Burkholder inequality

The Burkholder inequality (also Burkholder-Davis-Gundy inequality ) is an inequality from stochastics . It establishes the connection between the size of a martingale and its quadratic variation . It was named after Donald Burkholder , who was a professor emeritus at the University of Illinois .

formulation

Let be a continuous local martingale with , defined on a probability space . Then there exist constants for each, so that for each ${\displaystyle M}$ ${\displaystyle M_{0}=0\;}$ ${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t}),P)\ }$ ${\displaystyle p>0\;}$ ${\displaystyle c_{p},C_{p}\;}$ ${\displaystyle t>0\;}$

${\displaystyle c_{p}E[\langle M,M\rangle _{t}^{\frac {p}{2}}]\leq E[\sup _{s\leq t}|M_{s}|^{p}]\leq C_{p}E[\langle M,M\rangle _{t}^{\frac {p}{2}}]}$

applies. Here referred to the quadratic variation of . ${\displaystyle \langle M,M\rangle _{t}\;}$ ${\displaystyle M}$

use

The Burkholder inequality is an important aid in the derivation of limit theorems for stochastic processes .

literature

• DL Burkholder: Martingale transforms. In: Annals of Mathematical Statistics. Volume 37, No. 6, 1966, pp. 1494-1504, doi: 10.1214 / aoms / 1177699141 JSTOR 2238766 .
• M. Beiglböck, J. Sieorpaes: Pathwise versions of the Burkholder – Davis – Gundy inequality. In: Bernoulli. Volume 21, No. 1, 2015, pp. 360-373, doi: 10.3150 / 13-BEJ570