Time-consistent investment strategy under partial information

This paper considers a mean–variance portfolio selection problem under partial information, that is, the investor can observe the risky asset price with random drift which is not directly observable in financial markets. Since the dynamic mean–variance portfolio selection problem is time inconsistent, to seek the time-consistent investment strategy, the optimization problem is formulated and tackled in a game theoretic framework. Closed-form expressions of the equilibrium investment strategy and the corresponding equilibrium value function under partial information are derived by solving an extended Hamilton–Jacobi–Bellman system of equations. In addition, the results are also given under complete information, which are need for the partial information case. Furthermore, some numerical examples are presented to illustrate the derived equilibrium investment strategies and numerical sensitivity analysis is provided.     

A characterization of equilibrium strategies in continuous-time mean–variance problems for insurers

In this work, we study the equilibrium reinsurance/new business and investment strategy for mean–variance insurers with constant risk aversion. The insurers are allowed to purchase proportional reinsurance, acquire new business and invest in a financial market, where the surplus of the insurers is assumed to follow a jump–diffusion model and the financial market consists of one riskless asset and a multiple risky assets whose price processes are driven by Poisson random measures and independent Brownian motions. By using a version of the stochastic maximum principle approach, we characterize the open loop equilibrium strategies via a stochastic system which consists of a flow of forward–backward stochastic differential equations (FBSDEs in short) and an equilibrium condition. Then by decoupling the flow of FSBDEs, an explicit representation of an equilibrium solution is derived as well as its corresponding objective function value.     

Optimal choice of dividend barriers for a risk process with stochastic return on investments

We consider a risk process with stochastic return on investments and we are interested in expected present value of all dividends paid until ruin occurs when the company uses a simple barrier strategy, i.e. when it pays dividends whenever its surplus reaches a level b. It is shown that given the barrier b, this expected value can be found by solving a boundary value problem for an integro-differential equation. The solution is then found in two special cases; when return on investments is constant and the surplus generating process is compound Poisson with exponentially distributed claims, and also when both return on investments as well as the surplus generating process are Brownian motions with drift. Also in this latter case we are able to find the optimal barrier b^*, i.e. the barrier that gives the highest expected present value of dividends. Parallell with this we treat the problem of finding the Laplace transform of the distribution of the time to ruin when a barrier strategy is employed, noting that the probability of eventual ruin is 1 in this case. The paper ends with a short discussion of the same problems when a time dependent barrier is employed.     

Consumption and portfolio rules for time-inconsistent investors

This paper extends the classical consumption and portfolio rules model in continuous time [Merton, R.C., 1969. Lifetime portfolio selection under uncertainty: The continuous time case. Review of Economics and Statistics 51, 247–257, Merton, R.C., 1971. Optimum consumption and portfolio rules in a continuous time model. Journal of Economic Theory 3, 373–413] to the framework of decision-makers with time-inconsistent preferences. The model is solved for different utility functions for both, naive and sophisticated agents, and the results are compared. In order to solve the problem for sophisticated agents, we derive a modified HJB (Hamilton–Jacobi–Bellman) equation. It is illustrated how for CRRA functions within the family of HARA functions (logarithmic and power utilities) the optimal portfolio rule does not depend on the discount rate, but this is not the case for a general utility function, such as the exponential (CARA) utility function.     

Consumption–investment strategies with non-exponential discounting and logarithmic utility

In this paper, we revisit the consumption–investment problem with a general discount function and a logarithmic utility function in a non-Markovian framework. The coefficients in our model, including the interest rate, appreciation rate and volatility of the stock, are assumed to be adapted stochastic processes. Following Yong (2012a,b)’s method, we study an N-person differential game. We adopt a martingale method to solve an optimization problem of each player and characterize their optimal strategies and value functions in terms of the unique solutions of BSDEs. Then by taking limit, we show that a time-consistent equilibrium consumption–investment strategy of the original problem consists of a deterministic function and the ratio of the market price of risk to the volatility, and the corresponding equilibrium value function can be characterized by the unique solution of a family of BSDEs parameterized by a time variable.     

Dividend problems in the dual model with diffusion and exponentially distributed observation time

Consider dividend problems in the dual model with diffusion and exponentially distributed observation time where dividends are paid according to a barrier strategy. Assume that dividends can only be paid with a certain probability at each point of time, that is, on each observation, if the surplus exceeds the barrier, the excess is paid as dividend. In this paper, integro-differential equations for the expected discounted sum of dividends paid until ruin and the Laplace transform of ruin time are derived. When the gains are exponentially distributed, explicit expressions for the ruin probability, the expected discounted sum of dividends paid until ruin, the Laplace transform of ruin time and the expectation of ruin time are also obtained.     

