Here, I attempt to construct a counterexample to the hypotheses of the earlier post, *Do convex and decreasing functions preserve the semimartingale property?* There, it was asked, for any semimartingale *X* and function such that is convex in *x* and right-continuous and decreasing in *t*, is necessarily a semimartingale? It was explained how this is equivalent to the hypothesis: for any function such that is convex and Lipschitz continuous in *x* and decreasing in *t*, does it decompose as where and are convex in *x* and *increasing* in *t*. This is the form of the hypothesis which this post will be concerned with, so the example will only involve simple real analysis and no stochastic calculus. I will give some numerical calculations suggesting that the construction below is a counterexample, but do not have any proof of this. So, the hypothesis is still open.

Although the construction given here will be self-contained, it is worth noting that it is connected to the example of a martingale which moves along a deterministic path. If is the martingale constructed there, then

defines a function from to which is convex in *x* and increasing in *t*. The question is then whether *C* can be expressed as the difference of functions which are convex in *x* and *decreasing* in *t*. The example constructed in this post will be the same as *C* with the time direction reversed, and with a linear function of *x* added so that it is zero at . Continue reading “Do Convex and Decreasing Functions Preserve the Semimartingale Property — A Possible Counterexample”