Optimal dividend strategies in a Cramér–Lundberg model with capital injections

Kulenko, Natalie; Schmidli, Hanspeter

doi:10.1016/j.insmatheco.2008.05.013

Cited by 129 publications

(94 citation statements)

References 10 publications

(13 reference statements)

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“…Dickson and Waters [15], Gerber, Shiu and Smith [19], and Gerber, Lin and Yang [17] looked at related value functions and problems in the compound Poisson setting; the case S = 0 and K = 1 was discussed in [15,19] and a numerical study, for penalty functions taking the form of a polynomial of degree less than or equal to 2, was carried out in [17]. In the Lévy setting, Avram, Palmowski and Pistorius [7] studied a model where capital is injected (repeatedly) to keep the surplus process afloat (for the compound Poisson case, see also the paper of Kulenko and Schmidli [21]). This is close in spirit to our problem as it considers the fact that the surplus process can go negative eventually and that the shareholders should then be held responsible to cover the deficit.…”

Section: Problem Formulationmentioning

confidence: 99%

De Finetti’s optimal dividends problem with an affine penalty function at ruin

Loeffen

Renaud

2010

Insurance: Mathematics and Economics

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Abstract. In a Lévy insurance risk model, under the assumption that the tail of the Lévy measure is log-convex, we show that either a horizontal barrier strategy or the take-the-money-and-run strategy maximizes, among all admissible strategies, the dividend payments subject to an affine penalty function at ruin. As a key step for the proof, we prove that, under the aforementioned condition on the jump measure, the scale function of the spectrally negative Lévy process has a log-convex derivative.

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Section: Problem Formulationmentioning

confidence: 99%

De Finetti’s optimal dividends problem with an affine penalty function at ruin

Loeffen

Renaud

2010

Insurance: Mathematics and Economics

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“…The associated general maximization problem was recently solved in Kulenko & Schmidli [76] for the classical risk model, see also Avram et al [14]. It turns out that the optimal strategy is now for arbitrary claim size distributions a barrier strategy and injections should only take place when the process is negative.…”

Section: Value Functionsmentioning

confidence: 99%

Optimality results for dividend problems in insurance

Albrecher

Thonhauser

2009

Rev. R. Acad. Cien. Serie A. Mat.

145

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“…Otherwise the company holders have to raise new money to proceed with the business. Therefore, capital injections could be added to the surplus process, and the value of the discounted cash flow can be considered, see Kulenko and Schmidli [50]. However if injecting additional money is not penalised, the optimal strategy would be to pay the income as dividends and to finance the outflow by capital injections.…”

Section: Measuring Risksmentioning

confidence: 99%

On optimal control of capital injections by reinsurance and investments

Eisenberg¹

2010

Blätter DGVFM

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AcknowledgementsI would like to express my thanks to my supervisor Prof. Dr. Hanspeter Schmidli. I have greatly profited from his hints and suggestions generously given during our conversations. I am very grateful to Prof. Dr. Josef Steinebach who agreed to be coreferent.I would like to acknowledge Markus Schulz for his listening ear and Mrs Anderka for believing in me.In particular, I would like to thank my family who has supported me all the time and enabled a successful completion of this work. AbstractAn insurance company, having an initial capital x, cashes premiums continuously and pays claims of random sizes at random times. In addition to that, the company can buy reinsurance or/and invest money into a riskless or risky assets. The company holders are confronted with the problem of taking decisions on a business policy of the company. Thus, a measure for the risk connected with an insurance portfolio is sorely needed.The ruin probability, i.e. the probability that the surplus process becomes negative in finite time, is typically the measure for an insurance company's solvency. However, the ruin probability approach has been criticised among other things for not considering the severity of an insolvency and for ignoring the time value of money.An alternative to measure the risk of a surplus process is to consider the value of expected discounted capital injections, which are necessary to keep the process above zero. Naturally, it raises the question how to minimise this value. If the company holders prefer (or are indifferent) investing tomorrow to investing today, it is optimal to inject capital only when the surplus becomes negative and only as much as is necessary to keep the process above zero.In the first part of this work, we solve the problem of minimising the expected discounted capital injections over all dynamic reinsurance strategies for the classical risk model and its diffusion approximation. In the second part, we extend the concept by adding the possibility of investing money, if the surplus remains positive, into a riskless asset. In these two cases we are able to show the existence and uniqueness of the optimal reinsurance strategy and the value function as the minimising value of expected discounted capital injections.In the third part, we consider the surplus process, where the company holders can invest money into a risky asset modeled as a Black-Scholes model. The forth part extends the setup of the third part by possibility of reinsurance. In the last two cases we solve the problem explicitly only for the case of diffusion approximation. In the classical risk model the concept of viscosity solutions introduced by Crandall and Lions has been used.All the studies are illustrated by simulations, written in Java. Als alternatives Risikomaß betrachtet man den Wert der erwarteten diskontierten Kapitalzuführungen, welche notwendig sind damit derÜberschussprozess nichtnegativ bleibt. Es stellt sich die Frage, wie man diesen Wert minimiert. Wir nehmen an, dass die Inhaber der Versicherungsges...

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Optimal dividend strategies in a Cramér–Lundberg model with capital injections

Cited by 129 publications

References 10 publications

De Finetti’s optimal dividends problem with an affine penalty function at ruin

De Finetti’s optimal dividends problem with an affine penalty function at ruin

Optimality results for dividend problems in insurance

On optimal control of capital injections by reinsurance and investments

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