A simple approach to the L{\o}kka-Zervos dichotomy for absolutely continuous dividend strategies

TL;DR

This paper analyzes an optimal dividend and capital injection problem in a Brownian risk model, revealing a dichotomy where surplus is either never ruined or can be ruined, with explicit conditions and numerical insights.

Contribution

It introduces an affine-bounded absolutely continuous dividend strategy framework with ruin penalties, extending the L{46}kka-Zervos dichotomy to this setting and providing explicit solution conditions.

Findings

The solution exhibits a dichotomy: either no ruin occurs or surplus is never ruined.

Explicit conditions are derived to determine the dichotomy based on model parameters.

Numerical analysis illustrates the range of optimal strategies under the affine-bound structure.

Abstract

We revisit the optimization problem solved in L{\o}kka & Zervos (2008), i.e., the maximization of dividends, in a Brownian risk model, with the possibility (not the obligation) of making capital injections. Following the approach introduced in Alvarez & Shepp (1998), Renaud & Simard (2021), Renaud et al. (2023), we consider instead absolutely continuous (AC) dividend strategies with an affine bound on the payment rates, while singular capital injections are still allowed. In addition, we incorporate a parameter for the cost of ruin or, said differently, a penalty at ruin in the performance function. We show that the solution is a so-called L{\o}kka-Zervos dichotomy: the surplus is never ruined by making bail-out payments, or no capital is injected and bankruptcy can occur; in either case, dividends are paid at full rate when the surplus is above a threshold. Our framework allows us to provide explicit conditions to express the dichotomy, either using the cost of capital injections or the cost of ruin as a criterion, which also exposes the underlying structure of the solution. In particular, for some values of the parameters, we show that it is optimal to liquidate. Moreover, we perform a numerical analysis highlighting the range of values generated under this AC affine-bound structure.