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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, 22:3, 572–573

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