On the Novikov-Shiryaev Optimal Stopping Problems in Continuous Time

Andreas Kyprianou,Budhi Surya

doi:10.1214/ecp.v10-1144

Abstract

Novikov and Shiryaev (2004) give explicit solutions to a class of optimal stopping problems for random walks based on other similar examples given in Darling et al. (1972). We give the analogue of their results when the random walk is replaced by a Lévy process. Further we show that the solutions show no contradiction with the conjecture given in Alili and Kyprianou (2004) that there is smooth pasting at the optimal boundary if and only if the boundary of the stopping reigion is irregular for the interior of the stopping region.

Highlights

IntroductionNote that when q = 0 and lim supt↑∞ Xt = ∞ it is clear that it is never optimal to stop in (2) for the given choices of G
Let X = {Xt : t ≥ 0} be a Levy process defined on a filtered probability space (Ω, F, {Ft}, P) satisfying the usual conditions
Instead we work with fluctuation theory of Levy processes which is essentially the direct analogue of the random walk counterpart used in Novikov and Shiryaev (2004)

Summary

Introduction

Note that when q = 0 and lim supt↑∞ Xt = ∞ it is clear that it is never optimal to stop in (2) for the given choices of G This short note verifies that the results of Novikov and Shiryaev (2004) for random walks carry over into the context of the Levy process as predicted by the aforementioned authors. Instead we work with fluctuation theory of Levy processes which is essentially the direct analogue of the random walk counterpart used in Novikov and Shiryaev (2004). In this sense our proofs are loyal to those of of the latter. In addition we show that the solutions show no contradiction with the conjecture given in Alili and Kyprianou (2004) that there is smooth pasting at the optimal boundary if and only if the boundary of the stopping reigion is irregular for the interior of the stopping region

Results

Preliminary Lemmas

Proofs of Theorems