Chao, Tsung-Ming; Chou, Ching-Sung On some inequalities of multiple stochastic integrals for normal martingales. (English) Zbl 1043.60509 Stochastics Stochastics Rep. 64, No. 3-4, 161-176 (1998). Summary: Let \((X_ t)_ {t\geq0}\) be a càdlàg martingale with sharp bracket \(\langle X,X\rangle_ t=t\). Define \[ X_ n=\int^ \infty_ 0dX_ {s_ 1}\int^ {s^ -_ 1}_ 0dX_ {s_ 2}\cdots \int^ {s^ -_ {n-1}}_ 0dX_ {s_ n}f_ n(s_ 1,s_ 2,\cdots,s_ n), \]\[ Y_ n=\Big(\int^ \infty_ 0ds_ 1\int^ {s_ 1}_ 0ds_ 2\cdots \int^ {s_ {n-1}}_ 0ds_ nf^ 2_ n(s_ 1,s_ 2,\cdots,s_ n)\Big)^ {1/2}, \] where \(f_ n\): \({R}^ {n-1}_ +\times{R}_ +\times\Omega\to R\) is deterministic in the first \(n-1\) variables, predictable in the \(n\)th variable and \(E(X^ 2_ n)=E(Y^ 2_ n)<\infty\). Then there exist two constants \(c_ {n,p},C_ {n,p}\) depending only on \(n,p\) such that \(c_ {n,p}E(Y^ p_ n)\leq E(| X_ n| ^ p)\leq C_ {n,p}E(Y^ p_ n)\), with \(c_ {n,p}=(p/2)^ {-p_ n/2}, C_ {n,p}=4^ {p_ n/2}(e(1+p/2))^ {n(1+p/2)}\), where the right-hand inequality holds for any \(p>0\), and the left-hand inequality holds for any \(p\geq2\). MSC: 60G42 Martingales with discrete parameter 60E15 Inequalities; stochastic orderings 60G44 Martingales with continuous parameter 60H05 Stochastic integrals PDFBibTeX XMLCite \textit{T.-M. Chao} and \textit{C.-S. Chou}, Stochastics Stochastics Rep. 64, No. 3--4, 161--176 (1998; Zbl 1043.60509) Full Text: DOI