Optimal Control of Capital Injections by Reinsurance with a Constant Rate of Interest

Eisenberg, Julia; Schmidli, Hanspeter

doi:10.1239/jap/1316796911

Cited by 16 publications

(3 citation statements)

References 15 publications

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“…They studied this objective (roughly corresponding to unrestricted maximal severity of ruin a = ∞), as well as the classic de Finetti dividends with absorption problem (AP), separately (and left many outstanding open problems behind). A complete solution for both of these problems (AP) and (RP) for spectrally negative Lévy processes was given in [6]-see also [7][8][9][10][11] for further developments. Note though that all these papers deal with forced bailouts.…”

Section: Introductionmentioning

confidence: 99%

Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model

Avram

Goreac

et al. 2021

Mathematics

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We investigate a control problem leading to the optimal payment of dividends in a Cramér-Lundberg-type insurance model in which capital injections incur proportional cost, and may be used or not, the latter resulting in bankruptcy. For general claims, we provide verification results, using the absolute continuity of super-solutions of a convenient Hamilton-Jacobi variational inequality. As a by-product, for exponential claims, we prove the optimality of bounded buffer capital injections (−a,0,b) policies. These policies consist in stopping at the first time when the size of the overshoot below 0 exceeds a certain limit a, and only pay dividends when the reserve reaches an upper barrier b. An exhaustive and explicit characterization of optimal couples buffer/barrier is given via comprehensive structure equations. The optimal buffer is shown never to be of de Finetti (a=0) or Shreve-Lehoczy-Gaver (a=∞) type. The study results in a dichotomy between cheap and expensive equity, based on the cost-of-borrowing parameter, thus providing a non-trivial generalization of the Lokka-Zervos phase-transition Løkka-Zervos (2008). In the first case, companies start paying dividends at the barrier b*=0, while in the second they must wait for reserves to build up to some (fully determined) b*>0 before paying dividends.

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Section: Introductionmentioning

confidence: 99%

Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model

Avram

Goreac

et al. 2021

Mathematics

View full text Add to dashboard Cite

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“…For optimal reinsurance problems when the surplus process is approximated by a diffusion model, see Zeng & Li (2011), Hipp & Taksar (2010), Gu et al (2012), Eisenberg & Schmidli (2011), Meng & Zhang (2010), Zeng (2010), Guan & Liang (2014). For optimal reinsurance problems when the surplus is represented by the classical Cramér-Lundberg process, see Hipp & Vogt (2003), Hipp & Taksar (2010), Romera & Runggaldier (2012), Azcue & Muler (2005), Liang & Yuen (2014).…”

Section: Introductionmentioning

confidence: 99%

Optimal reinsurance: minimize the expected time to reach a goal

Luo

Wang

Zeng

2015

Scandinavian Actuarial Journal

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In this paper, we consider an optimal reinsurance problem. The surplus model of the insurance company is approximated by a diffusion model with the drift coefficient μ. The insurance company employs reinsurance to reduce the risk. π is the proportion of each claim paid by the company and the remainder proportion of the claim is paid by the reinsurer. λ(1 − π) is the rate at which the premiums are diverted to the reinsurer, thus it holds λ ≥ μ > 0 in general. We discuss two cases: (i) non-cheap reinsurance (when λ > μ), (ii) cheap reinsurance (when λ = μ). The objective of the insurance company is to make an optimal decision on reinsurance to reach a goal (a given surplus level) in minimal expected time. We disclose that for some case it is not suitable to search optimal decisions by minimizing the expected time to reach a goal. In order to deal with this case, we study two other methodologies (ruin probability minimization and expected discount factor maximization) for the optimal strategy selection problem in reinsurance decision.

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“…In another study, Eisenberg and Schmidli (2011) without taking into consideration insurance companies that have suffered from ruin with possibilities of recovery, they had aim of minimis-ing the expected discounted capital injections over all admissible reinsurance strategies. For the diffusion approximation case, they used HJB approach to obtain the optimal strategy and obtaining expression for the value function explicitly.…”

Section: Refinancingmentioning

confidence: 99%

Insurance Ccompanies portfolio optimisation with possibilities of recovery after ruin

Komunte

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This dissertation, is about a study on insurance companies that have experienced ruin but have a possibility of recovery from ruin. The study has proposed a perturbed mathematical model, analysed and used it for modelling the portfolio of insurance companies with possibilities of re covery after ruin. Return on investment and refinancing have been used as approaches for over coming ruin. The model was analysed for various cases of possibilities of recovery after ruin in the closed interval [0, 1]. The basic perturbed classical risk process was later compounded by refinancing and return on investment. The Hamilton-Jacobi-Bellman and Integro-Differential Equation of Volterra type were obtained. The Volterra Integro-Differential Equation for sur vival function of an insurance company was converted to a third order ordinary differential equation and later converted into a system of first order ordinary differential equations which was solved numerically using the fourth order Runge-Kutta method. The results indicate that the return on investment plays a vital role in reducing ultimate ruin and that as the possibility of recovery for insurance companies increases, the return on investment reduces ruin much faster. Also, the survival function increases with the increasing intensity of the counting pro cess but decreases with an increase in the instantaneous rate of stock return and return volatility. Because an insurance company faces more risks, these results also suggest that insurance com panies should increase their counting process since doing so will help the insurance companies in servicing more customers.

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Optimal Control of Capital Injections by Reinsurance with a Constant Rate of Interest

Cited by 16 publications

References 15 publications

Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model

Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model

Optimal reinsurance: minimize the expected time to reach a goal

Insurance Ccompanies portfolio optimisation with possibilities of recovery after ruin

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