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 Electronic Communications in Probability > Vol. 15(2010) > Paper 9 open journal systems 


On the Principle of Smooth Fit for Killed Diffusions

Farman Samee, University of Manchester


Abstract
We explore the principle of smooth fit in the case of the discounted optimal stopping problem
V(x)=supτ Ex[e-βτG(Xτ)].
We show that there exists a regular diffusion $X$ and differentiable gain function $G$ such that the value function $V$ above fails to satisfy the smooth fit condition $V'(b)=G'(b)$ at the optimal stopping point $b$. However, if the fundamental solutions $psi$ and $phi$ of the `killed' generator equation $LXu(x) - beta u(x) =0$ are differentiable at $b$ then the smooth fit condition $V'(b)=G'(b)$ holds (whenever $X$ is regular and $G$ is differentiable at $b$). We give an example showing that this can happen even when `smooth fit through scale' (in the sense of the discounted problem) fails.


Full text: PDF

Pages: 89-98

Published on: March 22, 2010
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Electronic Communications in Probability. ISSN: 1083-589X