Optimal Proportional Reinsurance Research Articles

Optimal investment and reinsurance to reach a bequest goal with random time solvency regulation

ABSTRACT This paper studies optimal investment and proportional reinsurance policies for an insurer with Markov regime-switching model and random time solvency regulation. The goal of the insurer is to maximise the probability that its wealth exceeds a predetermined level l before the regulatory time arrives. By constructing an auxiliary control problem without regime switching, together with the classical results on Hamilton-Jacobi-Bellman (HJB) equation and fixed-point method, we prove the regularity of the value function. When the current state of the Markov chain is given, we found that the optimal policies for an insurer in a multiple-regime market is the same as those in a single-regime market. Explicit optimal policies can be derived when the premium is calculated by the expectation principle. For more general cases, numerical schemes for value functions and feedback optimal policies are given by the Markov chain approximating method.

Optimal Claim-Dependent Proportional Reinsurance Under a Self-Exciting Claim Model

Optimal Claim-Dependent Proportional Reinsurance Under a Self-Exciting Claim Model

Optimal dividend and proportional reinsurance strategy for the risk model with common shock dependence

. This article focuses on the classic optimal dividend and reinsurance problems. Different from the existing literature, it assumes that the insurance company has two lines of business with a common shock dependence. It can purchase proportional reinsurance to reduce business risk and pay dividends to stay competitive. The goal is to find out the optimal dividend and reinsurance strategies for maximizing the company’s value. Under the diffusion approximation model, we decomposed the problem into several situations and gave the corresponding solutions by using the stochastic control method. Some numerical examples and economic explanations are presented to illustrate the results.

Optimal robust reinsurance contracts with investment strategy under variance premium principle

This paper investigates the optimal robust proportional reinsurance contracts with investment in a liquid financial market under variance premium principle in a principal-agent framework. The surplus process of the insurer (agent) is assumed to follow an approximating compound Poisson risk process. Both the insurer and reinsurer (principal) are allowed to invest in a risk-free asset and a risky asset whose price process is governed by the constant elasticity of variance model. The insurer and reinsurer aim to maximize the expected exponential utility of terminal wealth. The reinsurer is ambiguity-averse and has deterministic ambiguity aversion preferences against the diffusion risk caused by the financial market and the approximated diffusion risk which comes from the claim process. The reinsurance price is described by the reinsurer's safety loading which can be decided by the optimal strategies of the insurer and the reinsurer. By utilizing stochastic optimal control principle and HJB (or HJBI) equations, the optimal (robust) proportional reinsurance-investment strategies and the corresponding value functions are obtained explicitly. Numerical examples are provided.

Investigations to the optimal derivative-based investment and proportional reinsurance strategies

Investigations to the optimal derivative-based investment and proportional reinsurance strategies

Optimal proportional reinsurance to maximize an insurer’s exponential utility under unobservable drift

AbstractWe study an optimal reinsurance problem for a diffusion model, in which the drift of the claim follows an Ornstein–Uhlenbeck process. The aim of the insurer is to maximize the expected exponential utility of its terminal wealth. We consider two cases: full information and partial information. Full information occurs when the insurer directly observes the drift; partial information occurs when the insurer observes only its claims. By applying stochastic control and by solving the corresponding Hamilton–Jacobi–Bellman equations, we find the value function and the optimal reinsurance strategy under both full and partial information. We determine a relationship between the value function and reinsurance strategy under full information with the value function and reinsurance strategy under partial information.

Optimal proportional reinsurance and pairs trading under exponential utility criterion for the insurer

<p style='text-indent:20px;'>This paper studies the optimal proportional reinsurance and investment strategy for an insurer who invests one paired assets, where their price spread is described by Ornstein-Uhlenbeck (O-U) processes. The insurer's objective is to maximize the expected exponential utility of the terminal wealth in a finite time horizon under two risk models: a classical risk model and a diffusion model. Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation, we characterize the optimal strategies and provide a verification result for the value function via the exponential integrability of the square of an O-U process. Finally, numerical examples are performed to obtain sensitivity analysis.</p>

Robust optimal proportional reinsurance and investment strategy for an insurer and a reinsurer with delay and jumps

In this paper, we consider a robust optimal proportional reinsurance and investment problem in a model with delay and jumps for an insurer and a reinsurer, who are concerned about model uncertainty in model parameters and aim to develop the robust optimal reinsurance and investment strategy. The surplus process of the insurer is described by a diffusion model (which is an approximation of the classical Crmér-Lundberg model) and the insurer can purchase proportional reinsurance. Assume that the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset, respectively. Moreover, the insurer and the reinsurer can invest in different risky assets whose price processes are governed by the jump-diffusion models. Specially, the insurer's and the reinsurer's wealth processes are modeled by stochastic differential delay equations by introducing the performance-related capital inflow (or outflow). Taking both the insurer and the reinsurer into account, this paper's target is to maximize the minimal expected product of the insurer's and the reinsurer's exponential utilities of the combination of terminal wealth and average performance wealth. By applying the techniques of stochastic optimal control theory, we build the robust Hamilton-Jacobi-Bellman equation. Furthermore, we obtain the robust optimal strategy and the corresponding value function, which extends those for the existing models to slightly more general versions. Finally, we present some numerical examples to demonstrate the impacts of some model parameters on the optimal strategy.

