Une définition faible de BMO. (A weak definition of BMO) (Q1087225)

A local martingale \((M_ t)_ t\) belongs to BMO iff, by definition, there exists a constant c such that the inequality \[ E((M_{\infty}- M_{T-})^ 2| {\mathcal F}_ T)\leq c^ 2 \] holds for every stopping time T. The author proves that this definition is equivalent with any of the following four inequalities, supposed to hold for every stopping time T: (1) There exists an increasing function \(G: {\mathbb{R}}_+\to {\mathbb{R}}_+\) and a constant \(c<\sup G\) such that \(E(G(| M_{\infty}-M_{T- })| {\mathcal F}_ T)\leq c;\) (2) there exists an increasing function G and a constant c as before such that \(E(G([M,M]_{\infty}-[M,M]_{T-})| {\mathcal F}_ T)\leq c\), where [M,M] is the natural increasing process of the local martingale M; (3) there exist \(a>0\), \(\epsilon >0\) such that \(P([M,M]_{\infty}- [M,M]_{T-}>a| {\mathcal F}_ T)\leq 1-\epsilon;\) (4) there exist a and \(\epsilon\) as before such that P(\(\sup_{t}| M_{T+t}-M_{T-}| >a| {\mathcal F}_ T)\leq 1-\epsilon.\) The main tool in the proofs is the following simple and smart Lemma (we do not know if it is known): Suppose that \(A_ t\) is an increasing natural process satisfying the following two conditions for every stopping time \(T\): \[ P(A_{\infty}-A_{T-}>a| {\mathcal F}_ T)\leq \alpha \text{ for some } a,\alpha; \] \[ P(A_{\infty}-A_{T-}>b| {\mathcal F}_ T)\leq \beta \text{ for some } b,\beta. \] Then \(P(A_{\infty}-A_{T-}>a+b| {\mathcal F}_ T)\leq \alpha \beta\) for every stopping time T. We were not able to detect the connection between the paper and its motto.