Subgame Perfect Nash Equilibrium Strategy Research Articles

Optimal mean-variance reinsurance with delay and multiple\\ classes of dependent risks

In this paper, we consider an optimal proportional reinsurance problem for the compound Poisson risk model with delay and multiple dependent classes of insurance business. Under the criterion of maximizing the mean-variance utility of the terminal wealth, we formulate the time-inconsistent problem within a game theoretic framework and look for a subgame perfect Nash equilibrium strategy. Based on the technique of stochastic control theory and the corresponding extended Hamilton-Jacobi-Bellman equation, the explicit expressions of the optimal equilibrium strategy and the value function are derived for the case of insurance business $n=2$. Meanwhile, we derive the closed-form expressions of optimal solutions for the case of $n=3$ by the method of dimension reduction which can be used to get the optimal results for the case of $n>3$. Finally, some numerical examples are presented to show the impact of model parameters on the optimal strategies.

Time-consistent mean-variance reinsurance-investment in a jump-diffusion financial market

This paper study an optimal time-consistent reinsurance-investment strategy selection problem in a financial market with jump-diffusion risky asset, where the insurance risk model is modulated by a compound Poisson process. The aggregate claim process and the price process of risky asset are correlated by a common Poisson process. The objective of the insurer is to choose an optimal time-consistent reinsurance-investment strategy so as to maximize the expected terminal surplus while minimizing the variance of the terminal surplus. Since this problem is time-inconsistent, we attack it by placing the problem within a game theoretic framework and looking for subgame perfect Nash equilibrium strategy. We investigate the problem using the extended Hamilton–Jacobi–Bellman dynamic programming approach. Closed-form solutions for the optimal reinsurance-investment strategy and the corresponding value functions are obtained. Numerical examples and economic significance analysis are also provided to illustrate how the optimal reinsurance-investment strategy changes when some model parameters vary.

Portfolio optimization for jump‐diffusion risky assets with common shock dependence and state dependent risk aversion

SummaryAn optimal portfolio problem with one risk‐free asset and two jump‐diffusion risky assets is studied in this paper, where the two risky asset price processes are correlated through a common shock. Under the criterion of maximizing the mean‐variance utility of the terminal wealth with state dependent risk aversion, we formulate the time‐inconsistent problem within a game theoretic framework and look for subgame perfect Nash equilibrium strategy. Based on the technique of stochastic control theory and the corresponding extended Hamilton–Jacobi–Bellman equation, the closed‐form expressions of the optimal equilibrium strategy and value function are derived. Furthermore, we find that the optimal strategy, that is, the amount of money invested into the risky asset, is proportional to current wealth. Finally, some numerical examples are presented to show the impact of model parameters on the optimal results. Copyright © 2016 John Wiley & Sons, Ltd.

Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion

This paper studies the portfolio optimization of mean-variance utility with state-dependent risk aversion, where the stock asset is driven by a stochastic process. The sub-game perfect Nash equilibrium strategies and the extended Hamilton-Jacobi-Bellman equations have been used to derive the system of non-linear partial differential equations. From the economic point of view, we demonstrate the numerical evaluation of the suggested solution for a special case where the risk aversion rate is proportional to the wealth value. Our results show that the asset driven by the stochastic volatility process is more general and reasonable than the process with a constant volatility.

Spectrum Bargaining: A Model for Competitive Sharing of Unlicensed Radio Spectrum

In this paper, we address the problem of dynamic channel access by a set of cognitive radio-enabled nodes, where each node acting in a selfish manner tries to access and use as many channels as possible, subject to interference constraints. We model the dynamic channel access problem as a modified Rubinstein–Stahl bargaining game. In our model, each node (player) negotiates with the other nodes in the network reduce in a distributed manner to obtain an agreeable sharing rule of the available channels such that no two interfering nodes use the same channel. We solve the bargaining game by finding subgame perfect Nash equilibrium (SPNE) strategies of the game. First, we consider finite horizon version of the bargaining game and investigate its SPNE strategies, which allow each node to maximize its utility against the other nodes (opponents). We then extend these results to the infinite horizon bargaining game. Furthermore, we identify Pareto optimal equilibria of the game, which help enhance network throughput. We conduct simulations to study how the “self-gain” maximizing strategy of the players impact systemwide performance.

MEAN–VARIANCE PORTFOLIO OPTIMIZATION WITH STATE‐DEPENDENT RISK AVERSION

The objective of this paper is to study the mean–variance portfolio optimization in continuous time. Since this problem is time inconsistent we attack it by placing the problem within a game theoretic framework and look for subgame perfect Nash equilibrium strategies. This particular problem has already been studied in Basak and Chabakauri where the authors assumed a constant risk aversion parameter. This assumption leads to an equilibrium control where the dollar amount invested in the risky asset is independent of current wealth, and we argue that this result is unrealistic from an economic point of view. In order to have a more realistic model we instead study the case when the risk aversion depends dynamically on current wealth. This is a substantially more complicated problem than the one with constant risk aversion but, using the general theory of time‐inconsistent control developed in Björk and Murgoci, we provide a fairly detailed analysis on the general case. In particular, when the risk aversion is inversely proportional to wealth, we provide an analytical solution where the equilibrium dollar amount invested in the risky asset is proportional to current wealth. The equilibrium for this model thus appears more reasonable than the one for the model with constant risk aversion.