Robust Optimal Reinsurance and Investment Problem Under Markov Switching via Actor–Critic Reinforcement Learning

In this article, we analyze a robust optimal reinsurance and investment problem in a model with default risks and jumps for a general company which holds shares of an insurer and a reinsurer. The insurer’s surplus is characterized by a diffusion model and the insurer has the opportunity to buy proportional reinsurance from the reinsurer. Moreover, the insurer can invest in a risk-free asset (bank account), a risky asset (stock), and a defaultable bond. Meanwhile, the reinsurer can also invest in a risk-free asset and another risky asset. The price dynamics of the risky asset is assumed to follow a jump-diffusion model. What is more, the general company is concerned about the ambiguity and aims to find a robust optimal reinsurance and investment strategy. The objective of the general company is to maximize the minimal expected exponential utility of the weighted sums of the insurer’s and the reinsurer’s terminal wealth. Applying stochastic dynamic programming approach, we first establish the robust HJB (Hamilton-Jacobi-Bellman) equations for the post-default case and the pre-default case, respectively. Furthermore, we obtain the robust optimal strategy explicitly and present the corresponding verification theorem. Finally, we give some numerical examples to demonstrate the influences of model parameters on the robust optimal strategy.

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.

Robust optimal reinsurance and investment strategies for an AAI with multiple risks

. In this article, we consider the robust reinsurance and investment problem for two interdependent insurance businesses under two different objective criteria. The claim process is described by a Brownian motion with drift, and the insurer can invest in risk-free assets and risky assets described by the Ornstein-Uhlenbeck (O-U) process. After considering the uncertainty of the model, we obtain the robust optimization problem. We derive the α-robust equilibrium strategy under the α-maxmin mean-variance criterion by solving the extended Hamilton-Jacobi-Bellman (HJB) equation. Subsequently, the existence of the semi-explicit equilibrium investment strategy is demonstrated through the Riccati differential equation system. Additionally, we obtain a closed-form robust strategy that maximizes the expected exponential utility of terminal wealth under worst-case ambiguity uncertainty and compare it with the robust strategy under the mean-variance criterion. We find that robust reinsurance strategies are the same under both criteria when the risk aversion coefficients are the same. Numerical simulations are conducted to further compare the robust strategies under the two criteria.

Motivated by the financial crisis of 2007-2009 and the increasing demand for portfolio and risk management, we study optimal insurance and investment problems with regime switching in this thesis. We incorporate an insurable risk into the classical consumption and investment framework and consider an investor who wants to select optimal consumption, investment and insurance policies in a regime switching economy. We allow not only the financial market but also the insurable risk to depend on the regime of the economy. The objective of the investor is to maximize his/her expected total discounted utility of consumption over an infinite time horizon. For the case of hyperbolic absolute risk aversion (HARA) utility functions, we obtain the first explicit solutions for simultaneous optimal consumption, investment and insurance problems when there is regime switching. Next we consider an insurer who wants to maximize his/her expected utility of terminal wealth by selecting optimal investment and risk control policies. The insurer’s risk is modeled by a jump-diffusion process and is negatively correlated with the capital gains in the financial market. In the case of no regime switching in the economy, we apply the martingale approach to obtain optimal policies for HARA utility functions, constant absolute risk aversion (CARA) utility functions, and quadratic utility functions. When there is regime switching in the economy, we apply dynamic programming to derive the associated Hamilton-Jacobi-Bellman (HJB) equation. Optimal investment and risk control policies are then obtained in explicit forms by solving the HJB equation. ii We provide economic analyses for all optimal control problems considered in this thesis. We study how optimal policies are affected by the economic conditions, the financial and insurance markets, and investor’s risk preference.

Robust optimal control for an insurer with reinsurance and investment under Heston’s stochastic volatility model

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.

