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Singular optimal control models of proportional reinsurance problem
比例再保险问题的奇异型最优控制模型

YUAN Ji-hong,
袁继红

控制理论与应用 , 2010,
Abstract: Two factors, a constant payment of corporate debts and compensatory payment on the insurance companys bankruptcy, are introduced into a proportional reinsurance model with dividend payment, which generates a new class of singular optimal stochastic control models on proportional reinsurance. Effects of the three factors on the companys reserve fund are analyzed by virtue of the methods of stochastic analysis. Moreover, optimal control policies and the corresponding maximal yields in different cases are obtained in explicit forms according to the different relations between the income rate and the debts paid per year.
Band Control of Mutual Proportional Reinsurance  [PDF]
John Liu,Michael Taksar,Jiguang Yuan
Mathematics , 2011,
Abstract: In this paper, we investigate the optimization of mutual proportional reinsurance --- a mutual reserve system that is intended for the collective reinsurance needs of homogeneous mutual members, such as P&I Clubs in marine mutual insurance and reserve banks in the U.S. Federal Reserve. Compared to general (non-mutual) insurance models, which involve one-sided impulse control (i.e., either downside or upside impulse) of the underlying insurance reserve process that is required to be positive, a mutual insurance differs in allowing two-sided impulse control (i.e., both downside and upside impulse), coupled with the classical proportional control of reinsurance. We prove that a special band-type impulse control $(a,A,B,b)$ with $a=0$ and $a
Impulse Control of Proportional Reinsurance with Constraints  [PDF]
Hui Meng,Tak Kuen Siu
International Journal of Stochastic Analysis , 2011, DOI: 10.1155/2011/190603
Abstract: We consider an insurance company whose surplus follows a diffusion process with proportional reinsurance and impulse dividend control. Our objective is to maximize expected discounted dividend payouts to shareholders of the company until the time of bankruptcy. To meet some essential requirements of solvency control (e.g., bankruptcy not soon), we impose some constraints on the insurance company's dividend policy. Under two types of constraints, we derive the value functions and optimal control policies of the company. 1. Introduction Reinsurance is an effective tool for insurance companies to manage and control their exposure to risk, and distributions of dividends are used by firms as a vehicle for distributing some of their profits to their shareholders. The problem of determining an optimal dividend policy can be formulated as a singular/regular stochastic control problem in absence of fixed transaction costs, or an impulse control consisting of lump sum dividends distributed at discrete moments of time with fixed transaction cost. For details, interested readers may refer to Gerber [1], Asmussen and Taksar [2], Paulsen [3], Benkherouf and Bensoussan [4], and Cadenillas et al. [5]. Recently, optimizing dividends payouts with solvency constraints have received much attention. For example, Paulsen [6] and He et al. [7] studied optimal singular dividend problems under barrier constraints with no reinsurance and proportional reinsurance, respectively. Choulli et al. [8] investigated an optimal singular dividend problem under constrained proportional reinsurance. Bai et al. [9] and Ormeci et al. [10] considered optimal impulse dividend problems under different constraints. By these ideas we further discuss an optimal impulse control of an insurance company with proportional reinsurance policy under some different solvency constraints. The paper is organized as follows. In Section 2 we establish optimal impulse control problems of the insurance company with proportional reinsurance policy and discrete dividends. In Section 3, we derive the value function and an optimal policy under some constraints of liquid reserves at impulse times. With some constraints of dividends amounts, we obtain the value function and an optimal policy in Section 4. The final section gives concluding remarks. 2. The Model We fix a complete, filtered probability space on which a real-valued, -standard Brownian motion is defined, where is a real-world probability as usual. Consider the following controlled process: where , , , and is an increasing sequence of stopping times and is
Mixed Band Control of Mutual Proportional Reinsurance  [PDF]
Michael Taksar, John Liu, Jiguang Yuan
Journal of Mathematical Finance (JMF) , 2013, DOI: 10.4236/jmf.2013.32025
Abstract:

In this paper, we investigate the optimization of mutual proportional reinsurance—a mutual reserve system that is in- tended for the collective reinsurance needs of homogeneous mutual members, such as P&I Clubs in marine mutual in- surance and reserve banks in the US Federal Reserve, where a mutual member is both an insurer and an insured. Compared to general (non-mutual) insurance models, which involve one-sided impulse control (i.e., either downside or upside impulse) of the underlying insurance reserve process that is required to be positive, a mutual insurance differs in allowing two-sided impulse control (i.e., both downside and upside impulse), coupled with the classical proportional control of reinsurance. We prove that a special band-type impulse control (a, A, B, b) with a=0 and a<A<B<b, coupled with a proportional reinsurance policy (classical control), is optimal when the objective is to minimize the total maintenance cost. That is, when the reserve position reaches a lower boundary of a=0, the reserve should immedi- ately be raised to level A; when the reserve reaches an upper boundary of b, it should immediately be reduced to a level B. An interesting finding produced by the study reported in this paper is that there exists a situation such that if the up-

