Julia Eisenberg

Minimising expected discounted capital injections by reinsurance in a classical risk model

In this paper we consider a classical continuous time risk model, where the claims are reinsured by some reinsurance with retention level , where means ‘no reinsurance’ and b=0 means ‘full reinsurance’. The insurer can change the retention level continuously. To prevent negative surplus the insurer has to inject additional capital. The problem is to minimise the expected discounted cost over all admissible reinsurance strategies. We show that an optimal reinsurance strategy exists. For some special cases we will be able to give the optimal strategy explicitly. In other cases the method will be illustrated only numerically.

Optimal Consumption under Deterministic Income

We consider an individual or household endowed with an initial wealth, having an income and consuming goods and services. The wealth development rate is assumed to be a deterministic continuous function of time. The objective is to maximize the discounted consumption over a finite time horizon. Via the Hamilton–Jacobi–Bellman approach, we prove the existence and the uniqueness of the solution to the considered problem in the viscosity sense. Furthermore, we derive an algorithm for explicit calculation of the value function and optimal strategy. It turns out that the value function is in general not continuous. The method is illustrated by two examples.

A Note on the Optimal Dividends Paid in a Foreign Currency

Abstract We consider an insurance entity endowed with an initial capital and a surplus process modelled as a Brownian motion with drift. It is assumed that the company seeks to maximise the cumulated value of expected discounted dividends, which are declared or paid in a foreign currency. The currency fluctuation is modelled as a Lévy process. We consider both cases: restricted and unrestricted dividend payments. It turns out that the value function and the optimal strategy can be calculated explicitly.

Asymptotic optimal investment under interest rate for a class of subexponential distributions

We consider a classical risk model with the possibility of investment and positive interest rate for the riskless bond. The stock price movement is modelled as a geometric Brownian motion, the claim sizes are assumed to have a distribution belonging to a certain subclass of subexponential distributions. In this setting, we study the asymptotic behaviour of the optimal investment strategy under the ruin probability as a risk measure. This problem has been already considered before, but no results were obtained, for instance, for Weibull and Benktander-type-II distributions with certain parameters. We introduce a method which closes this gap.

Unrestricted consumption under a deterministic wealth and an Ornstein–Uhlenbeck process as a discount rate

ABSTRACT We consider an individual or household endowed with an initial capital and an income, modeled as a linear function of time. Assuming that the discount rate evolves as an Ornstein–Uhlenbeck process, we target to find an unrestricted consumption strategy such that the value of the expected discounted consumption is maximized. Differently than in the case with restricted consumption rates, we can determine the optimal strategy and the value function.