Optimal impulse dividend and capital injection model with proportional and fixed transaction costs

In this paper, a hybrid strategy with optimal proportional reinsurance, dividends, and capital injections is discussed by a diffusion model with impulse process of discrete dividend events and discrete capital injections. Due to the presence of fixed transaction costs, the problem is formulated as a more complicated impulse stochastic control problem. By using methodologies from impulse control theory and the techniques of translation transformation, we obtain the explicit solutions of the value function and the corresponding optimal strategy. Firstly, the original problem is transformed into two types of suboptimal impulse control problems, one of which is “the case without capital injections” and the other one of which is “the case with compulsory capital injections.” Then, by verification theorem and the expressions of value functions of these two suboptimal problems, we verify that the optimal solution of the original problem is either “keeping bankruptcy without injections” or “never bankruptcy with injections.” It depends on the parameters of the model.

Optimal investment and reinsurance on survival and growth problems for the risk model with common shock dependence

This paper investigates goal-reaching problems regarding optimal investment and proportional reinsurance with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. The optimization problems are formulated in a general form first, and then four criteria including maximum survival probability, minimum expected ruin penalty, minimum (maximum) expected time (reward) to reach a goal are fully discussed. By the technique of stochastic control theory and through the corresponding Hamilton–Jacobi–Bellman equation, the optimal results are derived and analyzed in different cases. In particular, when discussing the maximum survival probability with a target level U beyond the safe level (where ruin can be avoided with certainty once it is achieved), we construct ε-optimal (suboptimal) strategies to resolve the inaccessibility of the safe level caused by classical optimal strategies. Furthermore, numerical simulations and analysis are presented to illustrate the influence of typical parameters on the main results.

Stochastic asset allocation and reinsurance game under contagious claims

In this paper, we consider a stochastic asset allocation and reinsurance game between two insurance companies with contagious claims, where the insurance claim of one insurer can simultaneously affect the claim intensities of itself and its competitor. This clustering feature of claims is modelled by a mutual-excitation Hawkes process with exponential decays. Furthermore, we assume that the management of the insurance company wants to maximise the expected utility of the relative difference between its terminal surplus and that of its competitor at a fixed time point. The Nash equilibrium strategies have been constructed by solving the Hamilton–Jacobi–Bellman equations, where the explicit formulas of the optimal allocation policies have been derived to be independent of the claim intensities. We also introduce an iterative scheme based on the Feynman–Kac formula to compute the optimal proportional reinsurance policies numerically, where the existence and uniqueness of the solution to the fixed point equation and the convergence of the iterative numerical algorithm are proved rigorously. Finally, numerical examples are presented to show the effect of claim intensities on the optimal controls.

Robust optimal proportional reinsurance and investment problem for an insurer with delay and dependent risks

In this article, we consider a robust optimal proportional reinsurance and investment problem in a model with delay and dependent risks, in which the insurer’s surplus process is assumed to follow a risk model with two dependent classes of insurance business. The insurer is allowed to purchase proportional reinsurance and invest his surplus in a financial market, which contains one risk-free asset and one risky asset whose price process satisfies a jump-diffusion model. Under the consideration of the performance-related capital inflow or outflow, the insurer’s wealth process is modeled by a stochastic differential delay equation. The insurer’s aim is to develop the robust optimal reinsurance and investment strategy for the insurer by maximizing the expected exponential utility of the combination of the average historical performance and terminal wealth under the worst-case scenario of the alternative measures. By adopting the stochastic dynamic programming technique, the expressions of the robust optimal strategy and value function are explicitly obtained. Finally, we present some numerical examples to illustrate the effects of some model parameters on the optimal strategy.

Optimal reinsurance and investment under common shock dependence between financial and actuarial markets

We study optimal proportional reinsurance and investment strategies for an insurance company which experiences both ordinary and catastrophic claims and wishes to maximize the expected exponential utility of its terminal wealth. We propose a modeling setting where the insurance framework is affected by environmental factors, and aggregate claims and stock prices are subject to common shocks, i.e. drastic events such as earthquakes, extreme weather conditions, or even pandemics, that have an immediate impact on the financial market and simultaneously induce insurance claims. Using a classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation, we provide a verification result for the value function via classical solutions of two backward partial differential equations and characterize the optimal reinsurance and investment strategy. Finally, we provide a comparison analysis to discuss the effect of common shock dependence.

