A STOCHASTIC GRONWALL LEMMA

作者：Scheutzow Michael^*

来源：Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2013, 16(2): 1350019.

DOI：10.1142/S0219025713500197

• Stochastic Gronwall lemma
• martingale inequality

We prove a stochastic Gronwall lemma of the following type: if Z is an adapted non-negative 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 is an element of (0, 1) the pth moment of the supremum of Z is bounded by a constant kappa(p) (which does not depend on M) times the pth 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.

• 出版日期2013-6