On some inequalities of local times for Azéma

Abstract

Let$(u_t,{calG}_t{)}_{t\ge 0}$ be Azéma martingale and its filtration, and let$({\lambda }_t^x;x\in R,t\ge 0)$ be the local times of the Azéma martingale defined by the following Tanaka formula: $$u_t{1}_{\left\{u_t>x\right\}}={\int }_0^t{1}_{\left\{u_{s^{-}}>x\right\}}d{u}_s+\frac{1}{2}{\lambda }_t^x.$$ Then, for every$({calG}_t{)}_{t\ge 0}$ stopping time T and every p>0, there exist two universal constants c_p, C_p >0 depending only on p, such that $$c_p\parallel T^{1/2}{\parallel }_p\le \parallel {\lambda }_T^{*}{\parallel }_p\le C_p\parallel T^{1/2}{\parallel }_p,$$ where${\lambda }_t^{*}={sup}_{x\in R}{\lambda }_t^x$ .