Stochastic Differential Reinsurance Research Articles

Stochastic differential reinsurance game for two competitive insurers with ambiguity-aversion under mean-variance premium principle

Stochastic differential reinsurance game for two competitive insurers with ambiguity-aversion under mean-variance premium principle

Non-zero-sum reinsurance and investment game under thinning dependence structure: mean–variance premium principle

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.

Stochastic differential reinsurance and investment games with delay under VaR constraints⋆

This article investigates stochastic differential reinsurance and investment games with delay under Value-at-Risk (VaR) constraints. In our model, three players and two games are considered, i.e., a non-zero-sum stochastic differential game between two insurers and a stochastic Stackelberg differential game between the reinsurer and the two insurers are investigated. The reinsurer can set the reinsurance premium price and invest her surplus in a financial market consisting of a risk-free asset and a risky asset. The two insurers can purchase proportional reinsurance and invest in the same financial market. Since the delay is considered in this article, the wealth processes of three players are described by the stochastic delay differential equations. In this article, we establish two optimization problems, one is to maximize the expected utility of the reinsurer’s terminal wealth with delay, and the other is to maximize the expected utility of the combination of each insurer’s terminal wealth and relative performance with delay. Furthermore, in order to control risk, we incorporate the VaR constraints into the optimizations and derive the corresponding value functions and Nash equilibrium strategies by using optimal control theory, dynamic programming principle, backward induction, Lagrange function and Karush–Kuhn–Tucker condition. Finally, several numerical examples are given.

A hybrid stochastic differential reinsurance and investment game with bounded memory

This paper investigates a hybrid stochastic differential reinsurance and investment game between one reinsurer and two insurers, including a stochastic Stackelberg differential subgame and a non-zero-sum stochastic differential subgame. The reinsurer, as the leader of the Stackelberg game, can price reinsurance premium and invest her wealth in a financial market that contains a risk-free asset and a risky asset which is described by the constant elasticity of variance (CEV) model. The two insurers, as the followers of the Stackelberg game, can purchase proportional reinsurance from the reinsurer and invest in the same financial market. The competitive relationship between two insurers is modeled by the non-zero-sum game, and their decision-making will consider the relative performance measured by the difference in their terminal wealth. The bounded memory feature is characterized by the wealth process with delay. The purpose of the reinsurer is to maximize the expected utility of her own terminal wealth with delay. The two insurers aim to maximize the expected utility of the combination of her terminal wealth and the relative performance with delay. By using the idea of backward induction and the dynamic programming approach, we derive the equilibrium strategy and value functions explicitly. Then, we provide the corresponding verification theorem. Finally, some numerical examples and sensitivity analysis are presented to demonstrate the effects of model parameters on the equilibrium strategy. We find the delay factor discourages or stimulates investment depending on the length of delay. Moreover, competitive factors between two insurers make their optimal reinsurance-investment strategy interact, and reduce reinsurance demand and reinsurance premium price.

The Role of Risk Aversion in Stochastic Differential Reinsurance Games

The Role of Risk Aversion in Stochastic Differential Reinsurance Games

Stochastic differential reinsurance games in diffusion approximation models

In this paper, we study some noncooperative and cooperative stochastic differential games between two insurers with reinsurance controls. The surplus processes are modeled by diffusion approximation processes and the insurers can purchase reinsurance dynamically (i.e. control the drift and diffusion terms continuously over time). We consider two types of reinsurance: quota-share (QS) reinsurance and excess-of-loss (XL) reinsurance. In the noncooperative game, one insurer tries to minimize the probability that the surplus difference of the two insurers reaches a low target before it hits a high target, while the other aims to maximize the probability. We consider two cases of the game: in the first case, both insurers purchase XL reinsurance; and in the second case, one insurer purchases QS reinsurance while the other purchases XL reinsurance. In some parameter cases, we solve the game by finding the value function and Nash equilibrium strategy explicitly. In some parameter cases, the value function and Nash equilibrium strategy do not exist and we find the sup-value and sub-value functions of the game. We also establish and solve a cooperative game in which both insurers make joint efforts to minimize the probability that the sum of surplus processes reaches a low target before it hits a high target. Numerical examples and economic implications are given to illustrate the results.

