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: $$\mathbb{E}\underset{t\ge 0}{sup}|M_t{|}^p{\eqsim }_p\mathbb{E}[M{]}_{\mathrm{\infty }}^{p/2},1\le p<\mathrm{\infty },$$ 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.