A sharp and strict \(L^ p\)-inequality for stochastic integrals (Q1822132)

Let M be a right-continuous martingale, V be a predictable process, \(N=V\circ M\) (stochastic integral of V with respect to M). Let \(p^*\) be the maximum of p and q where \(1<p<\infty\) and \(1/p+1/q=1\). Set \(\| M\|_ p=\sup_{t}\| M_ t\|_ p.\) Theorem. If \(p=2\) and \(0<\| M\|_ p<\infty\), then \[ \| N\|_ p<(p^*-1)\| M\|_ p \] [strict inequality in contrast with a previous result of the author, see ibid. 12, 647-702 (1984; Zbl 0556.60021)].