## Electronic Communications in Probability

### On relaxing the assumption of differential subordination in some martingale inequalities

#### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 2, 9-21.

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

https://projecteuclid.org/euclid.ecp/1465261958

Digital Object Identifier
doi:10.1214/ECP.v16-1593

Mathematical Reviews number (MathSciNet)
MR2753300

Zentralblatt MATH identifier
1231.60036

Subjects
Primary: 60G44: Martingales with continuous parameter
Secondary: 60G42: Martingales with discrete parameter

Rights

#### Citation

Osekowski, Adam. On relaxing the assumption of differential subordination in some martingale inequalities. Electron. Commun. Probab. 16 (2011), paper no. 2, 9--21. doi:10.1214/ECP.v16-1593. https://projecteuclid.org/euclid.ecp/1465261958

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