Time-consistent Solution Research Articles

On the Time Consistent Solution to Optimal Stopping Problems with Expectation Constraint

We study the (weak) equilibrium problem arising from the problem of optimally stopping a one-dimensional diffusion subject to an expectation constraint on the time until stopping. The weak equilibrium problem is realized with a set of randomized but purely state dependent stopping times as admissible strategies. We derive a verification theorem and necessary conditions for equilibria, which together basically characterize all equilibria. Furthermore, additional structural properties of equilibria are obtained to feed a possible guess-and-verify approach, which is then illustrated by an example.

Strong Time-Consistent Solution for Cooperative Differential Games with Network Structure

One class of cooperative differential games on networks is considered. It is assumed that interaction on the network is possible not only between neighboring players, but also between players connected by paths. Various cooperative optimality principles and their properties for such games are investigated. The construction of the characteristic function is proposed, taking into account the network structure of the game and the ability of players to cut off connections. The conditions under which a strong time-consistent subcore is not empty are studied. The formula for explicit calculation of the Shapley value is derived. The results are illustrated by the example of one differential marketing game.

Continuous-time mean-variance portfolio selection with regime-switching financial market: Time-consistent solution

Abstract This paper studies optimal time-consistent strategies for the mean-variance portfolio selection problem. Especially, we assume that the price processes of risky stocks are described by regime-switching SDEs. We consider a Markov-modulated state-dependent risk aversion and we formulate the problem in the game theoretic framework. Then, by solving a flow of forward-backward stochastic differential equations, an explicit representation as well as uniqueness results of an equilibrium solution are obtained.

Hyperbolic Discounting and the Time-Consistent Solution of Three Canonical Environmental Problems

In this paper I propose a time-consistent method of discounting hyperbolically that contains the discount rate implied by Gamma discounting as a special case. I apply the discounting method to three canonical environmental problems: (i) optimal renewable resource use, (ii) the tragedy of the commons, (iii) economic growth and pollution. I then compare results with those for conventional exponential discounting using the normalization that both methods provide the same present value of an infinite constant flow. I show that, irrespective of potentially high initial discount rates, time-consistent hyperbolic discounting leads always to a steady state of maximum yield, or, if the environment enters the utility function, a steady state where the Green Golden Rule applies. While (asymptotic) extinction is a real threat under exponential discounting it is impossible under time-consistent hyperbolic discounting. This result is also confirmed for open access resources. In a model of economic growth and pollution, hyperbolic discounting establishes the Golden Rule of capital accumulation and the Modified Green Golden Rule.

OPTIMAL MEAN–VARIANCE PORTFOLIO SELECTION WITH NO-SHORT-SELLING CONSTRAINT

In this paper, the objective is to study the continuous mean–variance portfolio selection with a no-short-selling constraint and obtain a time-consistent solution. We assume that there is a self-financing portfolio with wealth process [Formula: see text], in which [Formula: see text] represents the fraction of wealth invested in the risk asset under the short selling prohibition. We investigate the mean–variance optimal constrained problem defined by obtaining the supremum over all admissible controls of the difference between the expectation of the value process at some designated terminal time [Formula: see text] and a positive constant times the variance of [Formula: see text]. To envisage the quadratic nonlinearity introduced by the variance, the method of Lagrangian multipliers reduces the nonlinear problem into a set of linear problems which can be solved by applying the Hamilton–Jacobi–Bellman equation and change of variables formula with local time on curves. Solving the HJB system provides the time-inconsistent solution and from there, we derive the time-consistent optimal control.

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.

Optimal asset allocation with heterogeneous discounting and stochastic income under CEV model

This article focuses on an optimal asset allocation problem with heterogeneous discounting and stochastic income under the constant elasticity of variance (CEV) model. Heterogeneous discounting is an important non-constant discounting model, which can describe the fact that a decision maker discounts in different ways the utility derived from consumption and that of the bequest or final function. It is consistent with the fact that the concern of a decision maker about the bequest left to her descendants when she is young is not the same as that when she is old. In our model, a decision maker with stochastic income can enjoy the consumption, purchase life insurance, and invest her wealth in a risk-free asset and a risky asset whose price process satisfies the CEV model. Meanwhile, the volatility of the stochastic income arises from the risky asset. Since the problem is time-inconsistent, the Bellman’s principle of optimality does not hold. To obtain the time-consistent solution, an equilibrium strategy is calculated. By applying the game theoretic framework and solving an extended Hamilton-Jacobi-Bellman system, we derive the time-consistent consumption, investment, and life insurance strategies for both exponential and logarithmic utility functions. Finally, we provide numerical simulations to illustrate our results.

