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, [GRAPHICS] where ln A(p,n) approximately ln B(p,n) approximately -(n/2)ln n as n --> infinity. Our proof requires the sharp constant b(p) in Burkholder-Davis-Gundy inequalities parallel-to M parallel-to p less-than-or-equal-to b(p) parallel-to <M, M>1/2 parallel-to p. In the Appendix we prove sup(p) greater-than-or-equal-to 1(b(p)/ square-root p) = 2. We apply our inequality to the study of the L(p) convergence of the Neuman series SIGMA-I(n)(t) for exponential martingales.