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
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona.
The preferred way is to send a scanned (jpeg or pdf) copy of the signed copyright form to the managing editor Philippe Carmona at ejpecpme@math.univ-nantes.fr.
If a paper has several authors, the corresponding author signs the copyright form
on behalf of all the authors.