Consider a regular martingale , , almost surely. Let and stand for and the quadratic variation , respectively.
The following inequality in -spaces was proved in [a2]:
where and are positive constants depending only on , .
In fact, this inequality was proved in three steps; D.L. Burkholder [a3] proved the cases ; Burkholder and R.F. Gundy [a4] proved the cases for a large class of martingales, and Gundy [a5] proved the case for all martingales.
where and are positive constants depending only on .
For a different proof of these inequalities, see, e.g., [a1].
|[a1]||N.L. Bassily, "A new proof of the right hand side of the Burkholder–Davis–Gundy inequality" , Proc. 5th Pannonian Symp. Math. Statistics, Visegrad, Hungary (1985) pp. 7–21|
|[a2]||D.L. Burkholder, B. Davis, R.F. Gundy, "Integral inequalities for convex functions of operators on martingales" , Proc. 6th Berkeley Symp. Math. Statistics and Probability , 2 (1972) pp. 223–240|
|[a3]||D.L. Burkholder, "Martingale transforms" Ann. Math. Stat. , 37 (1966) pp. 1494–1504|
|[a4]||D.L. Burkholder, R.F. Gundy, "Extrapolation and interpolation for convex functions of operators on martingales" Acta Math. , 124 (1970) pp. 249–304|
|[a5]||B. Davis, "On the integrability of the martingale square function" Israel J. Math. , 8 (1970) pp. 187–190|
Burkholder-Davis-Gundy inequality. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Burkholder-Davis-Gundy_inequality&oldid=22217