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On the Principle of Smooth Fit for Killed Diffusions

   
@article{ECP1531,
	author = {Farman Samee},
	title = {On the Principle of Smooth Fit  for Killed Diffusions},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {15},
	year = {2010},
	keywords = {Optimal stopping; discounted optimal stopping; principle of smooth fit; regular diffusion process; killed diffusion process; scale function; concave function},
	abstract = {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.},
	pages = {no. 9, 89-98},
	issn = {1083-589X},
	doi = {10.1214/ECP.v15-1531},    
        url = {http://ecp.ejpecp.org/article/view/1531}}