Almost Sure

12 October 16

Do Convex and Decreasing Functions Preserve the Semimartingale Property — A Possible Counterexample


Figure 1: The function f, convex in x and decreasing in t

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 {f\colon{\mathbb R}_+\times{\mathbb R}\rightarrow{\mathbb R}} such that {f(t,x)} is convex in x and right-continuous and decreasing in t, is {f(t,X_t)} necessarily a semimartingale? It was explained how this is equivalent to the hypothesis: for any function {f\colon[0,1]^2\rightarrow{\mathbb R}} such that {f(t,x)} is convex and Lipschitz continuous in x and decreasing in t, does it decompose as {f=g-h} where {g(t,x)} and {h(t,x)} 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 {\{M_t\}_{t\in[0,1]}} is the martingale constructed there, then

\displaystyle  C(t,x)={\mathbb E}[(M_t-x)_+]

defines a function from {[0,1]\times[-1,1]} to {{\mathbb R}} 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 {x=\pm1}. (more…)

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