Equilibrium Value Function Research Articles

Alpha-robust mean–variance reinsurance and investment strategies with transaction costs

This paper investigates the time-consistent reinsurance and investment strategies for insurers based on the alpha-robust mean–variance criterion. We assume that transaction costs with quadratic form exist in the financial market composed of a risk-free asset and n risky assets, and the insurance and financial markets are correlated. By solving a system of extended HJB equations, the equilibrium reinsurance and investment strategy and the corresponding value function are derived in terms of the solution to a system of matrix Riccati equations. In some special cases, more explicit expressions for the equilibrium strategies and value functions are provided. Numerical examples demonstrate that the growth rate of investment slows down as the transaction costs level or the correlation coefficient increases. In addition, we find that the transaction costs level has opposite effects on the utility losses due to ignoring jumps or ambiguity.

Robust equilibrium strategies for time-inconsistent stochastic optimal control problems with applications

In this paper, a new robust time-inconsistent stochastic optimal control problem is investigated under a general jump-diffusion model. Such a problem can be described as a sup-sup-inf problem consisting of the optimal control and the optimal selection of a probability measure that reflects the optimistic and pessimistic sentiments of the agent. Under a game-theoretic framework, the definition of the equilibrium control-measure strategy is given for the robust time-inconsistent stochastic optimal control problem and then an extended Hamilton–Jacobi–Bellman–Isaacs (HJBI) system and a verification theorem are derived for characterizing the equilibrium control-measure strategy and the corresponding equilibrium value function. Moreover, some financial optimization problems and numerical experiments are provided to illustrate the applicability of our newly derived results.

A non-zero-sum stochastic differential game between two mean-variance insurers with inside information

This paper is devoted to the study of a non-zero-sum investment-reinsurance game between two insurers with different opinions about some inside information. Each insurer is concerned about her terminal wealth level and the relative performance of wealth. We assume that the surplus processes of the insurers are correlated jump-diffusion processes. The insurers can invest in the financial market consisting of a risk-free asset and a risky asset. From the beginning of the transaction, the insurers have some inside information about the risky asset price in the future, which is disturbed by noise. However, one insurer trusts it while the other does not. We use the filtration expansion technique to transform the wealth processes for these two insurers with inside information. Then, under the dynamic mean-variance criterion, the explicit solutions for the equilibrium investment-reinsurance strategy and the equilibrium value function are derived by solving the extended HJB equations. Finally, the numerical examples are provided to analyze the effects of the inside information and the model parameters on the equilibrium strategy and the effective frontier.

Equilibrium investment strategy for multi-period DC pension funds with stochastic interest rate and regime switching

This paper studies the equilibrium investment strategy for multi-period defined contribution (DC) pension funds under the mean-variance criterion. We assume that the financial market consists of one risk-free asset and multiple risky assets, the interest rate is stochastic and characterized by the discrete-time Ho-Lee model, and the interest rate as well as the returns of the risky assets both depend on the market states, the evolution of the market states is described by a Markov chain. We regard this problem as a non-cooperative game whose equilibrium strategy is a time-consistent strategy, and derive the analytical expressions for the equilibrium strategy, the equilibrium value function and the equilibrium efficient frontier by using the extended Bellman equation and the matrix representation technique. Finally, a numerical example based on real data from the UK market is provided to shed light on the theoretical results established in this paper.

Slow persuasion

What are the value and form of optimal persuasion when information can be generated only slowly? We study this question in a dynamic model in which a “sender” provides public information over time subject to a graduality constraint, and a decision maker takes an action in each period. Using a novel “viscosity” dynamic programming principle, we characterize the sender's equilibrium value function and information provision. We show that the graduality constraint inhibits information provision relative to unconstrained persuasion. The gap can be substantial, but closes as the constraint slackens. Contrary to unconstrained persuasion, less‐than‐full information may be provided even if players have aligned preferences but different prior beliefs.

