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Contents of PMS, Vol. 36, Fasc. 2,
pages 311 - 333
 

A MAXIMAL INEQUALITY FOR STOCHASTIC INTEGRALS

Mateusz Rapicki

Abstract: Assume that X is a cądląg, real-valued martingale starting from zero, H is a predictable process with values in [- 1,1] and Y = ∫ HdX . This article contains the proofs of the following inequalities:

(i) If X has continuous paths, then

ℙ(sup Yt ≥ 1) ≤ 2E sup Xt, t≥0 t≥0

where the constant 2 is the best possible.

(ii) If X is arbitrary, then

ℙ(sup Yt ≥ 1) ≤ cE sup Xt, t≥0 t≥0

where c = 3.0446... is the unique positive number satisfying the equation 4 3 3c - 8c - 32 = 0 . This constant is the best possible.

2010 AMS Mathematics Subject Classification: Primary: 60G42; Secondary: 60G44.

Keywords and phrases: Martingale, sharp inequality.

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