Time-consistent Equilibrium Research Articles

The Time-Consistent Optimal Reinsurance Strategy of Insurance Group under the CEV Model

The article introduces proportional reinsurance contracts under the mean-variance criterion, studying the time-consistence investment portfolio problem considering the interests of both insurance companies and reinsurance companies. The insurance claims process follows a jump-diffusion model, assuming that the risk asset prices of insurance companies and reinsurance companies follow CEV models different from each other. In the framework of game theory, the time-consistent equilibrium reinsurance strategy is obtained by solving the extended HJB equation analytically. Finally, numerical examples are used to illustrate the impact of model parameters on equilibrium strategies and provide economic explanations. The results indicate that the decision weights of insurance companies and reinsurance companies do have a significant impact on both the reinsurance ratio and the equilibrium reinsurance strategy.

Time Consistency and Duration of Government Debt: A Model of Quantitative Easing

Abstract This article presents a model of quantitative easing (QE) at the zero lower bound (ZLB) on the short-term nominal interest rate. QE, which reduces the maturity of government debt, is effective at the ZLB because it generates expectations of future monetary expansion in a time-consistent equilibrium. Numerical experiments show that this effect can be substantial.

Time-consistent equilibria in dynamic models with recursive payoffs and behavioral discounting

We prove existence of time consistent equilibria in a class of dynamic models with recursive payoffs and generalized discounting involving both behavioral and normative applications. Our generalized Bellman equation method identifies and separates both: recursive and strategic aspects of the equilibrium problem and allows to determine the sufficient assumptions on preferences and stochastic transition to establish existence. In particular we show existence of minimal state space stationary Markov equilibrium (a time-consistent equilibrium) in a deterministic model of consumption-saving with beta-delta discounting and its generalized versions involving non-additive payoffs, general form certainty equivalents, as well as stochastic semi-hyperbolic and hyperbolic discounting models (over possibly unbounded state and unbounded above reward space). We also provide an equilibrium approximation method for a hyperbolic discounting model.

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.

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.

Peer Preference and Asset-Liability Management in InsurTech Market

Technology advances have enhanced competition in insurance industry. This paper investigates a class of dynamic asset and liability management(ALM) game among mean-variance insurers with relative log return performance. We obtain the unique explicit time-consistent equilibrium ALM strategies for both the finite non-zero-sum game and mean field game, respectively. The results show that competition leads the increases of insurers’ risky asset investment and the insurance liability. The equilibrium solution implies that “social util- ity”(relative concerns) leads to “social learning”(herd effect) in the complete information market. In particular, the peer relative concern can induce preference interaction. Finally, we find the Granger causal relationship between industry competition and herd behavior in the empirical data and conduct sensitivity analysis of the competition outcome with respect to the estimated risk parameters.

Lanchester duopoly model revisited: Advertising competition under time inconsistent preferences

In this paper, we study a finite time horizon advertising dynamic game under the assumption of time-inconsistent preferences. Specifically, we consider two types of discounting, heterogeneous discounting and hyperbolic discounting. In the case of heterogeneous discounting, the relative importance of the final function will increase/decrease as the end of the planning horizon approaches compared with current payoffs. Whereas when agents discount future payoffs hyperbolically, their discount rates diminish rapidly in earlier stages and then slowly in the long term. We compute time-inconsistent and time-consistent feedback Nash equilibrium strategies, and compare them with those of the standard discounting case. Our results reveal that heterogeneous discounting would lead to some last-minute changes, i.e., some adapting behaviours in the last years in accordance with their increasing/decreasing valuations of the final state. Under some circumstances, the change can be so radical that the pre-commitment solution takes the contrary path of time-consistent strategies. Concerning the competition under hyperbolic discounting, the temporal evolution of advertising efforts show a quite different nature. Different strategies exhibit disparity in the beginning, and encounter in the neighbourhood in the end, which is contrary to the heterogeneous discounting. Besides, a strong commitment power might induce over investment.

Open-loop equilibrium strategy for mean-variance asset-liability management with margin requirements

This paper considers a mean-variance asset-liability management problem in which short-selling is allowed, but accompanied by margin requirements. This is a mean-variance problem with a non linear state process. We derive a sufficient condition and a necessary condition for the time-consistent equilibrium strategy by using a system of forward backward stochastic differential equations. By decoupling this system, we obtain the unique equilibrium strategy. As a byproduct, we also get the equilibrium strategy for the mean-variance asset-liability management problem under short-selling prohibition.

Linear-Quadratic Time-Inconsistent Mean-Field Type Stackelberg Differential Games: Time-Consistent Open-Loop Solutions

In this article, we consider the linear-quadratic time-inconsistent mean-field type leader-follower Stackelberg differential game with an adapted open-loop information structure. The objective functionals of the leader and the follower include conditional expectations of state and control (mean field) variables, and the cost parameters could be general nonexponential discounting depending on the initial time. As stated in the existing literature, these two general settings of the objective functionals induce time inconsistency in the optimal solutions. Given an arbitrary control of the leader, we first obtain the follower's (time consistent) equilibrium control and its state feedback representation in terms of the nonsymmetric coupled Riccati differential equations (RDEs) and the backward stochastic differential equation (SDE). This provides the rational behavior of the follower, characterized by the forward-backward SDE (FBSDE). We then obtain the leader's explicit (time consistent) equilibrium control and its state feedback representation in terms of the nonsymmetric coupled RDEs under the FBSDE constraint induced by the follower. With the solvability of the nonsymmetric coupled RDEs, the equilibrium controls of the leader and the follower constitute the time-consistent Stackelberg equilibrium. Finally, the numerical examples are provided to check the solvability of the nonsymmetric coupled RDEs.