Time-consistent lifetime portfolio selection under smooth ambiguity

<p style='text-indent:20px;'>This paper studies the optimal consumption, life insurance and investment problem for an income earner with uncertain lifetime under smooth ambiguity model. We assume that risky assets have unknown market prices that result in ambiguity. The individual forms his belief, that is, the distribution of market prices, according to available information. His ambiguity attitude, which is similar to the risk attitude described by utility function <inline-formula><tex-math id="M1">\begin{document}$ U $\end{document}</tex-math></inline-formula>, is represented by an ambiguity preference function <inline-formula><tex-math id="M2">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>. Under the smooth ambiguity model, the problem becomes time-inconsistent. We derive the extended Hamilton-Jacobi-Bellman (HJB) equation for the equilibrium value function and equilibrium strategy. Then, we obtain the explicit solution for the equilibrium strategy when both <inline-formula><tex-math id="M3">\begin{document}$ U $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> are power functions. We find that a more risk- or ambiguity-averse individual will consume less, buy more life insurance and invest less. Moreover, we find that the Tobin-Markowitz separation theorem is no longer applicable when ambiguity attitude is taken into consideration. The investment strategy will change with the characteristics of the decision maker, such as risk attitude, ambiguity attitude and age.</p>

Deep Neural Network Solution for Finite State Mean Field Game with Error Estimation

We discuss the numerical solution to a class of continuous time finite state mean field games. We apply the deep neural network (DNN) approach to solving the fully coupled forward and backward ordinary differential equation system that characterizes the equilibrium value function and probability measure of the finite state mean field game. We prove that the error between the true solution and the approximate solution is linear to the square root of DNN loss function. We give an example of applying the DNN method to solve the optimal market making problem with terminal rank-based trading volume reward.

Uncertain Stochastic Optimal Control with Jump and Its Application in a Portfolio Game

This article describes a class of jump-uncertain stochastic control systems, and derives an Itô–Liu formula with jump. We characterize an optimal control law, that satisfies the Hamilton–Jacobi–Bellman equation with jump. Then, this paper deduces the optimal portfolio game under uncertain stochastic financial markets with jump. The information of players is symmetrical. The financial market is constituted of a risk-free asset and a risky asset whose price process is subjected to the jump-uncertain stochastic Black–Scholes model. The game is formulated by two utility maximization problems, each investor tries to maximize his relative utility, which is the weighted average of terminal wealth difference between his terminal wealth and that of his competitor. Finally, the explicit expressions of equilibrium investment strategies and value functions for the constant absolute risk-averse and constant relative risk-averse utility function are derived by using the dynamic programming principle.

Equilibrium reinsurance-investment strategy with a common shock under two kinds of premium principles

This paper investigates the optimal mean-variance reinsurance-investment problem for an insurer with a common shock dependence under two kinds of popular premium principles: the variance premium principle and the expected value premium principle. We formulate the optimization problem within a game theoretic framework and derive the closed-form expressions of the equilibrium reinsurance-investment strategy and equilibrium value function under the two different premium principles by solving the extended Hamilton–Jacobi–Bellman system of equations. We find that under the variance premium principle, the proportional reinsurance is the optimal reinsurance strategy for the optimal reinsurance-investment problem with a common shock, while under the expected value premium principle, the excess-of-loss reinsurance is the optimal reinsurance strategy. In addition, we illustrate the equilibrium reinsurance-investment strategy by numerical examples and discuss the impacts of model parameters on the equilibrium strategy.

Robust equilibrium investment and reinsurance strategy with bounded memory and common shock dependence

In this paper, we consider the robust investment and reinsurance problem with bounded memory and risk co-shocks under a jump-diffusion risk model. The insurer is assumed to be ambiguity-averse and make the optimal decision under the mean-variance criterion. The insurance market is described by two-dimensional dependent claims while the risky asset is depicted by the jump-diffusion model. By introducing the performance in the past, we derive the wealth process depicted by a stochastic delay differential equation (SDDE). Applying the stochastic control theory under the game-theoretic framework, together with stochastic control theory with delay, the robust equilibrium investment-reinsurance strategy and the corresponding robust equilibrium value function are derived. Furthermore, some numerical examples are provided to illustrate the effect of market parameters on the optimal investment and reinsurance strategy.

Optimal time-consistent investment-reinsurance strategy for state-dependent risk aversion with delay and common shocks

This article investigates the optimal investment-reinsurance problem for state-dependent risk aversion with delay and common shock dependence. We suppose that the insurer manages insurance risk which is characterized by a two-dimensional dependent claim process by purchasing proportional reinsurance, and the insurer can invest his wealth in a financial market that consists of a risk-free asset and a risky asset. On the basis of the consideration of the performance-related capital inflow/outflow, the wealth process of the insurer is modeled by a stochastic delay differential equation. Then we formulate the optimal investment and reinsurance mean-variance problem within a game-theoretic framework and then applying the stochastic control theory with delay, we derive the corresponding extended HJB equations with delay. Next by solving the extended HJB equations and constructing the exponential martingale process, the closed-form solution to the optimal investment-reinsurance strategy and the corresponding equilibrium value function are derived. Besides, we use a numerical example to analyze the influence of the model parameters on the optimal time-consistent strategy and give an economic explanation.

