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We study optimal reinsurance for an insurer and two reinsurers in the market through stochastic game theory. The relationship between the insurer and reinsurers is described by a Stackelberg model, where reinsurers, as market leaders, set prices for reinsurance treaties, and the insurer, as a price taker, determines reinsurance demand. Furthermore, we employ a Nash game to model the price competition between the two reinsurers who adopt different premium principles: the variance premium principle and the expected value premium principle. Both the insurer and reinsurers aim to maximize their respective mean-variance cost functions, leading to a time inconsistency control problem. This issue is resolved using a corresponding extended Hamilton-Jacobi-Bellman equation in the game-theoretic framework. We find that the insurer will adopt propositional and excess-of-loss reinsurance strategies with two reinsurers, respectively. Moreover, under an exponential claim size distribution, there exists a unique equilibrium reinsurance premium strategy. Our numerical analysis illuminates the effects of claim size, risk aversion, and the interest rates of the insurer and reinsurers on the equilibrium reinsurance and premium strategies, enhancing the understanding of competition in the reinsurance market. [ABSTRACT FROM AUTHOR]

We develop a model to analyze policymakers' incentives to install policy rules, comparing the case of no rule with a binding and a contingent policy rule that allows policymakers to suspend the rule in response to a sufficiently large shock. First, abstracting from political polarization, we show that the choice of the policy rule depends on policymakers' policy targets. Depending on the policy target, there is an unambiguous ranking going from a no‐rule regime to a contingent rule to a binding rule. Next, allowing for political polarization, the incentive to install the different types of rules changes with political polarization between different policymakers and their probability of being elected into office. Increasing political polarization when there is a sufficiently high election probability for policymakers with a high policy target increases the preference for more binding policy rules. It also leads to stricter rules in a contingent rule regime. [ABSTRACT FROM AUTHOR]

We study a mean-field zero-sum Dynkin game (MF-ZSDG) with time-inconsistent performance functionals adapted to the Brownian filtration. Despite the time-inconsistency of the MF-ZSDG, we show that it admits a value and that the pair of first times the value process hits the upper and lower obstacles, respectively, is a saddle point for the game. We solve the problem by approximating the associated lower and upper value processes with a sequence of value processes of interacting time-consistent zero-sum Dynkin games for which the saddle point of each of the value processes is the pair of first times each of those value processes hits the associated upper and lower obstacles, respectively. Under mild assumptions, we show that this sequence of saddle points converges in probability to the pair of first hitting times of the value process of the upper and lower obstacles, respectively, and that the limit is a saddle point for the time-inconsistent MF-ZSDG. [ABSTRACT FROM AUTHOR]

This paper investigates the moral hazard problem in finite horizon with both continuous and lump‐sum payments, involving a time‐inconsistent sophisticated agent and a standard utility maximizer principal: Building upon the so‐called dynamic programming approach in Cvitanić et al. (2018) and the recently available results in Hernández and Possamaï (2023), we present a methodology that covers the previous contracting problem. Our main contribution consists of a characterization of the moral hazard problem faced by the principal. In particular, it shows that under relatively mild technical conditions on the data of the problem, the supremum of the principal's expected utility over a smaller restricted family of contracts is equal to the supremum over all feasible contracts. Nevertheless, this characterization yields, as far as we know, a novel class of control problems that involve the control of a forward Volterra equation via Volterra‐type controls, and infinite‐dimensional stochastic target constraints. Despite the inherent challenges associated with such a problem, we study the solution under three different specifications of utility functions for both the agent and the principal, and draw qualitative implications from the form of the optimal contract. The general case remains the subject of future research. We illustrate some of our results in the context of a project selection contracting problem between an investor and a time‐inconsistent manager. [ABSTRACT FROM AUTHOR]

This article examines the relationship between agents' behavioral attributes (or time preferences) and the problem of obesity and, more generally, problems of both overnutrition and undernutrition. Through a primary survey in western Delhi data were gathered on participants' food choices and body mass index. Time preferences, as posited by the (β,δ${{\beta}},{{\delta}}$) quasi‐hyperbolic discounting model, were elicited using an incentivized, choice‐based experiment. Estimating a simultaneous two‐equation model, the article finds that individuals in the sample with lower β and/or lower δ (or higher time preferences) make unhealthy food choices, which, in turn, significantly increases their BMI. In addition, a supplementary empirical exercise analyzes a large, secondary unit record data set, with savings as a proxy for time preferences, to provide evidence that these behavioral attributes also explain the problem of underweight (germane in developing countries). [ABSTRACT FROM AUTHOR]

This paper considers time‐inconsistent problems when control and stopping strategies are required to be made simultaneously (called stopping control problems by us). We first formulate the time‐inconsistent stopping control problems under general multidimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair (û,C)$(\hat{u},C)$ of control‐stopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to verify or exclude equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a nonexponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general nonconstant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two‐dimensional example with a hyperbolic discount. [ABSTRACT FROM AUTHOR]