Time-consistent equilibrium reinsurance–investment strategy for [formula omitted] competitive insurers under a new interaction mechanism and a general investment framework

We propose a new interaction mechanism for n≥2 competitive insurers, under which we investigate the time-consistent equilibrium reinsurance–investment strategy for n insurers. Each insurer can purchase reinsurance for reducing the claim risk and invest in the financial market for increasing his wealth. We consider a general investment framework which includes typical diversified and concentrated investment patterns in the literature. The objective of each insurer is to find a time-consistent equilibrium reinsurance–investment strategy so as to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. By using the stochastic control technique, we obtain the explicit time-consistent equilibrium reinsurance–investment strategies and corresponding equilibrium value functions. Through analyzing the general investment strategy, we find the criterion of whether the insurer should invest in a specific risky asset. Furthermore, we examine theoretically and numerically how the price parameters of risky assets affect the concrete investment pattern. We find that when the appreciation rate of one risky asset exceeds some times as those of other risky assets, the insurer will only invest in this asset; otherwise, the insurer will adopt the diversified investment pattern. We also numerically examine the influences of the number of insurers and the surplus jump on the time-consistent equilibrium reinsurance strategy.

Optimal taxation: full-commitment versus time-consistent equilibrium

This paper examines the properties of the optimal tax mix under different types of policy equilibria and tries to narrow the gap between what the theory suggests and what the actual data show. To do so, we use a two-period deterministic version of the standard neoclassical growth model with heterogeneous agents. In particular, we search for the socially optimal mix of taxes on capital and labour income when the government is able to commit to future policies. Then, we compare this solution to the case in which the government cannot commit to future policies. We show that the optimal tax mix is closer to the actual data in the former case. Furthermore, we show that the lack of commitment creates the incentive for a degree of capitalist bias on the part of the government.

Mean-variance investment and contribution decisions for defined benefit pension plans in a stochastic framework

<p style='text-indent:20px;'>In this paper we investigate the management of a defined benefit pension plan under a model with random coefficients. The objective of the pension sponsor is to minimize the solvency risk, contribution risk and the expected terminal value of the unfunded actuarial liability. By measuring the solvency risk in terms of the variance of the terminal unfunded actuarial liability, we formulate the problem as a mean-variance problem with an additional running cost. With the help of a system of backward stochastic differential equations, we derive a time-consistent equilibrium strategy towards investment and contribution rate. The obtained equilibrium strategy turns out to be a good candidate for a stable contribution plan. When the interest rate is given by the Vasicek model and all other coefficients are deterministic, we obtain closed-form solutions of the equilibrium strategy and efficient frontier.

Time Consistent Equilibria in Dynamic Models With Recursive Payoffs and Behavioral Discounting

We prove existence of time consistent equilibria in a wide class of dynamic models with recursive payoffs and generalized discounting involving both behavioral and normative applications. Our generalized Bellman equation method identifies and separates both: recursive and strategic aspects of the equilibrium problem and allows to precisely determine the sufficient assumptions on preferences and stochastic transition to establish existence. In particular we show existence of minimal state space stationary Markov equilibrium (a time-consistent solution) in a deterministic model of consumption-saving with beta-delta discounting and its generalized versions involving magnitude effects, non-additive payoffs, semi-hyperbolic or hyperbolic discounting (over possibly unbounded state and unbounded above reward space). We also provide an equilibrium approximation method for a hyperbolic discounting model.

Equilibrium investment and risk control for an insurer with non-Markovian regime-switching and no-shorting constraints

In this paper, we study the time-consistent equilibrium investment and risk control policies for an insurer under the mean-variance criterion. The dynamics of liability and assets are given by non-Markovian regime-switching models driven by a Brownian motion and a continuous time finite-state Markov chain. It is assumed that the insurer has a time-varying preference of risk depending dynamically on current wealth, and is not allowed to short sell the risky assets. With the aid of backward stochastic differential equations and bounded mean oscillation martingales, we obtain feedback representations of the open-loop equilibrium strategies. Finally, the result is also applied to solve an example of the Markovian regime-switching model.

