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

• Autorzy: Litan Yan , Norihiko Kazamaki
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 60G44: Martingales with continuous parameter
• 60G46: Martingales and classical analysis

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2005
• Tom: 168
• Numer: 2
• Strony: 129-134

Daty

wydano

2005

Twórcy

autor

Litan Yan

• Department of Mathematics, College of Science, Donghua University, 1882 West Yan'an Rd., Shanghai 200051, P.R. China

autor

Norihiko Kazamaki

• Department of Mathematics, Toyama University, 3190 Gofuku,Toyama 930-8555, Japan

Identyfikatory

DOI

10.4064/sm168-2-3