We solve the problem of an investor who maximizes utility but faces random preferences. We propose a problem formulation based on expected certainty equivalents. We tackle the time‐consistency issues arising from that formulation by applying the equilibrium theory approach. To this end, we provide the proper definitions and prove a rigorous verification theorem. We complete the calculations for the cases of power and exponential utility. For power utility, we illustrate in a numerical example that the equilibrium stock proportion is independent of wealth, but decreasing in time, which we also supplement by a theoretical discussion. For exponential utility, the usual constant absolute risk aversion is replaced by its expectation. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a time-inconsistent linear-quadratic (LQ) mean-field game problem for large-population systems. By the Nash certainty equivalence methodology, we construct an auxiliary problem consisting of time-inconsistent LQ control problems for every agent, respectively. An equilibrium, instead of optimal solution, is presented explicitly in the form of linear feedback for each agent in this auxiliary problem. Inspired by the framework of tackling time-inconsistency for control problems, we generalize the definition of ϵ -Nash equilibrium for large-population games to the time-inconsistent case. Then, the set of linear feedback controls obtained above can be proved to be an ϵ -Nash equilibrium for our original problem. Moreover, we solve a resource investment problem and provide a numerical simulation to illustrate the application of our theoretic results. [ABSTRACT FROM AUTHOR]

We empirically test the hypothesis that, for married individuals with time-inconsistent preferences, leaving the management of family budgets to their spouse functions as a commitment device for savings. We conduct a cross-sectional analysis based on the micro-data of married Japanese couples obtained from waves 2009 and 2010 of Osaka University's Preference Parameters Study. Our results show that when wives have time-inconsistent preferences, their household savings are more likely to go as planned if they leave household budgets to their husbands in comparison to when they make these decisions themselves. However, time-inconsistent individuals do not actively use leaving the management of family budgets to their spouse as a commitment device for savings. Considering these, we conclude that individuals do not recognize the benefits of leaving the management of family budgets to their spouse, but it can work as a commitment device. [ABSTRACT FROM AUTHOR]

Yong J [Acta Math. Appl. Sin. Engl. Ser. 28 (2012), 1–30] [Math. Control Relat. Fields 1 (2011), 83–118] studied a standard linear quadratic time-inconsistent optimal control problem via a cooperative and non-cooperative approach, respectively. The authors extend his results to a singular case. To handle the singularity, the authors prove the solvability of a generalized Riccati equation, and introduce a notion of M P of matrix. It is shown that the authors can obtain a family of parameter equilibrium controls in both cases. Another interesting outcome is that a new type of parameter forward-backward Volterra integral equations is derived. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the open-loop equilibrium strategy for a mean-variance portfolio problem with investment constraints in a non-Markovian regime-switching jump-diffusion model. Specially, the investment strategies are constrained in a closed convex cone and all coefficients in the model are stochastic processes adapted to the filtration generated by a Markov chain. First, we provide a necessary and sufficient condition for an equilibrium strategy, which involves a system of forward and backward stochastic differential equations (FBSDEs, for short). Second, by solving these FBSDEs, we obtain a feedback representation of the equilibrium strategy. Third, we prove a theorem ensuring the almost everywhere uniqueness of the equilibrium solution. Finally, the results are applied to solve an example of the Markovian regime-switching model. [ABSTRACT FROM AUTHOR]

We investigate the stability of equilibrium-induced optimal values with respect to reward functions f and transition kernels Q for time-inconsistent stopping problems under nonexponential discounting in discrete time. First, with locally uniform convergence of f and Q equipped with total variation distance, we show that the optimal value is semicontinuous with respect to (f,Q). We provide examples showing that continuity may fail in general, and the convergence for Q in total variation cannot be replaced by weak convergence. Next we show that, with the uniform convergence of f and Q, the optimal value is continuous with respect to (f,Q) when we consider a relaxed limit over ε-equilibria. We also provide an example showing that for such continuity the uniform convergence of (f,Q) cannot be replaced by locally uniform convergence. [ABSTRACT FROM AUTHOR]

This paper studies open-loop equilibriums for a general class of time-inconsistent stochastic control problems under jump-diffusion SDEs with deterministic coefficients. Inspired by the idea of Four-Step-Scheme for forward-backward stochastic differential equations with jumps (FBSDEJs, for short), we derive two systems of integro-partial differential equations (IPDEs, for short). Then, we rigorously prove a verification theorem which provides a sufficient condition for open-loop equilibrium strategies. As special cases, a mean-variance portfolio selection problem and a time-inconsistent problem under non-exponential discounting are discussed. [ABSTRACT FROM AUTHOR]

We consider a general stochastic linear-quadratic differential game with time-inconsistency. The time-inconsistency arises from the presence of quadratic terms of the expected state as well as state-dependent term in the objective functionals. We define an equilibrium strategy, which is different from the classical one, and derive a sufficient condition for equilibrium strategies via a system of forward-backward stochastic differential equation. When the state is one-dimensional and the coefficients are all deterministic, we find an explicit equilibrium strategy. The uniqueness of such equilibrium strategy is also given.

