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

Litan Yan, and Norihiko Kazamaki. "On the distance between ⟨X⟩ and $L^{∞}$ in the space of continuous BMO-martingales." Studia Mathematica 168.2 (2005): 129-134. <http://eudml.org/doc/286666>.

@article{LitanYan2005,
abstract = {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.},
author = {Litan Yan, Norihiko Kazamaki},
journal = {Studia Mathematica},
keywords = {continuous martingales; BMO},
language = {eng},
number = {2},
pages = {129-134},
title = {On the distance between ⟨X⟩ and $L^\{∞\}$ in the space of continuous BMO-martingales},
url = {http://eudml.org/doc/286666},
volume = {168},
year = {2005},
}

TY - JOUR
AU - Litan Yan
AU - Norihiko Kazamaki
TI - On the distance between ⟨X⟩ and $L^{∞}$ in the space of continuous BMO-martingales
JO - Studia Mathematica
PY - 2005
VL - 168
IS - 2
SP - 129
EP - 134
AB - 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.
LA - eng
KW - continuous martingales; BMO
UR - http://eudml.org/doc/286666
ER -