Optimality of Refraction Strategies for Spectrally Negative Levy Processes

TL;DR

This paper demonstrates that refraction strategies are optimal for controlling spectrally negative Levy processes under convex costs, and shows convergence of strategies as control constraints are relaxed, supported by numerical validation.

Contribution

It establishes the optimality of refraction strategies under convex costs and analyzes their convergence when control restrictions are lifted.

Findings

Refraction strategies are proven optimal for the control problem.

Optimal strategies converge as control constraints are relaxed.

Numerical results support the analytical findings.

Abstract

We revisit a stochastic control problem of optimally modifying the underlying spectrally negative Levy process. A strategy must be absolutely continuous with respect to the Lebesgue measure, and the objective is to minimize the total costs of the running and controlling costs. Under the assumption that the running cost function is convex, we show the optimality of a refraction strategy. We also obtain the convergence of the optimal refraction strategies and the value functions, as the control set is enlarged, to those in the relaxed case without the absolutely continuous assumption. Numerical results are further given to confirm these analytical results.