Company Value with Ruin Constraint in Lundberg Models

Abstract

In this note we study the problem of company values with a ruin constraint in classical continuous time Lundberg models. For this, we adapt the methods and results for discrete de Finetti models to time and state continuous Lundberg models. The policy improvement method works also in continuous models, but it is slow and needs discretization. Better results can be obtained faster using the barrier method for discrete models which can be adjusted for Lundberg models. In this method, dividend strategies are considered which are based on barrier sequences. In our continuous state model, optimal barriers can be computed with the Lagrange method leading to a backward recursion scheme. The resulting dividend strategies will not always be optimal: in the case without ruin constraint, there are examples in which band strategies are superior. We also develop equations for optimal control of dynamic reinsurance to maximize the company value under a ruin constraint. These identify the optimal reinsurance strategy in no action regions and allow for an interactive computation of the value function. We apply the methods in a numerical example with exponential claims.

Optimal Dividend Payment in De Finetti Models: Survey and New Results and Strategies

“…As a result of many iterations given in (5), a value V(4, 0.2) = 12.817618 was obtained. The second value V(4, 0.2) = 12.9099 given in Hipp (2018) is wrong since the underlying dividend strategy has a ruin probability larger than 0.2.…”

“…We continue the numerical example in Hipp (2018) where p = 0.7, r = 1/1.03, s 0 = 4 and α = 0.2. As a result of many iterations given in (5), a value V(4, 0.2) = 12.817618 was obtained.…”

Time-inconsistent view on a dividend problem with penalty

“…This approach goes back to [Hipp, 2003]. Recent results in this direction are presented in [Hipp, 2018b, Hipp, 2018a, Hipp, 2020 and in talk by Brandon Israel García Flores 1 . On a finite time horizon such a constraint dividend maximization problem is analysed in [Grandits, 2015].…”