Optimal Proportional Reinsurance in a Bivariate Risk Model  [PDF]
Cristina Gosio, Ester C. Lari, Marina Ravera
Modern Economy (ME) , 2015, DOI: 10.4236/me.2015.66062
Abstract: The paper deals with the optimal proportional reinsurance in a collective risk theory model involving two classes of insurance business. These classes are dependent through the number of claims. The objective of the insurer is to choose an optimal reinsurance strategy that maximizes the expected exponential utility of terminal wealth. We are able to derive the evolution of the insurer surplus process under the assumption that the number of claims of the two classes of the insurance business has a Poisson bivariate distribution. We face the problem of finding the optimal strategy using the dynamic programming approach. Therefore, we determine the infinitesimal generator for the surplus process and for the value function, and we give the Hamilton Jacobi Bellmann (HJB) equation. Under particular assumptions, we obtain explicit form of the optimal reinsurance strategy on correspondent value function.
Optimal Dynamic Proportional and Excess of Loss Reinsurance under Dependent Risks  [PDF]
Cristina Gosio, Ester C. Lari, Marina Ravera
Modern Economy (ME) , 2016, DOI: 10.4236/me.2016.76075
Abstract: In this paper, we study an optimal reinsurance strategy combining a proportional and an excess of loss reinsurance. We refer to a collective risk theory model with two classes of dependent risks; particularly, the claim number of the two classes of insurance business has a bivariate Poisson distribution. In this contest, our aim is to maximize the expected utility of the terminal wealth. Using the control technique, we write the Hamilton-Jacobi-Bellman equation and, in the special case of the only excess of loss reinsurance, we obtain the optimal strategy in a closed form, and the corresponding value function.
Optimal Investment and Proportional Reinsurance with Risk Constraint  [PDF]
Jingzhen Liu, Ka Fai Cedric Yiu, Ryan C. Loxton, Kok Lay Teo
Journal of Mathematical Finance (JMF) , 2013, DOI: 10.4236/jmf.2013.34046
Abstract:

In this paper, we investigate the problem of maximizing the expected exponential utility for an insurer. In the problem setting, the insurer can invest his/her wealth into the market and he/she can also purchase the proportional reinsurance. To control the risk exposure, we impose a value-at-risk constraint on the portfolio, which results in a constrained stochastic optimal control problem. It is difficult to solve a constrained stochastic optimal control problem by using traditional dynamic programming or Martingale approach. However, for the frequently used exponential utility function, we show that the problem can be simplified significantly using a decomposition approach. The problem is reduced to a deterministic constrained optimal control problem, and then to a finite dimensional optimization problem. To show the effectiveness of the approach proposed, we consider both complete and incomplete markets; the latter arises when the number of risky assets are fewer than the dimension of uncertainty. We also conduct numerical experiments to demonstrate the effect of the risk constraint on the optimal strategy.

Optimal Portfolios of an Insurer and a Reinsurer under Proportional Reinsurance and Power Utility Preference
Silas A. Ihedioha, Bright O. Osu
Open Access Library Journal (OALib Journal) , 2015, DOI: 10.4236/oalib.1102033
Abstract: This study tackled portfolio selection problem for an insurer as well as a reinsurer aiming at maximizing the probability of survival of the Insurer and the Reinsurer, to assess the impact of proportional reinsurance on the survival of insurance companies as well as to determine the condition that would warrant reinsurance according to the optimal reinsurance proportion chosen by the insurer. It was assumed the insurer’s and the reinsurer’s surplus processes were approximated by Brownian motion with drift and the insurer could purchase proportional reinsurance from the reinsurer and their risk reserves followed Brownian motion with drift. Obtained were Hamilton-Jacobi-Bellman (HJB) equations which solutions gave the optimized values of the insurer’s and the reinsurer’s optimal investments in the risky asset and the value of the discount rate that would warrant reinsurance as a ratio of their portfolio weights in the risky asset.
Optimal proportional reinsurance-investment policies for an insurer under Capital-at-Risk constraint
风险资本约束下保险公司的最优比例再保险–投资策略

ZENG Yan,LI Zhong-fei,
曾燕
,李仲飞

控制理论与应用 , 2011,
Abstract: This paper investigates a reinsurance-investment problem for an insurer. Assume that the integral risk of the insurer is measured by Capital-at-Risk(CaR), the surplus process is described by a diffusion approximation model; the insurer is allowed to purchase proportional reinsurance(or acquire new business) and to invest on a risk-free asset and multiple risky assets at any time; the prices of risky assets are driven by the model of geometric Brownian motions. The target of the insurer is to maximize the expectation of the terminal wealth under a CaR constraint. Two mean-CaR models are established for the problem. Explicit expressions of the optimal policies and ef cient frontiers to the models are derived by using a hierarchical optimization method and the variational calculus approach.
Ruin Probabilities of Double Compound Poisson Risk Model under Proportional Reinsurance and Interest Force
Jia Niannian,Xia Yunfan
Journal of Modern Mathematics and Statistics , 2012, DOI: 10.3923/jmmstat.2012.10.13
Abstract: The researchers introduced interest force and reduced the risk of the insurance company with the proportional reinsurance under double compound Poisson risk model. Differential-Integral equations of ruin probabilities in finite and infinite time were provided. These conclusions have theoretical significance for the insurance company measuring ruin risk.
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