F.: Integrability of expected increments of point processes and a related random change of scale. Trans. Am. Math. Soc.165, 483–506 (1972)
Pitman, J., Yor, M.: Asymptotic laws of planar Brownian motion. Ann. Probab.14, 733–779 (1986)
Rogers, L.C.G., Williams, D.: Diffusions, Markov processes, and martingales, vol. 2. Chichester: Wiley 1987
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Research supported by NSF grant DMS-8703804
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Kallenberg, O. Random time change and an integral representation for marked stopping times. Probab. Th. Rel. Fields 86, 167–202 (1990). https://doi.org/10.1007/BF01474641
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DOI: https://doi.org/10.1007/BF01474641