On the optimal stopping of a one-dimensional diffusion

Abstract

We consider the one-dimensional diffusion $X$ that satisfies the stochastic differential equation
$$
dX_t = b(X_t) \, dt + \sigma (X_t) \, dW_t
$$

in the interior $int(I) = \mbox{} ]\alpha, \beta[$ of a given interval $I \subseteq [-\infty, \infty]$, where $b, \sigma: int(I)\rightarrow \mathbb{R}$ are Borel-measurable functions and $W$ is a standard one-dimensional Brownian motion. We allow for the endpoints $\alpha$ and $\beta$ to be inaccessibl or absorbing.

Given a Borel-measurable function $r: I \rightarrow \mathbb{R}_+$ that is uniformly bounded away from 0, we establish a new analytic representation of the $r(\cdot)$ potential of a continuous additive functional of $X$. Furthermore, we derive a complete characterisation of differences of two convex functions in terms of appropriate $r(\cdot)$-potentials, and we show that a function $F: I \rightarrow \mathbb{R}_+$ is $r(\cdot)$-excessive if and only if it is the difference of two convex functions and $- \bigl(\frac{1}{2} \sigma ^2 F'' + bF' - rF \bigr)$ is a positive measure. We use these results to study the optimal stopping problem that aims at maximising the performance index
$$
\mathbb{E}_x \left[ \exp \left( - \int _0^\tau r(X_t) \, dt \right) f(X_\tau)
{\bf 1} _{\{ \tau < \infty \}} \right] 
$$

over all stopping times $\tau$, where $f: I \rightarrow \mathbb{R}_+$ 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

$$
\max \left\{ \frac{1}{2}\sigma ^2 v'' + bv' - rv , \ \overline{f} - v \right\} = 0
$$

in the sense of distributions, where $\overline{f}$ 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.