Electronic Communications in Probability

On relaxing the assumption of differential subordination in some martingale inequalities

Adam Osekowski

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Martingale differential subordination moment inequality weak-type inequality

This work is licensed under aCreative Commons Attribution 3.0 License.


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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