Time-consistent proportional reinsurance and investment strategies under ambiguous environment

In this paper, we study the equilibrium proportional reinsurance and investment strategies for an insurer in an environment with parameter uncertainties. The insurer can buy proportional reinsurance business to hedge its insurance risks. However, the insurer is ambiguous about the insurance claims and risky assets. Specifically, the insurance claim is exponentially distributed and the rate parameter is uncertain. Besides, the return of a stock is uncertain. The insurer holds ambiguous beliefs over these states. The goal of the insurer is to maximize the smooth ambiguity utility proposed in Klibanoff et al. (2005). The equilibrium control is introduced to derive the time-consistent solution. In the end, a sensitivity analysis is presented to show the economic behaviors of the insurer under the smooth ambiguity. Results reveal that the uncertain beliefs play an important role in the equilibrium reinsurance and investment strategies. When the insurer is more risk averse towards ambiguity, the insurer will invest less in the ambiguous asset and more in the non-ambiguous asset.

Time-Consistent Strategies for Multi-Period Portfolio Optimization with/without the Risk-Free Asset

The pre-commitment and time-consistent strategies are the two most representative investment strategies for the classic multi-period mean-variance portfolio selection problem. In this paper, we revisit the case in which there exists one risk-free asset in the market and prove that the time-consistent solution is equivalent to the optimal open-loop solution for the classic multi-period mean-variance model. Then, we further derive the explicit time-consistent solution for the classic multi-period mean-variance model only with risky assets, by constructing a novel Lagrange function and using backward induction. Also, we prove that the Sharpe ratio with both risky and risk-free assets strictly dominates that of only with risky assets under the time-consistent strategy setting. After the theoretical investigation, we perform extensive numerical simulations and out-of-sample tests to compare the performance of pre-commitment and time-consistent strategies. The empirical studies shed light on the important question: what is the primary motivation of using the time-consistent investment strategy.

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.

Monetary Policy with Declining Deficits: Theory and an Application to Recent Argentine Monetary Policy

The authors study the nature of the optimal monetary policy in a regime of fiscal dominance when the monetary authority—which can print money or issue interest-earning debt—is required to finance an exogenous sequence of transfers to the Treasury. They show that the degree of commitment on the part of the monetary authority has a significant impact on the details of the optimal policy. They apply this model to the recent experience of Argentina and find that the inflation rate experienced by Argentina during the first year of the monetary program is close to the predictions of a weakly time-consistent solution. Moreover, consistent with both versions of their model—full commitment and weak time consistency—the Central Bank of Argentina has increased the ratio of interest-earning debt relative to the monetary base.

Discrete-Time Stochastic Linear-Quadratic Optimal Control with Time-Inconsistency

In this paper, the time-consistent solution of a discrete-time time-inconsistent stochastic linear-quadratic optimal control is investigated. Different from existing literature, the definiteness constraint is not posed on the state weight matrices and the control weight matrices of the cost functional. Necessary and suffcient conditions are obtained to the existence of the open-loop time-consistent equilibrium control, which contain the solvability of certain forward-backward stochastic difference equation systems, the stationary conditions and the convexity conditions. Under additional conditions, the closed-form of the open-loop equilibrium control is characterized via the solutions of systems of certain generalized difference Riccati equations. Interestingly, the system of generalized difference Riccati equations do not admit symmetry structure. Finally, for a special case of the considered problem, the existence of the open-loop equilibrium control for all the initial pairs is shown to be equivalent to the solvability of certain generalized difference Riccati equation.

An analysis of eurobonds

We analyse different forms of debt mutualisation in a union of countries. One country suffers from a political distortion and may resort to (partial) debt default. We consider a debt repayment guarantee, which can be “unlimited” or ”limited”, i.e. only be invoked when the guarantee threshold is not exceeded. We also explore the ”blue–red” bonds proposal, under which blue debt is guaranteed, while red debt is not guaranteed. Only a suitably chosen limited guarantee induces the government to reduce debt and raises union welfare. This result is upheld under the time-consistent solution when there are costs to the rest of the union of not providing financial rescue. Making the guarantee also conditional on sufficient structural reform may in addition stimulate reform effort, thereby raising union welfare.

