Maximal weak-type inequality for stochastic integrals. (English) Zbl 1318.60047

Based on the author’s abstract: In this paper, the following results are obtained. Assume that \(X\) is a real-valued martingale starting from \(0\), \(H\) is a predictable process with values in \([-1, 1]\), and \(Y\) is the stochastic integral of \(H\) with respect to \(X\). Then, the following sharp weak-type estimates are derived:
(i) if \(X\) has continuous paths, then \[ {\mathbb P} \bigg( \sup_{t \geq 0} |Y_t| \geq 1 \bigg) \leq 2 {\mathbb E} \sup_{t \geq 0} X_t; \] (ii) if \(X\) is arbitrary, then \[ {\mathbb P} \bigg( \sup_{t \geq 0} |Y_t| \geq 1 \bigg) \leq 3.477977 \ldots {\mathbb E} \sup_{t \geq 0} X_t. \] The proofs of these results are based on Burkholder’s method and use the existence of certain special functions possessing appropriate concavity and majorisation properties.


60G44 Martingales with continuous parameter
60H05 Stochastic integrals
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