TIME INCONSISTENCY AND REPUTATION IN MONETARY POLICY： A STRATEGIC MODELLING IN CONTINUOUS TIME

This article develops a model to examine the equilibrium behavior of the time inconsistency problem in a continuous time economy with stochastic and endogenized distortion. First, the authors introduce the notion of sequentially rational equilibrium, and show that the time inconsistency problem may be solved with trigger reputation strategies for stochastic setting. The conditions for the existence of sequentially rational equilibrium are provided. Then, the concept of sequentially rational stochastically stable equilibrium is introduced. The authors compare the relative stability between the cooperative behavior and uncooperative behavior, and show that the cooperative equilibrium in this monetary policy game is a sequentially rational stochastically stable equilibrium and the uncooperative equilibrium is sequentially rational stochastically unstable equilibrium. In the long run, the zero inflation monetary policies are inherently more stable than the discretion rules, and once established, they tend to persist for longer periods of the time.     

TIME INCONSISTENCY AND REPUTATION IN MONETARY POLICY: A STRATEGIC MODELLING IN CONTINUOUS TIME

This article develops a model to examine the equilibrium behavior of the time inconsistency problem in a continuous time economy with stochastic and endogenized dis-tortion. First, the authors introduce the notion of sequentially rational equilibrium, and show that the time inconsistency problem may be solved with trigger reputation strategies for stochastic setting. The conditions for the existence of sequentially rational equilibrium are provided. Then, the concept of sequentially rational stochastically stable equilibrium is introduced. The authors compare the relative stability between the cooperative behavior and uncooperative behavior, and show that the cooperative equilibrium in this monetary policy game is a sequentially rational stochastically stable equilibrium and the uncooper-ative equilibrium is sequentially rational stochastically unstable equilibrium. In the long run, the zero inflation monetary policies are inherently more stable than the discretion rules, and once established, they tend to persist for longer periods of the time.     

The perturbed compound Poisson risk model with linear dividend barrier

In this paper, we consider a diffusion perturbed classical compound Poisson risk model in the presence of a linear dividend barrier. Partial integro-differential equations for the moment generating function and the nth moment of the present value of all dividends until ruin are derived. Moreover, explicit solutions for the nth moment of the present value of dividend payments are obtained when the individual claim size distribution is exponential. We also provided some numerical examples to illustrate the applications of the explicit solutions. Finally we derive partial integro-differential equations with boundary conditions for the Gerber-Shiu function.     

Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier

Based on the matrix-analytic approach to fluid flows initiated by Ramaswami, we develop an efficient time dependent analysis for a general Markov modulated fluid flow model with a finite buffer and an arbitrary initial fluid level at time 0. We also apply this to an insurance risk model with a dividend barrier and a general Markovian arrival process of claims with possible dependencies in successive inter-claim intervals and in claim sizes. We demonstrate the implementability and accuracy of our algorithms through a set of numerical examples that could also serve as test cases for comparing other solution approaches.      

An anticipative hyperbolic discount utility on intertemporal decision making

Discount utility is facilitated to scrutinize human decision behaviors for different points in time under the consideration of time preference. Recent studies are extended from two basic models, which are the exponential and the hyperbolic discount models, and the more popular hyperbolic discount model is well recognized for resolving the effects of preference reversal and is less steeply discounting than the exponential discount model. However, to enhance the capability of explaining intertemporal decision making behavior, an anticipative hyperbolic discount utility model was proposed which revised the conventional hyperbolic discount utility by introducing the anticipative parameters under the consideration of anticipation of future gains or losses. An empirical investigation was employed to validate the effectiveness of the proposed anticipative hyperbolic discounting utility model, which was able to empirically go beyond the traditional hyperbolic discounting utility model. Also, the proposed anticipative hyperbolic model is capable of dealing with the anomalies of preference reverse and framing effects.     

An equilibrium efficiency frontier data envelopment analysis approach for evaluating decision-making units with fixed-sum outputs

Based on the minimal reduction strategy, Yang et al. (2011) developed a fixed-sum output data envelopment analysis (FSODEA) approach to evaluate the performance of decision-making units (DMUs) with fixed-sum outputs. However, in terms of such a strategy, all DMUs compete over fixed-sum outputs with “no memory” that will result in differing efficient frontiers’ evaluations. To address the problem, in this study, we propose an equilibrium efficiency frontier data envelopment analysis (EEFDEA) approach, by which all DMUs with fixed-sum outputs can be evaluated based on a common platform (or equilibrium efficient frontier). The proposed approach can be divided into two stages. Stage 1 constructs a common evaluation platform via two strategies: an extended minimal adjustment strategy and an equilibrium competition strategy. The former ensures that original efficient DMUs are still efficient, guaranteeing the existence of a common evaluation platform. The latter makes all DMUs achieve a common equilibrium efficient frontier. Then, based on the common equilibrium efficient frontier, Stage 2 evaluates all DMUs with their original inputs and outputs. Finally, we illustrate the proposed approach by using two numerical examples.