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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,  
 
  Abstract:

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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