Nonzero-Sum Stochastic Differential Reinsurance Games with Jump–Diffusion Processes

In this paper, a nonzero-sum stochastic differential reinsurance game is studied. A model including controls for the market share (advertising), investment, and reinsurance policies is considered. A jump–diffusion process is used to represent insurance claims. Necessary conditions that would lead to the Nash equilibrium can be found in the duopoly game we consider. Cases with and without controls on the reinsurance are discussed separately. Closed-form solutions for optimal strategies are derived by applying the Hamilton–Jacobi–Bellman equation. Numerical examples are given to validate the correctness of our results.

A Stackelberg reinsurance–investment game with asymmetric information and delay

ABSTRACT This paper investigates a Stackelberg stochastic differential reinsurance–investment game problem, in which the reinsurer is the leader and the insurer is the follower. The unequal status of the reinsurer and the insurer in the financial market is characterized by the asymmetric information model. We consider their wealth processes with delay to characterize bounded memory. The objective of the reinsurer is to find the optimal premium pricing strategy and investment strategy such that its constant absolute risk aversion (CARA) utility of the combination of terminal wealth and average performance wealth is maximized. The objective of the insurer is to find the optimal reinsurance strategy and investment strategy such that its CARA utility of the relative performance is maximized. We derive the equilibrium strategy explicitly for the game by solving corresponding Hamilton–Jacobi–Bellman equations sequentially. Then, we establish the verification theorem. The equilibrium investment strategy indicates that the insurer with less information completely imitates the investment strategy of the reinsurer who has more information on the financial market. Further, we find that the effect of the delay weight on the equilibrium strategy is related to the length of delay time. Finally, we present some numerical examples to demonstrate the findings.

Time-consistent non-zero-sum stochastic differential reinsurance and investment game under default and volatility risks

This paper investigates a non-zero-sum stochastic differential game between two mean–variance insurers. These two insurers are concerned about their terminal wealth and the relative performance compared with each other. We assume that they can buy proportional reinsurance and invest in a financial market consisting of a risk-free asset, a stock and a defaultable bond. The price process of stock is driven by the constant elasticity of variance (CEV) model and the defaultable bond recovers a proportion of value at default. So, these two insurers are faced with insurance risk, volatility risk and default risk. The non-zero-sum goal of these insurers is to maximize the mean–variance utility of a weighted value of their terminal and relative wealths. We solve the mean–variance problem in the time-consistent case and establish the extended Hamilton−Jacobi−Bellman systems for the post-default case and the pre-default case, respectively. Furthermore, we derive the closed form solutions of the Nash equilibrium reinsurance and investment strategies for these two insurers. In the end of this paper, we calibrate the parameters based on real data and several numerical examples are provided to illustrate the effects of economic parameters on the equilibrium strategies.

Stochastic differential reinsurance games with capital injections

This paper investigates a class of reinsurance game problems between two insurance companies under the framework of non-zero-sum stochastic differential games. Both insurers can purchase proportional reinsurance contracts from reinsurance markets and have the option of conducting capital injections. We assume the reinsurance premium is calculated under the generalized variance premium principle. The objective of each insurer is to maximize the expected value that synthesizes the discounted utility of his surplus relative to a reference point, the penalties caused by his own capital injection interventions, and the gains brought by capital injections of his competitor. We prove the verification theorem and derive explicit expressions of the Nash equilibrium strategy by solving the corresponding quasi-variational inequalities. Numerical examples are also conducted to illustrate our results.

Non-zero-sum stochastic differential reinsurance and investment games with default risk

This paper investigates the implications of strategic interaction (i.e., competition) between two CARA insurers on their reinsurance-investment policies. The two insurers are concerned about their terminal wealth and the relative performance measured by the difference in their terminal wealth. The problem of finding optimal policies for both insurers is modelled as a non-zero-sum stochastic differential game. The reinsurance premium is calculated using the variance premium principle and the insurers can invest in a risk-free asset, a risky asset with Heston’s stochastic volatility and a defaultable corporate bond. We derive the Nash equilibrium reinsurance policy and investment policy explicitly for the game and prove the corresponding verification theorem. The equilibrium strategy indicates that the best response of each insurer to the competition is to mimic the strategy of its opponent. Consequently, either the reinsurance strategy or the investment strategy of an insurer with the relative performance concern is riskier than that without the concern. Numerical examples are provided to demonstrate the findings of this study.

