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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.

MSC:

60G44 Martingales with continuous parameter
60G15 Gaussian processes

Citations:

Zbl 1098.60042
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References:

[1] D. L. Burkholder, Distribution function inequalities for martingales , Ann. Probab. 1 (1973), 19–42. · Zbl 0301.60035 · doi:10.1214/aop/1176997023
[2] D. L. Burkholder, The best constant in the Davis inequality for the expectation of the martingale square function , Trans. Amer. Math. Soc. 354 (2001), 91–105. JSTOR: · Zbl 0984.60041 · doi:10.1090/S0002-9947-01-02887-2
[3] B. Davis and J. Suh, On Burkholder’s supermartingales , Illinois J. Math. 50 (2006), 313–322. · Zbl 1098.60042
[4] I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus , Springer, New York, 1988. · Zbl 0638.60065
[5] D. Revuz and M. Yor, Continuous martingales and Brownian motion , Springer, Berlin, 1999. · Zbl 0917.60006
[6] M. Yor, Sur le balayage des semi-martingales continues , Séminaire de Probabilités XIII, Springer, New York, 1979, pp. 453–471. · Zbl 0409.60042
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