Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On a subclass of square integrable martingales


Author: Dean Isaacson
Journal: Proc. Amer. Math. Soc. 34 (1972), 521-526
MSC: Primary 60J65
DOI: https://doi.org/10.1090/S0002-9939-1972-0295432-1
MathSciNet review: 0295432
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathcal{M}_2^ \ast $ denote the class of continuous, nowhere constant, square integrable martingales, $ M(t) = X({\langle M\rangle _t})$, for which $ {\langle M\rangle _t}$ is a time change on the $ \sigma $-fields generated by the Brownian motion $ X(t)$. It is shown that if $ M(t) \in \mathcal{M}_2^ \ast $, then the family of $ \sigma $-fields generated by $ M(t)$ is a right continuous family. If $ M(t) \in \mathcal{M}_2^ \ast $ and if $ \sigma \{ M(s):s \leqq t\} = \sigma \{ X(s):s \leqq t\} $ for some Brownian motion $ X(t)$, then $ M(t) = \smallint_0^t {\Phi (s)dX(s)} $ and $ X(t) = \smallint_0^t {(1/\Phi (s))dM(s)} $ for some process $ \Phi (s)$ with $ \Phi (s) \ne 0$ a.e. $ dt \times dP$.


References [Enhancements On Off] (What's this?)

  • [1] R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Appl. Math., vol. 29, Academic Press, New York, 1968. MR 41 #9348. MR 0264757 (41:9348)
  • [2] K. È. Dambis, On decomposition of continuous submartingales, Teor. Verojatnost. i Primenen. 10 (1965), 438-448 = Theor. Probability Appl. 10 (1965), 401-410. MR 34 #2052. MR 0202179 (34:2052)
  • [3] J. L. Doob, Stochastic processes, Wiley, New York; Chapman and Hall, London, 1953. MR 15, 445. MR 0058896 (15:445b)
  • [4] L. E. Dubins and G. Schwarz, On continuous martingales, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 913-916. MR 31 #2756. MR 0178499 (31:2756)
  • [5] D. L. Fisk, Sample quadratic variation of sample continuous, second order martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 6 (1966), 273-278. MR 35 #1078. MR 0210184 (35:1078)
  • [6] D. Isaacson, Stochastic integrals and derivatives, Ann. Math. Statist. 40 (1969), 1610-1616. MR 40 #5038. MR 0251811 (40:5038)
  • [7] -, Continuous martingales with discontinuous marginal distributions, Ann. Math. Statist. 42 (1971), 2139-2142. MR 0303596 (46:2733)
  • [8] P. A. Meyer, Sur un théorème de Deny, Séminaire de Probabilités (Univ. Strasbourg, 1966/67), vol. I, Springer, Berlin, 1967, pp. 163-165. MR 38 #2332. MR 0234011 (38:2332)
  • [9] -, Probability and potentials, Blaisdell, Waltham, Mass., 1966. MR 34 #5119.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60J65

Retrieve articles in all journals with MSC: 60J65


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0295432-1
Keywords: Continuous square integrable nowhere constant martingales, Brownian motion, time change, stochastic integral
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society