Robust non-zero-sum stochastic differential reinsurance game

This paper considers the non-zero-sum stochastic differential game problem between two ambiguity-averse insurers (AAIs) who encounter model uncertainty and seek the optimal reinsurance decision under relative performance concerns. Each AAI manages her own risks by purchasing reinsurance with the objective of maximizing the expected utility of her relative terminal surplus with respect to that of her counterparty. The two AAIs’ decisions influence each other through the insurers’ relative performance concerns and the correlation between their surplus processes. We establish a general framework of Nash equilibrium for the associated non-zero-sum game with model uncertainty. For the representative case of exponential utilities, we solve the equilibrium strategies explicitly. Numerical studies are conducted to draw economic interpretations.

Non-Zero-Sum Stochastic Differential Reinsurance and Investment Games with Default Risk

This paper investigates implications of strategic interaction (competition) between two CARA insurers on their reinsurance and investment policies. The two insurers are concerned about their terminal wealth as well as the relative performance measured by the difference between their terminal wealth. The problem of finding optimal policies for the both insurers is modeled by a non-zero-sum stochastic differential game. We assume that the insurers can invest in a risky asset with Heston’s stochastic volatility and a defautable corporate bond, and the reinsurance premium is calculated by the variance premium principle. We derive the Nash equilibrium reinsurance policy and investment policy explicitly for the game and prove the corresponding verification theorem. The equilibrium strategy indicates that the best response of each insurer to the competition is to mimic the strategy of its opponent. As a consequence, the reinsurance and investment strategy of an insurer with the relative performance concern is riskier than that without the concern. We illustrate our results by numerical examples.

A reinsurance game between two insurance companies with nonlinear risk processes

In this paper, we consider a stochastic differential reinsurance game between two insurance companies with nonlinear (quadratic) risk control processes. We assume that the goal of each insurance company is to maximize the exponential utility of the difference between its terminal surplus and that of its competitor at a fixed terminal time T. First, we give an explicit partition (including nine subsets) of time interval [0,T]. Further, on every subset, an explicit Nash equilibrium strategy is derived by solving a pair of Hamilton–Jacobi–Bellman equations. Finally, for some special cases, we analyze the impact of time t and quadratic control parameter on the Nash equilibrium strategy and obtain some simple partition of [0,T]. Based on these results, we apply some numerical analysis of the time t, quadratic control parameter and competition sensitivity parameter on the Nash equilibrium strategy and the value function.

A stochastic differential reinsurance game

We study a stochastic differential game between two insurance companies who employ reinsurance to reduce the risk of exposure. Under the assumption that the companies have large insurance portfolios compared to any individual claim size, their surplus processes can be approximated by stochastic differential equations. We formulate competition between the two companies as a game with a single payoff function which depends on the surplus processes. One company chooses a dynamic reinsurance strategy in order to maximize this expected payoff, while the other company simultaneously chooses a dynamic reinsurance strategy so as to minimize the same quantity. We describe the Nash equilibrium of this stochastic differential game and solve it explicitly for the case of maximizing/minimizing the exit probability.

A stochastic differential reinsurance game

We study a stochastic differential game between two insurance companies who employ reinsurance to reduce the risk of exposure. Under the assumption that the companies have large insurance portfolios compared to any individual claim size, their surplus processes can be approximated by stochastic differential equations. We formulate competition between the two companies as a game with a single payoff function which depends on the surplus processes. One company chooses a dynamic reinsurance strategy in order to maximize this expected payoff, while the other company simultaneously chooses a dynamic reinsurance strategy so as to minimize the same quantity. We describe the Nash equilibrium of this stochastic differential game and solve it explicitly for the case of maximizing/minimizing the exit probability.