Burkholder-Davis-Gundy inequalities and stochastic integration in Banach spaces

In 1970s Burkholder, Davis, and Gundy proved the following inequalities which connect the $L^p$-norm of a martingale with its quadratic variation: \[ \mathbb E \sup_{t\geq 0}|M_t|^p \eqsim_p \mathbb E [M]_{\infty}^{p/2},\;\;\; 1\leq p<\infty, \] where $M$ is a real-valued martingale and $[M]$ is the quadratic variation of $M$. These inequalities are known to be exceptionally important for stochastic integration theory as they yield sharp estimates for real-valued stochastic integrals.

In this talk we present Burkholder-Davis-Gundy inequalities for martingales with values in general Banach spaces. As a corollary, we extend the theory of stochastic integration with respect to a general martingale to infinite dimensions.