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XV: 19, 278-284, LNM 850 (1981)

**ÉMERY, Michel**

Le théorème de Garnett-Jones, d'après Varopoulos (Martingale theory)

Let $M$ be a martingale belonging to $BMO$. The John-Nirenberg theorem implies that, for some constant $0<\lambda<\infty$, the conditional expectations $E[\exp( {1\over\lambda}(M_{\infty} -M_{T_-}))\, |\,{\cal F}_T]$ belongs to $L^{\infty}$ for all stopping times $T$, with a norm independent of $T$. The Garnett-Jones theorem (proved by Varopoulos in the probabilistic set-up) asserts that the smallest such $\lambda$ is ``equivalent'' to the $BMO$ distance of $M$ to the subspace $L^\infty$. One half of the equivalence is general, while the other half requires all martingales of the filtration to be continuous. The examples given in the second part show that this hypothesis is essential

Keywords: $BMO$

Nature: Exposition, Original additions

Retrieve article from Numdam

Le théorème de Garnett-Jones, d'après Varopoulos (Martingale theory)

Let $M$ be a martingale belonging to $BMO$. The John-Nirenberg theorem implies that, for some constant $0<\lambda<\infty$, the conditional expectations $E[\exp( {1\over\lambda}(M_{\infty} -M_{T_-}))\, |\,{\cal F}_T]$ belongs to $L^{\infty}$ for all stopping times $T$, with a norm independent of $T$. The Garnett-Jones theorem (proved by Varopoulos in the probabilistic set-up) asserts that the smallest such $\lambda$ is ``equivalent'' to the $BMO$ distance of $M$ to the subspace $L^\infty$. One half of the equivalence is general, while the other half requires all martingales of the filtration to be continuous. The examples given in the second part show that this hypothesis is essential

Keywords: $BMO$

Nature: Exposition, Original additions

Retrieve article from Numdam