Consumption, investment and life insurance strategies with heterogeneous discounting

In this paper we analyze how the optimal consumption, investment and life insurance rules are modified by the introduction of a class of time-inconsistent preferences. In particular, we account for the fact that an agent’s preferences evolve along the planning horizon according to her increasing concern about the bequest left to her descendants and about her welfare at retirement. To this end, we consider a stochastic continuous time model with random terminal time for an agent with a known distribution of lifetime under heterogeneous discounting. In order to obtain the time-consistent solution, we solve a non-standard dynamic programming equation. For the case of CRRA and CARA utility functions we compare the explicit solutions for the time-inconsistent and the time-consistent agent. The results are illustrated numerically.

Markowitz’s mean–variance asset–liability management with regime switching: A time-consistent approach

In this article, we provide the first study in the time consistent solution of the mean–variance asset–liability management (MVALM). The framework is even considered under a continuous time Markov regime-switching setting. Using the extended Hamilton–Jacobi–Bellman equation (HJB) (see Björk and Murgoci (2010)), we show that the time consistent equilibrium control is state dependent in the sense that it depends on the uncontrollable liability process, which is in substantial contrast with the time consistent solution of the similar problem in Björk and Murgoci (2010), in which it is independent of the state. Finally, we give a numerical comparison between our work with the corrected version (as obtained here) of pre-commitment strategy in Chen et al. (2008).

Transformation of Characteristic Function in Dynamic Games

AbstractThe problem of transformation of characteristic function, using a transformation matrix and connected with associated consistency of a solution concept, was investigated for the class of so-called “one-shot” games. The same problem also arises in dynamic games when players move step by step along the cooperative trajectory. In this case the evolution of characteristic function is, in general, unpredictable and may lead to time-inconsistency of a fixed cooperative solution concept. There are different approaches to overcome this problem based on transformation of characteristic function. In this paper, the step-wise transformation is used, which is a generalization of similar transformation of characteristic function for dynamic games with perfect information, and it leads to the time-consistent solution. The general form of such transformation is proposed.

A Cooperative Differential Game Model for Multiuser Rate-Based Flow Control

Flow control is one typical resource management tool. Its objectives are to adjust the traffic in the network, and resolve data traffic congestion, maximize the utilization of available bandwidth resource and realize fairness among different sources. Taking more consideration on the interaction and the dynamic characteristics of the flow, in this paper, we propose a cooperative differential game of flow control. Our goal is to assign sources with appropriate transmission rate levels so as to the queue length in the bottleneck link is optimal for the network performance and their benefits are maximal. We derive a payment distribution mechanism that would constitute a time-consistent solution and guarantee individual rationality.

Consumption, Investment and Life Insurance Strategies with Heterogeneous Discounting

In this paper we analyze how the optimal consumption, investment and life insurance rules are modified by the introduction of a class of time-inconsistent preferences. In particular, we account for the fact that an agent's preferences evolve along the planning horizon according to her increasing concern about the bequest left to her descendants and about her welfare at retirement. To this end, we consider a stochastic continuous time model with random terminal time for an agent with a known distribution of lifetime under heterogeneous discounting. In order to obtain the time-consistent solution, we solve a non-standard dynamic programming equation. For the case of CRRA and CARA utility functions we compare the explicit solutions for the time-inconsistent and the time-consistent agent. The results are illustrated numerically.

On the Theory of Continuous-Time Recursive Utility

We establish a connection between continuous-time recursive utility and the notion of consistency studied, in particular, in connection with the non-linear objective mean-variance. We propose a time-global optimization problem and show that the optimal time-consistent solution to this time-global problem is closely related to the standard recursive utility solution which is defined via differential equations. The two approaches are, to some extent, equivalent. Standard continuous-time recursive utility was developed for Brownain markets, and a generalization is complicated. Our approach contributes with insight in the notion of recursive utility and lightens up a relatively simple path that potentially gives access to results for general Markovian markets.

Consumption and Portfolio Rules with Stochastic Quasi-Hyperbolic Discounting

We investigate the joint consumption-saving and portfolio-selection problem under capital risk, assuming sophisticated but time-inconsistent agents. We introduce stochastic hyperbolic preferences as specified in Harris and Laibson (2008) and find closed-form solutions for the classic Merton (1969, 1971) optimal consumption and portfolio selection problem in continuous time. The portfolio rule remains identical to the time-consistent solution with power utility with no borrowing constraints. However, the marginal propensity to consume out of wealth is unambiguously greater than the time-consistent, exponential case.