Is an decreasing and convex function over super-martingale also super-martingale?

Question

I learned from class that if $\phi(x)$ is an increasing and convex function and $X$ is a sub-martingale, thus $\phi(X_{t})$ is sub-martingale.

Is there any similar theorem about the super-martingale? Like decreasing function on super-martingale also super-martingale?

Answer 1

If $\phi$ is concave and decreasing then $(\phi (X_n)$ is a supermartingale whenever $(X_n)$ is a supermartingale and $E|\phi(X_n)| <\infty$ for all $n$. (The proof is elementary. Let me know if you need the details).