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

Osȩkowski, Adam

Electron. Commun. Probab. 19, Paper No. 25, 13 p. (2014).

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.

Reviewer: Pavel Gapeev (London)

MSC:

60G44	Martingales with continuous parameter
60H05	Stochastic integrals

Keywords:

martingales; running maximum; weak type inequality; stochastic integrals