A stochastic Gronwall lemma

DOI10.1142/S0219025713500197zbMATH Open1271.60056arXiv1304.5424WikidataQ124856252 ScholiaQ124856252MaRDI QIDQ2842039

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Publication date: 30 July 2013

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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 $pin(0,1)$ the $p$-th moment of the supremum of $Z$ is bounded by a constant $kapp{a}_p$ (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.

Full work available at URL: https://arxiv.org/abs/1304.5424

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