On the existence of optional versions for martingales

L. I. Gal'čuk

Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 3, pp. 620-622

Résumé

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 $T$, $$ X_TI_{(T<\infty)}=\mathbf M[XI_{(T<\infty)}\mid\mathscr F_t]\ a.\,s. $$