Adaptive Strategic Cyber Defense for Advanced Persistent Threats in Critical Infrastructure Networks

Advanced Persistent Threats (APTs) have created new security challenges for critical infrastructures due to their stealthy, dynamic, and adaptive natures. In this work, we aim to lay a game-theoretic foundation by establishing a multi-stage Bayesian game framework to capture incomplete information of deceptive APTs and their multistage multi-phase movement. The analysis of the perfect Bayesian Nash equilibrium (PBNE) enables a prediction of attacker's behaviors and a design of defensive strategies that can deter the adversaries and mitigate the security risks. A conjugate-prior method allows online computation of the belief and reduces Bayesian update into an iterative parameter update. The forwardly updated parameters are assimilated into the backward dynamic programming computation to characterize a computationally tractable and timeconsistent equilibrium solution based on the expanded state space. The Tennessee Eastman (TE) process control problem is used as a case study to demonstrate the dynamic game under the information asymmetry and show that APTs tend to be stealthy and deceptive during their transitions in the cyber layer and behave aggressively when reaching the targeted physical plant. The online update of the belief allows the defender to learn the behavior of the attacker and choose strategic defensive actions that can thwart adversarial behaviors and mitigate APTs. Numerical results illustrate the defender's tradeoff between the immediate reward and the future expectation as well as the attacker's goal to reach an advantageous system state while making the defender form a positive belief.

A manufacturer-retailers dynamic cooperative advertising with retail competition

This work deals with the cooperative advertising in a manufacturer retailers supply chain. It considers the manufacturer as the Stackelberg leader and the retailers as the followers playing a Nash game with each other. Using differential game theory it obtains a time consistent feedback Stackelberg equilibrium for the optimal advertising strategies and payoffs for a situation where retail advertising is subsidised and where it is not subsidised. The work shows that while the manufacturer's advertising effort reduces with subsidy, the retailers' advertising efforts and product awareness increase. These consequently lead to increase in all the players' payoffs. While a fair player would increase his advertising effort with increasing payoff, the others reduce their efforts. Further a retailer's advertising effort should be subsidised only if the rate of increase of the manufacturer's payoff through that retailer's margin to the manufacturer is twice greater than the rate of increase of that retailer's payoff.

Mean-Variance Portfolio Selection under a Non-Markovian Regime-Switching Model: Time-Consistent Solutions

This paper aims to find the time-consistent equilibrium strategy for a mean-variance portfolio selection problem under a non-Markovian regime-switching model, in which the coefficients are adapted to the filtration generated by a Markov chain. By introducing and investigating systems of coupled backward stochastic differential equations driven by the Markov chain, we obtain feedback representations of both open-loop equilibrium strategies and linear closed-loop equilibrium strategies. We also make further comparisons with the existing literature and reveal several interesting facts arising from the non-Markovian regime-switching model.

A paradox in time-consistency in the mean–variance problem?

We establish new conditions under which a constrained (no short-selling) time-consistent equilibrium strategy, starting at a certain time, will beat the unconstrained counterpart, as measured by the magnitude of their corresponding equilibrium mean–variance value functions. We further show that the pure strategy of solely investing in a risk-free bond can sometimes simultaneously dominate both constrained and unconstrained equilibrium strategies. With numerical experiments, we also illustrate that the constrained strategy can dominate the unconstrained one for most of the commencement dates (even more than 90%) of a prescribed planning horizon. Under a precommitment approach, the value function of an investor increases with the size of the admissible sets of strategies. However, this may fail to be true under the game-theoretic paradigm, as the constraint of time-consistency itself affects the value function differently when short-selling is and is not prohibited.

Time-Inconsistent Mean-Field Stochastic LQ Problem: Open-Loop Time-Consistent Control

This paper is concerned with the open-loop time-consistent solution of time-inconsistent mean-field stochastic linear-quadratic (LQ) optimal control. Different from standard stochastic linear-quadratic problems, both the system matrices and the weighting matrices are depending on the initial times, and the conditional expectations of the control and state enter quadratically into the cost functional. Such features will ruin Bellman's principle of optimality and result in the time inconsistency of optimal control. Based on the dynamical nature of the systems involved, a kind of open-loop time-consistent equilibrium control is investigated in this paper. It is shown that the existence of open-loop equilibrium control for a fixed initial pair is equivalent to the solvability of a set of forward–backward stochastic difference equations with stationary condition and convexity condition. By decoupling the forward–backward stochastic difference equations, necessary and sufficient conditions in terms of linear difference equations and generalized difference Riccati equations are given for the existence of open-loop equilibrium control for a fixed initial pair. Moreover, the existence of open-loop time-consistent equilibrium controls for all the initial pairs is shown to be equivalent to the solvability of a set of coupled constrained generalized difference Riccati equations and two sets of constrained linear difference equations.

Exponential utility maximization for an insurer with time-inconsistent preferences

This paper studies the optimal consumption–investment–reinsurance problem for an insurer with a general discount function and exponential utility function in a non-Markovian model. The appreciation rate and volatility of the stock, the premium rate and volatility of the risk process of the insurer are assumed to be adapted stochastic processes, while the interest rate is assumed to be deterministic. The object is to maximize the utility of intertemporal consumption and terminal wealth. By the method of multi-person differential game, we show that the time-consistent equilibrium strategy and the corresponding equilibrium value function can be characterized by the unique solutions of a BSDE and an integral equation. Under appropriate conditions, we show that this integral equation admits a unique solution. Furthermore, we compare the time-consistent equilibrium strategies with the optimal strategy for exponential discount function, and with the strategies for naive insurers in two special cases.