Abstract
We consider a one-dimensional diffusion which solves a stochastic differential equation with Borel-measurable coefficients in an open interval. We allow for the endpoints to be inaccessible or absorbing. Given a Borel-measurable function $r$ that is uniformly bounded away from 0, we establish a new analytic representation of the $r$-potential of a continuous additive functional of the diffusion. We also characterize the value function of an optimal stopping problem with general reward function as the unique solution of a variational inequality (in the sense of distributions) with appropriate growth or boundary conditions. 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".
Highlights
We consider the one-dimensional diffusion X that satisfies the SDE (1) in the interior int I = ]α, β[ of a given interval I ⊆ [−∞, ∞]
In the presence of Assumption 1, a weak solution to (1) can be obtained by first timechanging a standard one-dimensional Brownian motion and making an appropriate state space transformation. This construction can be used to prove all of the results that we obtain by first establishing them assuming that the diffusion X identifies with a standard one-dimensional Brownian motion
Deriving the general results, which are important because many applications assume specific functional forms for the data b and σ, by means of this approach would require several time changes and state space transformations, which would lengthen the paper significantly
Summary
Dayanik and Karatzas [13] and Dayanik [12], who considers random discounting instead of discounting at a constant rate r, addressed the solution of the optimal stopping problem by means of a certain concave characterisation of excessive functions They established a generalisation of the so-called “principle of smooth fit” that is similar to, though not the same as, the one we derive here. We address the issue of pasting weak solutions to (1), or, more, generally, the issue of pasting stopping strategies for the optimal stopping problem that we consider, at an appropriate stopping time (see Theorem (20) and Corollary 21) Such a rather intuitive result is fundamental to dynamic programming and has been assumed by several authors in the literature (e.g., see the proof of Proposition 3.2 in Dayanik and Karatzas [13]). We develop the theory concerned with pasting weak solutions to (1) in the Appendix
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
