LP Estimates on Iterated Stochastic Integrals

For a continuous martingale M, let M, M denote the increasing process. Let I₀, I₁,… denote the iterated stochastic integrals of M. We prove the inequalities of Burkholder-Davis-Gundy type, Ap,n\| M, M 1/2ₜ\|ⁿₙₚ ≤ \|Iₙ(t)\|ₚ ≤ Bp,n\| M, M 1/2ₜ\|ⁿₙₚ, where ln Ap,n ~ ln Bp,n ~ -(n/2)ln n as n → ∞. Our proof requires the sharp constant bₚ in Burkholder-Davis-Gundy inequalities \|M\|ₚ ≤ bₚ\| M, M1/2\|ₚ. In the Appendix we prove p≥ 1(bₚ/√ p) = 2. We apply our inequality to the study of the Lᵖ convergence of the Neuman series ∑ Iₙ(t) for exponential martingales.