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

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Yan L., Kazamaki N.

Studia Mathematica

2005

tom 168

nr 2

129-134

Let X = (Xₜ,ℱₜ) be a continuous BMO-martingale, that is, $||X||_{BMO} ≡ sup_{T}|| E[|X_{∞}-X_{T}| | ℱ_{T}] ||_{∞} < ∞$, where the supremum is taken over all stopping times T. Define the critical exponent b(X) by $b(X) = {b > 0: sup_{T}|| E[exp(b²(⟨X⟩_{∞} - ⟨X⟩_{T})) | ℱ_{T}] ||_{∞} < ∞}$, where the supremum is taken over all stopping times T. Consider the continuous martingale q(X) defined by $q(X)ₜ = E[⟨X⟩_{∞} | ℱₜ] - E[⟨X⟩_{∞} | ℱ₀]$. We use q(X) to characterize the distance between ⟨X⟩ and the class $L^{∞}$ of all bounded martingales in the space of continuous BMO-martingales, and we show that the inequalities $1/4d₁(q(X),L^{∞}) ≤ b(X) ≤ 4/d₁(q(X),L^{∞})$ hold for every continuous BMO-martingale X.

Ograniczanie wyników

1 Studia Mathematica

1 Kazamaki N.

1 Yan L.

1 2005

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