An approach for solving perpetual optimal stopping problems driven by Lévy processes

Abstract

In this paper, we expose an approach for solving perpetual optimal stopping problems for a general class of payoff functions under Lévy processes. This approach was inspired by the work of Boyarchenko and Levendorski. In contrast to Boyarchenko and Levendorsikii, our approach does not appeal to a free boundary problem associated to the optimal stopping problem nor to the theory of pseudodifferential operators to solve the problem. Instead, we introduce an averaging problem from which we obtain using the Wiener–Hopf factorization a fluctuation identity for overshoots of a Lévy process. This identity constitutes the main principle in solving the optimal stopping problem. If a solution to the averaging problem can be found and has certain monotonicity properties, we show using the fluctuation identity that an optimal solution to the optimal stopping problem can be written in terms of such monotone function. Using the optimal solution, we give sufficient and necessary conditions for the smooth pasting condition to occur in the considered problem. Our conclusion over the smooth pasting condition extends further the recent result of Alili and Kyprianou into a more general payoff function.

Cite this work

No public copy is stored for this record, so only the metadata is cited.

Stored only when both papers are in the corpus — 37 citations exist upstream.

Papers that cite the same references — the relational fallback while embeddings are still backfilling.