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Teor. Veroyatnost. i Primenen., 1965, Volume 10, Issue 3, Pages 438–448 (Mi tvp540)  

This article is cited in 2 scientific papers (total in 2 papers)

On decomposition of continuous submartingales

K. È. Dambis

Moscow

Abstract: Doob (see [2], p. 267) proved that every discrete parameter submartingale $X=(x_n,\mathfrak F_n)$, $1\le n<\infty$, may be decomposed as a sum $X=\Psi+\Gamma$ where $\Psi=(\psi_n,\mathfrak F_n)$ is a non-decreasing process and $\Gamma=(\gamma_n,\mathfrak F_n)$ is a martingale. Meyer (see [4], p. 199) found necessary and sufficient conditions for a right continuous submartingale $X=(x_t,\mathfrak F_t)$, $0\le t<\infty$ to have Doob's decomposition. In the present paper a generalisation of Doob's decomposition is obtained which is applicable to every continuous submartingale. The second main result of this paper consist in the fact that every continuous martingale $X=(x_t,\mathfrak F_t)$, $0\le t<\infty$, with $X_0=0$ has an equivalent one $X'=(x'_t,\mathfrak F'_t)$, $0\le t<\infty$, which may be obtained from some Wiener process by means of a continuous random time change. Finally we prove that sample functions of a continuous submartingale (martingale) either have infinite variation or nondecrease (are constant) on every interval.

Full text: PDF file (2013 kB)

English version:
Theory of Probability and its Applications, 1965, 10:3, 401–410

Bibliographic databases:

Received: 04.02.1964

Citation: K. È. Dambis, “On decomposition of continuous submartingales”, Teor. Veroyatnost. i Primenen., 10:3 (1965), 438–448; Theory Probab. Appl., 10:3 (1965), 401–410

Citation in format AMSBIB
\Bibitem{Dam65}
\by K.~\`E.~Dambis
\paper On decomposition of continuous submartingales
\jour Teor. Veroyatnost. i Primenen.
\yr 1965
\vol 10
\issue 3
\pages 438--448
\mathnet{http://mi.mathnet.ru/tvp540}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=202179}
\zmath{https://zbmath.org/?q=an:0141.15102}
\transl
\jour Theory Probab. Appl.
\yr 1965
\vol 10
\issue 3
\pages 401--410
\crossref{https://doi.org/10.1137/1110048}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Zhou H.-l., Wang Sh.-ya., “A Computation of the Price of Convertible Bonds with Changes of Numeraire and Changes of Time”, Advances in Business Intelligence and Financial Engineering, Advances in Intelligent Systems Research, 5, 2008, 142–148  isi
    2. Engelbert H.-J., Peskir G., “Stochastic Differential Equations For Sticky Brownian Motion”, Stochastics, 86:6 (2014), 993–1021  crossref  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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