Abstract
The concept of finitely additive supermartingales, originally due to Bochner, is revived and developed. We exploit it to study measure decompositions over filtered probability spaces and the properties of the associated Doléans-Dade measure. We obtain versions of the Doob–Meyer decomposition and, as an application, we establish a version of the Bichteler and Dellacherie theorem with no exogenous probability measure.
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Agnew, R.P., Morse, A.P.: Extensions of linear functionals, with applications to limits, integrals, measures, and densities. Ann. Math. 39, 20–30 (1938)
Armstrong, T.: Finitely additive F-processes. Trans. Am. Math. Soc. 279, 271–295 (1983)
Armstrong, T.: Finitely additive supermartingales and differences of martingales and adapted increasing processes. Proc. Am. Math. Soc. 95, 619–625 (1985)
Berti, P., Rigo, P.: Convergence in distribution of non-measurable random elements. Ann. Probab. 32, 365–379 (2004)
Bhaskara Rao, K.P.S., Bhaskara Rao, M.: Theory of Charges. Academic Press, London (1983)
Bochner, S.: Stochastic processes. Ann. Math. 48, 1014–1061 (1947)
Bochner, S.: Partial ordering in the theory of stochastic processes. Proc. Natl. Acad. Sci. 36, 439–443 (1950)
Bochner, S.: Partial ordering in the theory of martingales. Ann. Math. 62, 162–169 (1955)
Cassese, G.: Decomposition of supermartingales indexed by a linearly ordered set. Stat. Probab. Lett. 77, 795–802 (2007)
Cassese, G.: Asset pricing with no exogenous probability measure. Math. Fin. 18, 23–54 (2008)
Dellacherie, C., Meyer, P.A.: Probabilities and Potential B. North-Holland, Amsterdam (1982)
Doléans-Dade, C.: Existence du processus croissant naturel associé à un potentiel de la classe (D). Z. Wahrscheinlichkeitstheor. Verw. Geb. 9, 309–314 (1968)
Dubins, L.: Finitely additive conditional probabilities, conglomerability and disintegrations. Ann. Probab. 3, 79–99 (1975)
Dudley, R.M.: Nonmetric compact spaces and nonmeasurable processes. Ann. Probab. 108, 1001–1005 (1990)
Dunford, N., Schwartz, J.: Linear Operators. Wiley, New York (1988)
Föllmer, H.: On the representation of semimartingales. Ann. Probab. 5, 580–589 (1973)
Kadane, J.B., Schervish, M.J., Seidenfeld, T.: Reasoning to a foregone conclusion. J. Am. Stat. Assoc. 91, 1228–1235 (1996)
Karatzas, I., Zitkovic, G.: Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31, 1821–1858 (2003)
Komlos, J.: A generalization of a problem of Steinhaus. Acta Math. Hung. 18, 217–229 (1967)
Metivier, M., Pellaumail, J.P.: On Doléans–Föllmer measure for quasi-martingales, Ill. J. Math. 19, 491–504 (1975)
Meyer, P.A.: Un cours sur les integrales stochastiques. Sém. Probab. 10, 245–400 (1976)
Schwartz, M.: New proofs of a theorem of Komlos. Acta Math. Hung. 45, 181–185 (1986)
Sion, M.: On general minimax theorems. Pac. J. Math. 8, 171–176 (1958)
van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. Springer, New York (1996)
Yosida, K., Hewitt, E.: Finitely additive measures. Trans. Am. Math. Soc. 72, 46–66 (1952)
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I am indebted to an anonymous referee for several helping suggestions.
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Cassese, G. Finitely Additive Supermartingales. J Theor Probab 21, 586–603 (2008). https://doi.org/10.1007/s10959-008-0164-8
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DOI: https://doi.org/10.1007/s10959-008-0164-8
Keywords
- Bichteler–Dellacherie theorem
- Conditional expectation
- Doléans-Dade measure
- Doob–Meyer decomposition
- Finitely additive measures
- Supermartingales
- Yosida–Hewitt decomposition