In this paper, we investigate the optimal reinsurance and investment problem from joint interests of the insurer and the reinsurer in the framework of the mixed leadership game, where both of them can invest their wealth in the financial market, and the liabilities are correlated with the risky asset. More specifically, we assume that the insurer is the leader to determine the amount he/she invests into the risky asset but acts as the follower to decide the optimal reinsurance retention level, while the reinsurer is the leader to set the reinsurance premium price and acts as a follower to determine his/her investment strategy. Both the insurer and the reinsurer aim to maximize the expected utility of their terminal wealth, and the explicit Stackelberg-Nash equilibrium is derived by solving the corresponding Hamilton-Jacobi-Bellman (HJB) equations. In order to better understand the effect of the mixed leadership game on the optimal reinsurance and investment problem, we also consider the optimization problem within the framework of the stochastic Stackelberg differential game shown in [Bai, Y., Zhou, Z., Xiao, H., Gao, R. & Zhong, F. (2021). A Stackelberg reinsurance-investment game with asymmetric information and delay. Optimization 70(10), 2131–2168.], where the reinsurer is the leader while the insurer acts as the follower for both the reinsurance and investment strategies. Some theoretical analyses and numerical examples are provided to show the economic intuition and insights of the results.

This paper studies the optimal reinsurance and investment problem with multiple risky assets and correlation risk. The claim process is described by a Brownian motion with drift. The insurer is allowed to invest in a risk-free asset and multiple risky assets and the instantaneous return rate of each risky asset follows the Ornstein-Uhlenbeck (O-U) model. Moreover, the correlation between risk model and the risky assets’ price is taken into account. We first consider the optimal investment problem for the insurer. Subsequently, we assume that the insurer can purchase proportional reinsurance and invest in the financial market. In both cases, the insurer’s objective is to maximize the expected exponential utility of the terminal wealth. By applying stochastic control approach, we derive the optimal reinsurance and investment strategies and the corresponding value functions explicitly. Finally, numerical simulations are presented to illustrate the effects of model parameters on the optimal reinsurance and investment strategies.

Robust optimal proportional reinsurance and investment strategy for an insurer with defaultable risks and jumps

Nonlinear robust integral based actor–critic reinforcement learning control for a perturbed three-wheeled mobile robot with mecanum wheels

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.

In this paper, we investigate the optimal consumption and investment problem under the expected utility maximization criterion. It is supposed that the financial market consists of a risky asset and a risk-free asset, and the risky asset prices follow the 4/2 Cox–Ingersoll–Ross (CIR) stochastic hybrid model. The investment objective is to obtain an optimal consumption–investment strategy by maximizing the objective function. The closed-form expression of the optimal consumption–investment strategy is obtained by using optimal control theory and the corresponding Hamilton–Jacobi–Bellman (HJB) equation under the power utility function. In addition, we present a numerical example to illustrate the influence of model parameters on the optimal consumption–investment strategy.

In this paper, the optimal investment and reinsurance problem is investigated for a class of the jump-diffusion risk model. Here, the insurer can purchase excess-of-loss reinsurance and invest his or her surplus into a financial market consisting of one risk-free asset and one risk asset whose price is modeled by constant elasticity of variance (CEV) model, the net profit condition and the criterion of maximizing the expected exponential utility of terminal wealth are considered in the CEV financial market. By using stochastic control theory to solve the Hamilton-Jacobi-Bellman (HJB) equation, and the explicit form of the optimal policies and value functions can be obtained. Finally, numerical examples are presented to show the impacts of model parameters on the optimal strategies.

This article investigates the optimal investment problem with multiple risky assets and the correlation between risk model and financial market for an insurer. The insurer’s claim process is described by a Brownian motion with drift under the mean-variance premium principle. The insurer is allowed to invest in one risk-free asset and multiple risky assets whose price processes follow the constant elasticity of variance (CEV) model. Moreover, the correlation between the claim process and each risky asset’s price is taken into account. The insurer’s objective is to maximize the exponential utility of the terminal wealth. By applying dynamic programming approach, we propose a new form of the solution to the Hamilton-Jacobi-Bellman (HJB) equation and derive the optimal investment strategy explicitly. This is the first time that an explicit result of the optimal investment strategy is given under the framework of the exponential utility function for the general CEV model based on the correlation between the financial market and the insurance market. In addition, we provide some special cases of our model, i.e., the optimal investment problem with one risky asset and that with two risky assets. We also consider the case that the risk model and the financial market is independent. The results show that ignoring the correlation between the claim process and the risky asset’s price will misestimate the value of risky assets and has a significant effect on the insurer’s investment decision. Finally, numerical simulations are presented to analyze the effects of model parameters on the optimal investment strategy.