## 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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