Some years ago, I spent considerable effort trying to prove the hypothesis below. After failing at this, I spent time trying to find a counterexample, but also with no success. I did post this as a question on mathoverflow, but it has so far received no conclusive answers. So, as far as I am aware, the following statement remains unproven either way.
Hypothesis H1 Let be such that is convex in x and right-continuous and decreasing in t. Then, for any semimartingale X, is a semimartingale.
It is well known that convex functions of semimartingales are themselves semimartingales. See, for example, the Ito-Tanaka formula. More generally, if was increasing in t rather than decreasing, then it can be shown without much difficulty that is a semimartingale. Consider decomposing as
for some process V. By convexity, the right hand derivative of with respect to x always exists, and I am denoting this by . In the case where f is twice continuously differentiable then the process V is given by Ito’s formula which, in particular, shows that it is a finite variation process. If is convex in x and increasing in t, then the terms in Ito’s formula for V are all increasing and, so, it is an increasing process. By taking limits of smooth functions, it follows that V is increasing even when the differentiability constraints are dropped, so is a semimartingale. Now, returning to the case where is decreasing in t, Ito’s formula is only able to say that V is of finite variation, and is generally not monotonic. As limits of finite variation processes need not be of finite variation themselves, this does not say anything about the case when f is not assumed to be differentiable, and does not help us to determine whether or not is a semimartingale.
Hypothesis H1 can be weakened by restricting to continuous functions of continuous martingales.
Hypothesis H2 Let be such that is convex in x and continuous and decreasing in t. Then, for any continuous martingale X, is a semimartingale.
Hypothesis H3 Let be such that is convex in x and decreasing in t. Then, where and are convex in x and increasing in t.