Timeline for Do convex and decreasing functions preserve the semimartingale property?
Current License: CC BY-SA 2.5
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| Oct 31, 2010 at 7:50 | comment | added | has2 | Another thing that seem relevant is the following: the region where $x \rightarrow f(x,0) - f(x,t)$ is not convex must be small, because otherwise $f(x,0)$ could not dominate $f(x,t)$. | |
| Oct 31, 2010 at 7:45 | comment | added | has2 | You are welcome; your questions above and your related results are very interesting, thanks for sharing. What makes the problem difficult seems to be that there are many things that influence how $g$ and $h$ are to evolve in time and there seems to be many choices. The optimization problem in the discrete formulation (the choice of $S_2$) referred to in my answer seems to be nontrivial (I think it can be formulated as a control problem). | |
| Oct 30, 2010 at 21:42 | comment | added | George Lowther | Also, if f is either twice differentiable in x or once in t, then it will have the required decomposition. The problem appears with nondifferentiable functions. You can always approximate by smooth functions but, then, it is possible that the decompositions of these smooth approximations diverge when you take the limit. | |
| Oct 30, 2010 at 21:38 | comment | added | George Lowther | Thanks for looking at this. I agree that discretizing it is a good first step (it also helps to restrict to a compact interval for x). However, it works out a bit better if you solve for g,h by starting at the last time and working backwards. Doing this, you can show that it converges to the continuous time decomposition if and only if there is a continuous time decomposition. I'm going to post an answer myself, when I have time, describing a few different ways to reformulate this problem. | |
| Oct 30, 2010 at 18:44 | history | edited | has2 | CC BY-SA 2.5 |
added 99 characters in body
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| Oct 30, 2010 at 18:37 | history | answered | has2 | CC BY-SA 2.5 |