|
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
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
Linking options:
https://www.mathnet.ru/eng/tvp3264 https://www.mathnet.ru/eng/tvp/v22/i3/p620
|
Statistics & downloads: |
Abstract page: | 228 | Full-text PDF : | 81 |
|