Two inequalities for iterated stochastic integrals

Abstract.

Let $M = (M_{t}, {\mathcal F}_{t})_{t \geq 0}$ be a continuous local martingale with quadratic variation process $\langle M \rangle$ and $M_{0} = 0$ . In this paper, we consider the corresponding sequence of the iterated stochastic integrals $I_{n}(M) = (I_{n}(t, M), {\mathcal F}_{t})(n \geq 0)$ , defined inductively by

$I_{n}(t, M) = \int\limits_{0}^{t} I_{n-1}(s, M)dM_{s}$

with $I_{0}(t, M) = 1$ and $I_{1}(t, M) = M_{t}$ . We obtain a maximal inequality at any time and a local time inequality.