Open Access
2010 On the Principle of Smooth Fit for Killed Diffusions
Farman Samee
Author Affiliations +
Electron. Commun. Probab. 15: 89-98 (2010). DOI: 10.1214/ECP.v15-1531


We explore the principle of smooth fit in the case of the discounted optimal stopping problem $$ V(x)=\sup_\tau\, \mathsf{E}_x[e^{-\beta\tau}G(X_\tau)]. $$ 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 $L_X u(x) - \beta u(x) =0$ 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.


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Farman Samee. "On the Principle of Smooth Fit for Killed Diffusions." Electron. Commun. Probab. 15 89 - 98, 2010.


Accepted: 22 March 2010; Published: 2010
First available in Project Euclid: 6 June 2016

zbMATH: 1201.60038
MathSciNet: MR2606506
Digital Object Identifier: 10.1214/ECP.v15-1531

Primary: 60G40
Secondary: 60J60

Keywords: Concave function , discounted optimal stopping , killed diffusion process , Optimal stopping , principle of smooth fit , regular diffusion process , scale function

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