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On relaxing the assumption of differential subordination in some martingale inequalities


 
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1. Title Title of document On relaxing the assumption of differential subordination in some martingale inequalities
 
2. Creator Author's name, affiliation, country Adam Osekowski; University of Warsaw
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Martingale; differential subordination; moment inequality; weak-type inequality
 
3. Subject Subject classification 60G44; 60G42
 
4. Description Abstract

Let $X$, $Y$ be continuous-time martingales taking values in a se\-pa\-rable 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.
 
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7. Date (YYYY-MM-DD) 2011-01-02
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1593
 
10. Identifier Digital Object Identifier 10.1214/ECP.v16-1593
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 16
 
12. Language English=en
 
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