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  Volume 9, Issue 4, Article 93
Sharp Norm Inequality for Bounded Submartingales and Stochastic Integrals

    Authors: Adam Osekowski,  
    Keywords: Martingale, Submartingale, Stochastic integral, Norm inequality, Differential subordination.  
    Date Received: 25/02/08  
    Date Accepted: 22/10/08  
    Subject Codes:

Pri: 60642; Sec: 60H05.

    Editors: Sever S. Dragomir,  

Let $ lpha in [0,1]$ be a fixed number and $ f=(f_n)$ be a nonnegative submartingale bounded from above by $ 1$. Assume $ g=(g_n)$ is a process satisfying, with probability $ 1$,

$displaystyle ert dg_nertleq ert df_nert,   ertmathbb{E}(dg_{n+1... ... lpha mathbb{E}(df_{n+1}ertmathcal{F}_{n}),qquad  n=0, 1, 2, ldots.$    

We provide a sharp bound for the first moment of the process $ g$. A related estimate for stochastic integrals is also established.

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