Asymptotic solution of optimal reinsurance and investment problem with correlation risk for an insurer under the CEV model

In this paper, we consider an optimal proportional reinsurance and investment problem for an insurer whose objective is to maximise the expected exponential utility of terminal wealth. Suppose that the insurer's surplus process follows a Brownian motion with drift. The insurer is allowed to purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset and a risky asset whose price process is described by the constant elasticity of variance (CEV) model. The correlation between risk model and the risky asset's price is considered. By applying dynamic programming approach, we derive the corresponding Hamilton–Jacobi–Bellman (HJB) equation. The asymptotic expansions of the solution to the partial differential equation (PDE) derived from the HJB equation are presented as the parameter appearing in the exponent of the diffusion coefficient tends to 0. We use perturbation theory for partial differential equations (PDEs) to obtain the asymptotic solutions of the optimal reinsurance and investment strategies. Finally, we provide numerical examples and sensitivity analyses to illustrate the effects of model parameters on the optimal reinsurance and investment strategies.

Robust reinsurance contract with asymmetric information in a stochastic Stackelberg differential game

In this paper, we determine a robust reinsurance contract from joint interests of the insurer and reinsurer under the framework of Stackelberg differential game. More specifically, the reinsurer is the leader of the game and decides on an optimal reinsurance premium to charge, while the insurer is the follower of the game and chooses an optimal proportional reinsurance to purchase. In order to defend the large shocks of wealth process, a loss-dependent premium principle is applied to the insurer. Meanwhile, we incorporate model uncertainty into the reinsurer's controlled surplus due to the asymmetric information. Under the time-consistent mean-variance criterion, we derive the robust reinsurance contract explicitly by solving the coupled extended Hamilton–Jacobi–Bellman systems. It is interesting to prove that the optimal premium control for the reinsurer is determined by a time-adjusted variance principle. In addition, we find that the reinsurer would like to raise the reinsurance price to guard against the model uncertainty, which consequently decreases the insurer's reinsurance demand. Finally, further analyses are provided to show the necessity of considering the model uncertainty; otherwise, the reinsurance company will suffer a great loss of utility.

Optimal Investment and Proportional Reinsurance in a Regime-Switching Market Model under Forward Preferences

In this paper, we study the optimal investment and reinsurance problem of an insurance company whose investment preferences are described via a forward dynamic exponential utility in a regime-switching market model. Financial and actuarial frameworks are dependent since stock prices and insurance claims vary according to a common factor given by a continuous time finite state Markov chain. We construct the value function and we prove that it is a forward dynamic utility. Then, we characterize the optimal investment strategy and the optimal proportional level of reinsurance. We also perform numerical experiments and provide sensitivity analyses with respect to some model parameters.

Optimal Reinsurance and Investment Strategy with Delay in Heston’s SV Model

In this paper, we consider an optimal investment and proportional reinsurance problem with delay, in which the insurer’s surplus process is described by a jump-diffusion model. The insurer can buy proportional reinsurance to transfer part of the insurance claims risk. In addition to reinsurance, she also can invests her surplus in a financial market, which is consisted of a risk-free asset and a risky asset described by Heston’s stochastic volatility (SV) model. Considering the performance-related capital flow, the insurer’s wealth process is modeled by a stochastic differential delay equation. The insurer’s target is to find the optimal investment and proportional reinsurance strategy to maximize the expected exponential utility of combined terminal wealth. We explicitly derive the optimal strategy and the value function. Finally, we provide some numerical examples to illustrate our results.

Martingale method for optimal investment and proportional reinsurance

Numerous researchers have applied the martingale approach for models driven by Levy processes to study optimal investment problems. This paper considers an insurer who wants to maximize the expected utility of terminal wealth by selecting optimal investment and proportional reinsurance strategies. The insurer’s risk process is modeled by a Levy process and the capital can be invested in a security market described by the standard Black-Scholes model. By the martingale approach, the closed-form solutions to the problems of expected utility maximization are derived. Numerical examples are presented to show the impact of model parameters on the optimal strategies.

跳扩散模型下稳健最优再保险和CDS投资策略

跳扩散模型下稳健最优再保险和CDS投资策略

Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition

<p style='text-indent:20px;'>In this study, under the criterion of maximizing the expected exponential utility of terminal wealth, the optimal proportional reinsurance and investment strategy for an insurer is examined with the compound Poisson claim process. To make the model more realistic, the price process of the risky asset is modelled by the Brownian motion risk model with dividends and transaction costs, where the instantaneous of investment return follows as a mean-reverting Ornstein-Uhlenbeck process. At the same time, the net profit condition and variance reinsurance premium principle are also considered. Using stochastic control theory, explicit expressions for the optimal policy and value function are derived, and various numerical examples are given to further demonstrate the effectiveness of the model.</p>