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Two inequalities for iterated stochastic integrals. (English) Zbl 1048.60043

Summary: Let \(M=(M_t, {\mathcal F}_t)_{t\geq 0}\) be a continuous local martingale with quadratic variation process \(\langle M\rangle\) and \(M_0=0\). 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 \[ I_n(t,M)= \int^t_0I_{n-1} (s,M)dM_s \] 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.

MSC:

60H05 Stochastic integrals
60G44 Martingales with continuous parameter
60J55 Local time and additive functionals
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