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Teoriya Veroyatnostei i ee Primeneniya, 1976, Volume 21, Issue 4, Pages 718–726
(Mi tvp3418)
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This article is cited in 1 scientific paper (total in 2 paper)
On a representation of local martingales
R. Š. Lipcer Moscow
Abstract:
Let $(\Omega,\mathscr F,\mathbf P)$ be a probability space, $(\mathscr F_t)$, $t\ge 0$, be an increasing and right-continuous family of $\sigma$-subalgebras of $\mathscr F$ , and $(\xi_t,\mathscr F_t)$, $t\ge 0$, be a random process on $(\Omega,\mathscr F,\mathbf P)$ with continuous trajectories such that the process $(\xi_t-\xi_0,\mathscr F_t)$ , $t\ge 0$, is a local martingale. Denote by $(\mathscr F_t^{\xi})$, $t\ge 0$, the family of $\sigma$-algebras $\sigma(\xi_s,s\le t)$ and by $\mathbf Q$ the restriction of the measure $\mathbf P$ onto the $\sigma$-algebra $\mathscr F_{\infty}^{\xi}$. Let $\mathbf Q'$ be another probability measure on the measurable space $(\Omega,\mathscr F_{\infty}^{\xi})$ such that
(I) $\mathbf Q'\ll\mathbf Q$,
(II) the process $(\xi_t-\xi_0,\mathscr F_t^{\xi},\mathbf Q')$, $t\ge 0$, is a local martingale,
(III) the restrictions of the measures $\mathbf Q$ and $\mathbf Q'$ onto the $\sigma$-algebra $\mathscr F_0^{\xi}$ coincide.
The main result of this paper is: if every measure $\mathbf Q'$, which satisfies conditions (I)–(III), coincides with $\mathbf Q$, then any local martingale $(y_t,\mathscr F_t^{\xi})$, $t\ge 0$, has a representation of the form
$$
y_t=y_0+\int_0^t f(s)\,d\xi_s.
$$
Received: 30.01.1976
Citation:
R. Š. Lipcer, “On a representation of local martingales”, Teor. Veroyatnost. i Primenen., 21:4 (1976), 718–726; Theory Probab. Appl., 21:4 (1977), 698–705
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https://www.mathnet.ru/eng/tvp3418 https://www.mathnet.ru/eng/tvp/v21/i4/p718
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