Lamberton, Damien and Zervos, Mihail (2013) On the optimal stopping of a onedimensional diffusion. Electronic Journal of Probability, 18 . p. 34. ISSN 10836489

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Abstract
We consider the onedimensional diffusion X that satisfies the stochastic differential equation dXt=b(Xt)dt+σ(Xt)dWt in the interior int(I)=]α,β[ of a given interval I⊆[∞,∞], where b,σ:int(I)→R are Borelmeasurable functions and W is a standard onedimensional Brownian motion. We allow for the endpoints α and β to be inaccessibl or absorbing. Given a Borelmeasurable function r:I→R+ that is uniformly bounded away from 0, we establish a new analytic representation of the r({dot operator}) potential of a continuous additive functional of X. Furthermore, we derive a complete characterisation of differences of two convex functions in terms of appropriate r({dot operator})potentials, and we show that a function F:I→R+ is r({dot operator})excessive if and only if it is the difference of two convex functions and (1/2σ2F″′+bF′rF) is a positive measure. We use these results to study the optimal stopping problem that aims at maximising the performance index over all stopping times τ, where f:I→R+ is a Borelmeasurable function that may be unbounded. We derive a simple necessary and sufficient condition for the value function v of this problem to be real valued. In the presence of this condition, we show that v is the difference of two convex functions, and we prove that it satisfies the variational inequality max{1/2σ2v″′+bv′rv, fv}=0 in the sense of distributions, where f identifies wit the upper semicontinuous envelope of f in the interior int(I) of I. Conversely, we derive a simple necessary and sufficient condition for a solution to the equation above to identify with the value function v. Furthermore, we establish several other characterisations of the solution to the optimal stopping problem, including a generalisation of the socalled "principle of smooth fit". In our analysis, we also make a construction that is concerned with pasting weak solutions to the SDE at appropriate hitting times, which is an issue of fundamental importance to dynamic programming.
Item Type:  Article 

Official URL:  http://ejp.ejpecp.org 
Additional Information:  © 2013 The Authors 
Library of Congress subject classification:  Q Science > QA Mathematics 
Sets:  Departments > Mathematics 
Rights:  http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx 
Funders:  Engineering and Physical Sciences Research Council 
Projects:  GR/S22998/01 
Date Deposited:  05 Apr 2013 15:26 
URL:  http://eprints.lse.ac.uk/49625/ 
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