XV: 25, 347-350, LNM 850 (1981)
MAISONNEUVE, Bernard
Surmartingales-mesures (
Martingale theory)
Consider a discrete filtration $({\cal F}_n)$ and let ${\cal A}$ be the algebra, $\cup_n {\cal F}_n$, generating a $\sigma$-algebra ${\cal F}_\infty$. A positive supermartingale $(X_n)$ is called a supermartingale measure if the set function $A\mapsto\lim_n\int_A X_n\,dP$ on $A$ is $\sigma$-additive, and thus can be extended to a measure $\mu$. Then the Lebesgue decomposition of this measure is described (theorem 1). More generally, the Lebesgue decomposition of any measure $\mu$ on ${\cal F}_\infty$ is described. This is meant to complete theorem III.1.5 in Neveu,
Martingales à temps discret Comment: The author points out at the end that theorem 2 had been already proved by Horowitz (
Zeit. für W-theorie, 1978) in continuous time. This topic is now called Kunita decomposition, see
1005 and the corresponding references
Keywords: Supermartingales,
Kunita decompositionNature: Original
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