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2011 On relaxing the assumption of differential subordination in some martingale inequalities
Adam Osekowski
Author Affiliations +
Electron. Commun. Probab. 16: 9-21 (2011). DOI: 10.1214/ECP.v16-1593

Abstract

Let $X$, $Y$ be continuous-time martingales taking values in a separable Hilbert space $\mathcal{H}$.

(i) Assume that $X$, $Y$ satisfy the condition $[X,X]_t\geq [Y,Y]_t$ for all $t\geq 0$. We prove the sharp inequalities $$ \sup_t||Y_t||_p\leq (p-1)^{-1}\sup_t||X_t||_p,\qquad 1 < p\leq 2,$$ $$ \mathbb{P}(\sup_t|Y_t|\geq 1)\leq \frac{2}{\Gamma(p+1)}\sup_t||X_t||_p^p,\qquad 1\leq p\leq 2,$$ and for any $K>0$ we determine the optimal constant $L=L(K)$ depending only on $K$ such that $$ \sup_t ||Y_t||_1\leq K\sup_t\mathbb{E}|X_t|\log|X_t|+L(K).$$

(ii) Assume that $X$, $Y$ satisfy the condition $[X,X]_\infty-[X,X]_{t-}\geq [Y,Y]_\infty-[Y,Y]_{t-}$ for all $t\geq 0$. We establish the sharp bounds $$ \sup_t||Y_t||_p\leq (p-1)\sup_t||X_t||_p,\qquad 2\leq p < \infty$$ and $$ \mathbb{P}(\sup_t|Y_t|\geq 1)\leq \frac{p^{p-1}}{2}\sup_t||X_t||_p^p,\qquad 2\leq p < \infty.$$

This generalizes the previous results of Burkholder, Suh and the author, who showed the above estimates under the more restrictive assumption of differential subordination. The proof is based on Burkholder's technique and integration method.

Citation

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Adam Osekowski. "On relaxing the assumption of differential subordination in some martingale inequalities." Electron. Commun. Probab. 16 9 - 21, 2011. https://doi.org/10.1214/ECP.v16-1593

Information

Accepted: 2 January 2011; Published: 2011
First available in Project Euclid: 7 June 2016

zbMATH: 1231.60036
MathSciNet: MR2753300
Digital Object Identifier: 10.1214/ECP.v16-1593

Subjects:
Primary: 60G44
Secondary: 60G42

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