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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.

MSC:

60G44 Martingales with continuous parameter
60H05 Stochastic integrals
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