Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T01:29:27.897Z Has data issue: false hasContentIssue false
  • English
  • Français

The Doob-Meyer Decomposition Revisited

Published online by Cambridge University Press:  20 November 2018

Richard F. Bass
Richard F. Bass*
Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington 98195, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A new proof is given of the Doob-Meyer decomposition of a supermartingale into martingale and decreasing parts. Although not the most concise proof, the proof is elementary in the sense that nothing more sophisticated than Doob's inequality is used. If the supermartingale is bounded and the jump times are totally inaccessible, then it is shown that discrete time approximations converge to the decreasing part in L2. The general case is handled by reduction to the above special case.

Keywords

Primary: 60G07secondary: 60G0560G44supermartingalesDoob-Meyer decompositionpredictabletotally inaccessiblemartingales
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 2 , 01 June 1996 , pp. 138 - 150
Copyright
Copyright © Canadian Mathematical Society 1996

References

[B] Bass, R. F., Probabilistic Techniques in Analysis, New York, Springer, 1995.Google Scholar
[DM1] Dellacherie, C. and Meyer, R-A., Probabilités et Potentiel, Vol. 7, Paris, Hermann, 1975.Google Scholar
[DM2] Dellacherie, C. and Meyer, R-A., Probabilités et Potentiel: Théorie des Martingales, Vol. 2, Paris, Hermann, 1980.Google Scholar
[D] Doob, J .L., Stochastic Processes, New York, Wiley, 1953.Google Scholar
[DD] Doléans-Dade, C., Existence du processus croissant naturel associé à un potentiel de la classe (D), Zeit. Wahrschein. 9(1968), 309–314.Google Scholar
[IW] Ikeda, N. and Watanabe, S., Stochastic Differential Equations and Diffusion Processes, North Holland Kodansha, Amsterdam, 1981.Google Scholar
[Ml] Meyer, P.-A., A decomposition theorem for supermartingales, Illinois J. Math. 6(1962), 193–205.Google Scholar
[M2] Meyer, P.-A., Decomposition of supermartingales: the uniqueness theorem, Illinois J. Math. 7(1963), 1—17.Google Scholar
[P] Protter, P., Stochastic Integration and Differential Equations, New York, Springer, 1990.Google Scholar
[R] Rao, K. M., On decomposition theorems of Meyer, Math. Scand. 24(1969), 66–78.Google Scholar