Burkholder’s submartingales from a stochastic calculus perspective. (English) Zbl 1176.60033
Summary: We provide a simple proof, as well as several generalizations, of a recent result by B. Davis and J. Suh [Ill. J. Math. 50, No. 1–4, 313–322 (2006; Zbl 1098.60042)], characterizing a class of continuous submartingales and supermartingales that can be expressed in terms of a squared Brownian motion and of some appropriate powers of its maximum. Our techniques involve elementary stochastic calculus, as well as the Doob-Meyer decomposition of continuous submartingales. These results can be used to obtain an explicit expression of the constants appearing in the Burkholder-Davis-Gundy inequalities. A connection with some balayage formulae is also established.
Keywords:
Doob-Meyer decomposition; continuous submartingales; Burkholder-Davis-Gundy inequalities; balayage formulaeCitations:
Zbl 1098.60042References:
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[3] | B. Davis and J. Suh, On Burkholder’s supermartingales , Illinois J. Math. 50 (2006), 313–322. · Zbl 1098.60042 |
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