A Stochastic Gronwall Lemma
Abstract
We prove a stochastic Gronwall lemma of the following type: if $Z$ is an adapted nonnegative continuous process which satisfies a linear integral inequality with an added continuous local martingale $M$ and a process $H$ on the right hand side, then for any $p \in (0,1)$ the $p$th moment of the supremum of $Z$ is bounded by a constant $\kappa_p$ (which does not depend on $M$) times the $p$th moment of the supremum of $H$. Our main tool is a martingale inequality which is due to D. Burkholder. We provide an alternative simple proof of the martingale inequality which provides an explicit numerical value for the constant $c_p$ appearing in the inequality which is at most four times as large as the optimal constant.
 Publication:

arXiv eprints
 Pub Date:
 April 2013
 arXiv:
 arXiv:1304.5424
 Bibcode:
 2013arXiv1304.5424S
 Keywords:

 Mathematics  Probability;
 60G44
 EPrint:
 To appear in {\em Infin. Dimens. Anal. Quantum Probab. Relat. Top.}