Continuous-time optimal reinsurance strategy with nontrivial curved structures

Multiple per-claim reinsurance based on maximizing the Lundberg exponent

Optimal reinsurance with premium constraint under distortion risk measures

Optimal reinsurance under the mean–variance premium principle to minimize the probability of ruin

In this paper we investigate optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies. The original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income. We assume that the insurer uses a dynamic reinsurance policy to minimize the probability of absolute ruin, where the traditional ruin can be viewed as a special case of absolute ruin. In terms of approximation theory of stochastic process, the controlled diffusion model with a general reinsurance policy is established strictly. In such a risk model, absolute ruin is said to occur when the drift coefficient of the surplus process turns negative, when the insurer has no profitability any more. Under the expected value premium principle, we rigorously prove that a dynamic excess-of-loss reinsurance is the optimal form of reinsurance among a class of general reinsurance strategies in a dynamic control framework. Moreover, by solving the Hamilton-Jacobi-Bellman equation, we derive both the explicit expression of the optimal dynamic excess-of-loss reinsurance strategy and the closed-form solution to the absolute ruin probability under the optimal reinsurance strategy. We also illustrate these explicit solutions using numerical examples.

ABSTRACTIn this paper, we consider the ruin probability minimization of an insurance company that buys proportional reinsurance and invests in markets where borrowing is constrained. We use a diffusion approximation model for the surplus process of this company and assume that the company invests its surplus into a riskless and multiple risk assets that are modeled as geometric Brownian motions. To find the results, we introduce an auxiliary market parametrized with fictitious processes to relax the borrowing constraint and apply the techniques of stochastic optimal control. In this way, we find the optimal proportional reinsurance and investment strategy of an insurance company investing into multiple risky assets to minimize its ruin probability under the borrowing constraint. Furthermore, from our solutions, we show how our results connect to economic survival analysis and how investment and reinsurance strategies are related.

We consider a (non) zero-sum differential game between two insurers and determine a robust reinsurance contract from joint interests of the insurers and the reinsurer under the framework of Stackelberg differential game, that is, a hybrid stochastic differential game framework is introduced in this article. We assume that each party can invest in a financial market consisting of a risk-free asset and a risky asset whose price evolution follows a stochastic volatility model, and the reinsurer charges the premium according to both the expected value premium principle and the variance premium principle. Since the performance-related capital inflow or outflow feature is introduced, the insurers and reinsurer all aim to maximize their own time-inconsistent mean-variance criterion with delay. Using the dynamic programming approach, we derive the equilibrium investment strategy, the robust equilibrium reinsurance contract, and the corresponding equilibrium value functions under two premium principles with certain conditions, respectively. Some interesting properties of the equilibrium investment strategy are proved explicitly, and several numerical examples and sensitivity analysis are presented to demonstrate the effects of model parameters on the equilibrium reinsurance contract.

In this work, we study the optimal risk sharing problem for an insurer between himself and a reinsurer in a dynamical insurance model known as the Kramer–Lundberg risk process, which, unlike known models, models not per claim reinsurance but rather periodic reinsurance of damages over a given time interval. Here we take into account a natural upper bound on the risk taken by the reinsurer. We solve optimal control problems on an infinite time interval for mean-variance optimality criteria: a linear utility functional and a stationary variation coefficient. We show that optimal reinsurance belongs to the class of total risk reinsurances. We establish that the most profitable reinsurance is the stop-loss reinsurance with an upper limit. We find equations for the values of parameters in optimal reinsurance strategies.

This paper is concerned with stochastic optimization in continuous time and its application in reinsurance. It deals with a model of optimization reinsurance which makes it possible to maximize the technical benefit of an insurance company and to minimize the risk for a given period. Indeed, here we discuss a stochastic optimal control problem to determine the optimal risk management strategy, the purpose of which is to identify the appropriate level of transfer rate for "quota share" reinsurance contracts, which are most appropriate to the risk profile of an insurance company. In addition, we use a resolution approach based on dynamic programming and the HJB equation in order to reach the aforementioned aim, which is finding the optimal strategy for reinsurance to adapt better for insurance company.

General tax structures for a Lévy insurance risk process under the Cramér condition

The role of a representative reinsurer in optimal reinsurance

Optimal reinsurance-investment strategy with thinning dependence and delay factors under mean-variance framework

This paper considers a non-zero-sum stochastic differential reinsurance and investment game between two competitive insurers under the expected exponential utility. It is assumed that the surplus process of each insurer is characterized by a compound Poisson model and their claim businesses are correlated through thinning dependence structure. Each insurer can purchase proportional reinsurance and is allowed to invest in a risk-free asset and a risky asset. The insurers compete with each other and are concerned with the relative performance. By using the stochastic control technique, not only the existence and uniqueness of the optimal strategies are proved, but also the specific analyses for the constraint reinsurance strategies are provided. We find that (i) the thinning dependence structure has significant impact on optimal reinsurance strategies for both insurers and can help them reduce risks; (ii) the behavior of non-zero-sum game will reduce the proportion of purchasing reinsurance for each insurer.

Optimal reinsurance and investment problem for an insurer with counterparty risk

Optimal Hedging and Reinsurance Strategies under Risk Measures

A periodic-review insurance model is considered under the following assumptions. In order to avoid ruin the insurer maintains the company surplus above a chosen level a by capital injections at the end of each period. One-period insurance claims form a sequence of independent identically distributed nonnegative random variables with finite mean. A nonproportional reinsurance is applied for minimization of total expected discounted injections during a given planning horizon of n periods. Insurance and reinsurance premiums are calculated using the expected value principle. Optimal reinsurance strategy is established. Numerical results illustrating the theoretical ones are provided for three claims distributions.