On the optimal stopping of a one-dimensional diffusion

We consider the one-dimensional diffusion X that satisfies the SDE dX(t) = b (X-t)d(t) + sigma(X-t)dW(t) (SDE) in the interior int I =]alpha,beta[of a given interval I subset of [1; 1], where b;sigma : int I -> R are Borel-measurable functions and W is a standard one-dimensional Brownian motion. We allow for the endpoints alpha and beta to be inaccessible 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 (.) -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 (.)-potentials, and we show that a function F : I -> R + is r (.)-excessive if and only if it is the difference of two convex functions and -(1/2 sigma(2) F '' + bF' - rF) is a positive measure. We use these results to study the optimal stopping problem that aims at maximising the performance index E-x [exp(-integral(tau)(0) r(X-t) dt) f(X-tau)1({tau<infinity})] over all stopping times tau, 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 {1/2 sigma(2)+ bv '' + bv' - rv; <(f)over bar> - v}= 0 in the sense of distributions, where f identifies with 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 this variational inequality 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 (SDE) at appropriate hitting times, which is an issue of fundamental importance to dynamic programming.