Use this tag for questions related to bounded mean oscillation (BMO) martingales.
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values.
A continuous, uniformly-integrable martingale $(M_t, \mathcal F_t)$ with $M_0 = 0$ is said to be from the class BMO if $\sup_{\tau} ||E[\langle M \rangle_T − \langle M \rangle_{\tau} | \mathcal F_{\tau}]^{1/2}||_{\infty} < \infty $ where the supremum is taken over all stopping times $\tau \in [0, T]$ and $\langle M \rangle$ is the square characteristic of $M.$ (That definition comes from the paper "New proofs of some results on BMO martingales using BSDEs.")
BMO martingale theory is used to study backward stochastic differential equations