On a representation of local martingales

R. Š. Lipcer

Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 4, pp. 718-726

Résumé

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 $$ y_t=y_0+\int_0^t f(s)\,d\xi_s. $$