$L^P$ Estimates on Iterated Stochastic Integrals

Abstract

For a continuous martingale $M$, let $\langle M, M\rangle$ denote the increasing process. Let $I_0, I_1,\ldots$ denote the iterated stochastic integrals of $M$. We prove the inequalities of Burkholder-Davis-Gundy type, $A_{p,n}\|\langle M, M \rangle^{1/2}_t\|^n_{np} \leq \|I_n(t)\|_p \leq B_{p,n}\|\langle M, M \rangle^{1/2}_t\|^n_{np}$, where $\ln A_{p,n} \sim \ln B_{p,n} \sim -(n/2)\ln n$ as $n \rightarrow \infty$. Our proof requires the sharp constant $b_p$ in Burkholder-Davis-Gundy inequalities $\|M\|_p \leq b_p\|\langle M, M\rangle^{1/2}\|_p$. In the Appendix we prove $\sup_{p\geq 1}(b_p/\sqrt p) = 2$. We apply our inequality to the study of the $L^p$ convergence of the Neuman series $\sum I_n(t)$ for exponential martingales.