Optimal Time-Consistent Investment Strategy for a Random Household Expenditure with Default Risk under Relative Performance

Considering the mind of rivalry between families, each family focuses not only on its own wealth but also on other families, especially neighbors. In this paper, we investigate the non-zero-sum mean-variance game between two families with a random household expenditure under the default risk and relative performance. Applying the stochastic control theory within the framework of the game theory, the extended Hamilton–Jacobi–Bellman equation equations are derived. By solving this equation, we obtain the Nash equilibrium strategies of the two families and the corresponding equilibrium value functions. We also provide a numerical example to analyze the effects of relevant parameters on Nash equilibrium strategy and on the utility loss due to the mind of rivalry.

Robust equilibrium strategy for DC pension plan with the return of premiums clauses in a jump-diffusion model

This paper studies a robust equilibrium strategy for a mean-variance defined contribution (DC) pension plan with ambiguity aversion and return of premiums clauses in a jump-diffusion model. Ambiguity-averse members worry about model misspecification and aim to find a robust optimal strategy. In most of DC pension plans, the return of premiums clauses is been often set to protect the rights of plan members. A portion of premiums is usually refunded to members who died during the accumulation phase, while the difference between the sizes of the accumulated fund and the returned premiums is evenly divided among the surviving members. Assume that the financial market contains a risk-free asset and a risky asset, and the price process of risky asset is driven by a jump-diffusion model. We establish a mean-variance model with ambiguity aversion. By using the extended Hamilton–Jacobi–Bellman (HJB) equation, the robust time-consistent equilibrium strategy and the corresponding equilibrium value function are derived. In addition, some special cases are provided in detail. Finally, a numerical example illustrates the impact of model parameters on the robust equilibrium strategy and utility losses.

Equilibrium investment strategy for a DC pension plan with learning about stock return predictability

This paper investigates a time-consistent investment strategy under the mean-variance criterion for an investor who accumulates retirement savings through a defined contribution (DC) pension plan with stock and bond investment opportunities. The expected return rate on the stock is modulated by an unobservable predictor which follows a mean-reverting stochastic process. The evolution of the instantaneous interest rate is described by the Vasicek model. In addition, the contribution rate of the DC pension plan is stochastic and correlated with financial risks coming from the stochastic interest rate and stock price. In a game theoretic framework, we derive a closed-form equilibrium investment strategy and corresponding equilibrium value function for the mean-variance criterion by adopting the filtering technique and the stochastic control method. Furthermore, we provide an equilibrium investment strategy and equilibrium value function when the expected return rate of the stock is completely observable. Finally, some numerical examples are presented to demonstrate the sensitivity analysis of the equilibrium investment strategy and equilibrium efficient frontier. Numerical analysis confirms that there is non-negligible information loss on the equilibrium investment strategy and equilibrium value function due to partial observation in the stock price dynamics.

Equilibrium behavioral strategy for a DC pension plan with piecewise linear state-dependent risk tolerance

We investigate the equilibrium behavioral strategy for a defined contribution (DC) pension plan during the accumulation phase in this paper. We adopt a state-dependent function to describe the risk tolerance attitude of a mean-variance (MV) investor, which is a piecewise linear function of the current wealth level with the reference point being the discounted investment target. Meanwhile, the stochastic labor income is taken into account, whose risk sources arise from both financial and non-financial markets. The extended Hamilton-Jacobi-Bellman (HJB) system of equations for our problem is presented. Explicit expressions for the suboptimal equilibrium behavioral strategy and its corresponding equilibrium value function are obtained in the suboptimal sense by the stochastic control technique. We find that the suboptimal equilibrium behavioral strategy takes a piecewise linear feedback form of the current wealth level with respect to the discounted investment target, and it also depends on the current labor income. Finally, some numerical analyses are provided to illustrate the effects of model parameters on the suboptimal equilibrium behavioral strategy and the variance-expectation curve, which shed light on our theoretical results.

