Martingales

This chapter summarizes the essentials of sequential conditioning and martingale theory. After a review with examples of the basic properties of martingales and semi‐martingales, including the Doob decomposition, the upcrossing inequality and martingale convergence are studied and also the role of the conditional variances in establishing convergence. The important martingale inequalities of Kolmogorov, Doob, Burkholder, and Azuma are proved.

New variable martingale Hardy spaces

We investigate various variable martingale Hardy spaces corresponding to variable Lebesgue spaces $\mathcal {L}_{p(\cdot )}$ defined by rearrangement functions. In particular, we show that the dual of martingale variable Hardy space $\mathcal {H}_{p(\cdot )}^{s}$ with $0<p_{-}\leq p_{+}\leq 1$ can be described as a BMO-type space and establish martingale inequalities among these martingale Hardy spaces. Furthermore, we give an application of martingale inequalities in stochastic integral with Brownian motion.