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.
As continuous martingales are special cases of semimartingales, hypothesis H1 implies H2. In fact, the reverse implication also holds so that hypotheses H1 and H2 are equivalent.
Hypotheses H1 and H2 can also be recast as a simple real analysis statement which makes no reference to stochastic processes.
Hypothesis H3 Let be convex in x and decreasing in t. Then, where and are convex in x and increasing in t.