EUDML

Let X = (Xₜ,ℱₜ) be a continuous BMO-martingale, that is, ${| | X | |}_{B M O} \equiv s u p_{T} | | E [| X_{\infty} - X_{T} | | ℱ_{T} {] | |}_{\infty} < \infty$ , where the supremum is taken over all stopping times T. Define the critical exponent b(X) by $b (X) = b > 0 : s u p_{T} | | E [e x p (b ² (⟨ X ⟩_{\infty} - ⟨ X ⟩_{T})) | ℱ_{T}] {| |}_{\infty} < \infty$ , where the supremum is taken over all stopping times T. Consider the continuous martingale q(X) defined by $q (X) ₜ = E [⟨ X ⟩_{\infty} | ℱ ₜ] - E [⟨ X ⟩_{\infty} | ℱ ₀]$ . We use q(X) to characterize the distance between ⟨X⟩ and the class $L^{\infty}$ of all bounded martingales in the space of continuous BMO-martingales, and we show that the inequalities $1 / 4 d ₁ (q (X), L^{\infty}) \leq b (X) \leq 4 / d ₁ (q (X), L^{\infty})$ hold for every continuous BMO-martingale X.

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Krickeberg's decomposition for local martingales

A simple remark on the conditioned square functions for martingale transforms

A characterization of $B M O$ martingales

Une note sur les martingales faibles

A remark on the class of martingales with bounded quadratic variation

A remark on a problem of Girsanov

A counterexample related to $A_{p}$ -weights in martingale theory

Note on a stochastic integral equation

Examples on local martingales

Some properties of martingale integrals

Some remarks on ratio inequalities for continuous martingales

On the distance between ⟨X⟩ and $L^{\infty}$ in the space of continuous BMO-martingales