Open Access
2013 On the optimal stopping of a one-dimensional diffusion
Damien Lamberton, Mihail Zervos
Author Affiliations +
Electron. J. Probab. 18: 1-49 (2013). DOI: 10.1214/EJP.v18-2182


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 inaccessible 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 $$\mathbb{E}_x \left[ \exp \left( - \int _0^\tau r(X_t) \, dt \right) f(X_\tau)<br />{\bf 1} _{\{ \tau < \infty \}} \right]$$ 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 with the upper semicontinuous envelope of $f$ in the interior $int(I)$ sof $I$. 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.


Download Citation

Damien Lamberton. Mihail Zervos. "On the optimal stopping of a one-dimensional diffusion." Electron. J. Probab. 18 1 - 49, 2013.


Accepted: 9 March 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1296.60101
MathSciNet: MR3035762
Digital Object Identifier: 10.1214/EJP.v18-2182

Primary: 60G40
Secondary: 49L20 , 60J55 , 60J60

Keywords: Additive functionals , one-dimensional diffusions , Optimal stopping , potentials , variational inequalities

Vol.18 • 2013
Back to Top