On optimal joint reflective and refractive dividend strategies in spectrally positive L\'evy models

TL;DR

This paper determines the optimal dividend strategy in spectrally positive Lévy models, showing that a two-layer (a,b) strategy combining continuous and lump sum dividends is optimal, with explicit formulas derived using scale functions.

Contribution

It introduces a novel optimal dividend strategy combining reflective and refractive payments in spectrally positive Lévy models, with explicit formulas and verification of optimality.

Findings

Two-layer (a,b) dividend strategy is optimal.

Explicit formulas for expected dividends are derived.

Strategy adapts to different transaction rates.

Abstract

The expected present value of dividends is one of the classical stability criteria in actuarial risk theory. In this context, numerous papers considered threshold (refractive) and barrier (reflective) dividend strategies. These were shown to be optimal in a number of different contexts for bounded and unbounded payout rates, respectively. In this paper, motivated by the behaviour of some dividend paying stock exchange companies, we determine the optimal dividend strategy when both continuous (refractive) and lump sum (reflective) dividends can be paid at any time, and if they are subject to different transaction rates. We consider the general family of spectrally positive L\'evy processes. Using scale functions, we obtain explicit formulas for the expected present value of dividends until ruin, with a penalty at ruin. We develop a verification lemma, and show that a two-layer (a,b) strategy is optimal. Such a strategy pays continuous dividends when the surplus exceeds level a>0, and all of the excess over b>a as lump sum dividend payments. Results are illustrated.