On the optimal stopping of a one-dimensional diffusion

We consider the one-dimensional 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 Borel-measurable functions and W is a standard one-dimensional Brownian motion. We allow for the endpoints α and β to be inaccessibl or absorbing. Given a Borel-measurable 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 Borel-measurable 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, f--v}=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 so-called "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.