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Teoriya Veroyatnostei i ee Primeneniya, 1977, Volume 22, Issue 3, Pages 620–622 (Mi tvp3264)  

This article is cited in 11 scientific papers (total in 11 papers)

Short Communications

On the existence of optional versions for martingales

L. I. Gal'čuk

Moscow
Abstract: Let $(\Omega,\mathscr F,\mathbf P)$ be a complete probability space and $(\mathscr F_t)$, $t\in[0,\infty)$, be an increasing family of $\sigma$-subalgebras of $\mathscr F$, $\mathscr F_t$ being not necessarily complete and right continuous.
A stochastic process $(X_t)$, $t\in [0,\infty)$, is said to be optional if it is measurable with respect to the $\sigma$-algebra $\mathscr O$ in $\Omega\times[0,\infty)$ generated by all the processes which are well adapted with respect to $(\mathscr F_t)$, right continuous and have limits from the left at each point.
The purpose of this paper is to prove the following
Theorem. Let $X$ be an integrable random variable. Then there exists a unique (to within indistinguishability) version $(X_t)$ of the martingale $(\mathbf M[X\mid\mathscr F_t])$ such that $(X_t)$ is optional and, for any stopping time $T$,
$$ X_TI_{(T<\infty)}=\mathbf M[XI_{(T<\infty)}\mid\mathscr F_t]\ a.\,s. $$
Received: 02.04.1976
English version:
Theory of Probability and its Applications, 1978, Volume 22, Issue 3, Pages 572–573
DOI: https://doi.org/10.1137/1122068
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: L. I. Gal'čuk, “On the existence of optional versions for martingales”, Teor. Veroyatnost. i Primenen., 22:3 (1977), 620–622; Theory Probab. Appl., 22:3 (1978), 572–573
Citation in format AMSBIB
\Bibitem{Gal77}
\by L.~I.~Gal'{\v{c}}uk
\paper On the existence of optional versions for martingales
\jour Teor. Veroyatnost. i Primenen.
\yr 1977
\vol 22
\issue 3
\pages 620--622
\mathnet{http://mi.mathnet.ru/tvp3264}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=471067}
\zmath{https://zbmath.org/?q=an:0397.60038}
\transl
\jour Theory Probab. Appl.
\yr 1978
\vol 22
\issue 3
\pages 572--573
\crossref{https://doi.org/10.1137/1122068}
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  • https://www.mathnet.ru/eng/tvp/v22/i3/p620
  • This publication is cited in the following 11 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
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