On page 14 of "Brownian motion and stochastic calculus" by Karatzas and Shreve, (iii) of theorem 3.8, the book has the downcrossing inequalities for a submartingale
$$ED_{[\sigma,\tau]}(\alpha,\beta;X(\omega))\leq\frac{E(X_\tau-\alpha)^+}{\beta-\alpha}$$ where $\{X_t,\mathcal{F}_t;0\leq t<\infty\}$ is a submartingale whose every path is right continuous, $[\sigma,\lambda]$ is a subinterval of $[0,\infty)$, $\alpha<\beta$ and $\lambda>0$ are real numbers.
The book claims Chung's probability theory book theorem 9.4.2 proves the discrete time version. But when I looked it up, Chung's book only has upcrossing for submartingales and downcrossing for supermartingales (see theorem 9.4.3). Can anyone give a reference for downcrossing of submartingales? Or perhaps this is a mistake of the book?
There are fewer downcrossings than upcrossings if $X_0 < \alpha$ and at most one more if $X_0 \ge \alpha$, so once you bound upcrossings you've also bound downcrossings.