Sharp maximal inequalities for the moments of martingales and non-negative submartingales
arXiv:1201.1089 · doi:10.3150/10-BEJ314
Abstract
In the paper we study sharp maximal inequalities for martingales and non-negative submartingales: if , are martingales satisfying \[|\mathrm{d}g_n|\leq|\mathrm{d}f_n|,\qquad n=0,1,2,...,\] almost surely, then \[\Bigl\|\sup_{n\geq0}|g_n|\Bigr\|_p\leq p\|f\|_p,\qquad p\geq2,\] and the inequality is sharp. Furthermore, if , is a non-negative submartingale and satisfies \[|\mathrm{d}g_n|\leq|\mathrm{d}f_n|\quad and\quad |\mathbb{E}(\mathrm{d}g_{n+1}|\mathcal {F}_n)|\leqα\mathbb{E}(\mathrm{d}f_{n+1}|\mathcal{F}_n),\qquad n=0,1,2,...,\] almost surely, then \[\Bigl\|\sup_{n\geq0}|g_n|\Bigr\|_p\leq(α+1)p\|f\|_p,\qquad p\geq2,\] and the inequality is sharp. As an application, we establish related estimates for stochastic integrals and Itô processes. The inequalities strengthen the earlier classical results of Burkholder and Choi.
arXiv commentPublished in at http://dx.doi.org/10.3150/10-BEJ314 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)