Equilibrium strategy for a multi-period weighted mean-variance portfolio selection in a Markov regime-switching market with uncertain time-horizon and a stochastic cash flow

This article considers a multi-period weighted mean-variance portfolio selection problem with uncertain time-horizon and a stochastic cash flow in a Markov regime-switching market. The random returns of risky assets and amount of the cash flow all depend on the states of a stochastic market which are assumed to follow a discrete-time Markov chain. Based on the conditional distribution of uncertain time-horizon caused by exogenous factors, we construct a more general mean-variance investment model. Within a game theoretic framework, we derive the equilibrium strategy and equilibrium value function in closed-form by applying backward induction approach. In addition, we show the equilibrium efficient frontier and discuss some degenerate cases. Finally, some numerical examples and sensitivity analysis are presented to illustrate equilibrium efficient frontiers and the effects of uncertain time-horizon on the equilibrium strategy and equilibrium efficient frontier as well as regime-switching and stochastic cash flow on the equilibrium efficient frontier.

Equilibrium investment-reinsurance strategy with delay and common shock dependence under Heston's SV model

This paper investigates an investment and reinsurance problem with delay and common shock dependence under the mean-variance utility framework. We first use Heston's SV model to depict the financial market and two-dimensional Poisson process with common shock dependence to characterize the surplus process. We then introduce the past performance and use it to derive the wealth process depicted by the stochastic delay differential equation. Applying the stochastic control theory within the framework of the game theory, and stochastic control theory with delay, we derive an extended Hamilton-Jacobi-Bellman equation with delay. By solving the equation, we obtain the equilibrium strategy and the corresponding equilibrium value function. We also provide a numerical example to analyze the effects of delay parameters and co-shocks parameters on equilibrium strategy and explain why such effects occur.

Robust Stochastic Stackelberg Differential Reinsurance and Investment Games for an Insurer and a Reinsurer with Delay

This paper investigates the asset-liability management problem for an ordinary robust insurance system between a reinsurer and an insurer under the Heston model with delay. The reinsurer, as the leader of the Stackelberg game, can price reinsurance premium and invest its wealth in a financial market that contains a risk-free asset and a risky asset whose price process is described by the Heston model. The insurer, as the follower of the Stackelberg game, can purchase proportional reinsurance from the reinsurer and invest in the same financial market. Under the consideration of the performance-related capital inflow/outflow, the wealth processes of the insurer and reinsurer are modeled by stochastic differential delay equations (SDDEs). This paper aims to find the equilibrium strategies for the reinsurer and insurer by maximizing the expected utility of the player’s terminal wealth with delay under the worst-case scenario of the alternative measures. By using the idea of backward induction and dynamic programming approach, the explicit expressions of the robust equilibrium strategies and value functions are derived. Finally, some numerical examples and sensitivity analysis to illustrate the effects of model parameters on the robust optimal reinsurance and investment strategies are performed.

Optimal Time-Consistent Investment and Reinsurance Strategies with Default Risk and Delay under Heston’s SV Model

Considering the influence of past information on the decision-making of insurers, the correlation between the insurance businesses owned by insurers, and the possible default faced by insurers, we investigate the mean-variance investment and reinsurance problem with the default risk, delay, and common shock dependence. We characterize the insurance market by two-dimensional dependent claims, the financial market by the Heston SV model, and default risk by reduced-form approach and then obtain the evolution equation of the insurer’s wealth. Based on the introduction of time delay, the insurer’s wealth dynamics characterized by a stochastic delay differential equation are obtained. Furthermore, applying stochastic control theory within the game-theoretic framework and stochastic control theory with delay, we derive optimal time-consistent investment and reinsurance strategies, as well as equilibrium value function and equilibrium efficient frontier. Finally, we use a numerical example to analyze the influence of parameters on the time-consistent equilibrium strategies and give an economic explanation.

Equilibrium excess-of-loss reinsurance and investment strategies for an insurer and a reinsurer

In this paper, we consider the equilibrium excess-of-loss reinsurance and investment problem for both an insurer and a reinsurer. The risk process of the insurer is described by a classical Cramér-Lundberg (C-L) risk model and the insurer can purchase excess-of-loss reinsurance from the reinsurer. Both the insurer and the reinsurer are allowed to invest in a financial market consisting of a risk-free asset and two risky assets. The market price of risk depends on a Markovian, affine-form and square-root stochastic factor process. Dynamic mean-variance criterion is considered in this paper. We aim to maximize the weighted sum of the insurer’s and the reinsurer’s objectives with different risk averse coefficients. By solving the corresponding extended Hamilton-Jacobi-Bellman (HJB) equations, we derive the equilibrium reinsurance and investment strategies and the corresponding equilibrium value function. Finally, the economic implications of our findings are illustrated.