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Sharp inequalities for martingales and stochastic integrals. (English) Zbl 0656.60055
Les processus stochastiques, Coll. Paul Lévy, Palaiseau/Fr. 1987, Astérisque 157-158, 75-94 (1988).
[For the entire collection see Zbl 0649.00017.]
Sharp inequalities are derived for differentially subordinate martingales taking values in a real or complex Hilbert space, H. Two general inequalities are derived, one for \(\| f\|_ p=\sup_{n} \| f_ n\|_ p,\) where \(f=(f_ n)_{n\geq 0}\), is a martingale and the other for \(\sup_{n} E \Phi (| f_ n|),\) where \(\Phi\) is in a class of convex functions. These results are then applied to obtain sharp inequalities between stochastic integrals in which either the martingale integrators or the predictable integrands are H-valued. Furthermore, they lead to square function inequalities for H-valued martingales and provide a proof of a conjecture by G. Klincsek [Ann. Probab. 5, 823-825 (1978; Zbl 0375.60064)] for real martingales.
Reviewer: N.Weber

MSC:
60G42 Martingales with discrete parameter
60H05 Stochastic integrals