  Stroock and Varadhan in 1997 and Geiss in 2005 independently introduced
stochastic processes with bounded mean oscillation (BMO) and established their
exponential integrability with some unspecified exponential constant. This
result is an analogue of the John--Nirenberg inequality for functions of
bounded mean oscillation. In this work, we quantify the size of the exponential
constant by the modulus of mean oscillation. Some new applications of BMO
processes in rough stochastic differential equations, numerical approximations
and regularization by noise are discussed.