The paper presents a characterization of equilibrium in a game-theoretic description of discounting conditional stochastic linear-quadratic (LQ for short) optimal control problem, in which the controlled state process evolves according to a multidimensional linear stochastic differential equation, when the noise is driven by a Poisson process and an independent Brownian motion under the effect of a Markovian regime-switching. The running and the terminal costs in the objective functional are explicitly dependent on several quadratic terms of the conditional expectation of the state process as well as on a nonexponential discount function, which create the time-inconsistency of the considered model. Open-loop Nash equilibrium controls are described through some necessary and sufficient equilibrium conditions. A state feedback equilibrium strategy is achieved via certain differential-difference system of ODEs. As an application, we study an investment–consumption and equilibrium reinsurance/new business strategies for mean-variance utility for insurers when the risk aversion is a function of current wealth level. The financial market consists of one riskless asset and one risky asset whose price process is modeled by geometric Lévy processes and the surplus of the insurers is assumed to follow a jump-diffusion model, where the values of parameters change according to continuous-time Markov chain. A numerical example is provided to demonstrate the efficacy of theoretical results.

We consider a general stochastic linear-quadratic differential game with time-inconsistency. The time-inconsistency arises from the presence of quadratic terms of the expected state as well as state-dependent term in the objective functionals. We define an equilibrium strategy, which is different from the classical one, and derive a sufficient condition for equilibrium strategies via a system of forward-backward stochastic differential equation. When the state is one-dimensional and the coefficients are all deterministic, we find an explicit equilibrium strategy. The uniqueness of such equilibrium strategy is also given. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study a kind of time-inconsistent linear-quadratic non-zero sum stochastic differential game problems with random jumps. The time-inconsistency arises from the presence of a quadratic term of the expected state and a state-dependent term, as well as the time-dependent weight of each term in the cost functionals. We define the time-consistent Nash equilibrium point for this kind of problems and establish a general sufficient condition for it through a flow of forward-backward stochastic differential equations with random jumps. In the situation of one-dimensional state and deterministic coefficients, a Nash equilibrium point is given explicitly by some flows of Riccati-like and linear ordinary differential equations. We apply the truncation method to obtain the existence and uniqueness of solutions to them for a specific case. Moreover, an investment and consumption problem is solved and some numerical examples are further provided to illustrate the application of our theoretic results. [ABSTRACT FROM AUTHOR]

A moment constraint that limits the number of dividends in an optimal dividend problem is suggested. This leads to a new type of time-inconsistent stochastic impulse control problem. First, the optimal solution in the precommitment sense is derived. Second, the problem is formulated as an intrapersonal sequential dynamic game in line with Strotz's consistent planning. In particular, the notions of pure dividend strategies and a (strong) subgame-perfect Nash equilibrium are adapted. An equilibrium is derived using a smooth fit condition. The equilibrium is shown to be strong. The uncontrolled state process is a fairly general diffusion. [ABSTRACT FROM AUTHOR]

This paper defines a general framework to study infinitely repeated games with time-dependent discounting in which we distinguish and discuss both time-consistent and -inconsistent preferences. To study the long-term properties of repeated games, we introduce an asymptotic condition to characterize the fact that players become more and more patient; that is, the discount factors at all stages uniformly converge to one. Two types of folk theorems are proven without the public randomization assumption: the asymptotic one, that is, the equilibrium payoff set converges to the feasible and individual rational set as players become patient, and the uniform one, that is, any payoff in the feasible and individual rational set is sustained by a single strategy profile that is an approximate subgame perfect Nash equilibrium in all games with sufficiently patient discount factors. We use two methods for the study of asymptotic folk theorem: the self-generating approach and the constructive proof. We present the constructive proof in the perfect-monitoring case and show that it can be extended to time-inconsistent preferences. The self-generating approach applies to the public-monitoring case but may not extend to time-inconsistent preferences because of a nonmonotonicity result. [ABSTRACT FROM AUTHOR]

The paper presents a characterization of equilibrium in a game-theoretic description of discounting conditional stochastic linear-quadratic (LQ for short) optimal control problem, in which the controlled state process evolves according to a multidimensional linear stochastic differential equation, when the noise is driven by a Poisson process and an independent Brownian motion under the effect of a Markovian regime-switching. The running and the terminal costs in the objective functional are explicitly dependent on several quadratic terms of the conditional expectation of the state process as well as on a nonexponential discount function, which create the time-inconsistency of the considered model. Open-loop Nash equilibrium controls are described through some necessary and sufficient equilibrium conditions. A state feedback equilibrium strategy is achieved via certain differential-difference system of ODEs. As an application, we study an investment-consumption and equilibrium reinsurance/new business strategies for mean-variance utility for insurers when the risk aversion is a function of current wealth level. The financial market consists of one riskless asset and one risky asset whose price process is modeled by geometric Lévy processes and the surplus of the insurers is assumed to follow a jump-diffusion model, where the values of parameters change according to continuous-time Markov chain. A numerical example is provided to demonstrate the efficacy of theoretical results. [ABSTRACT FROM AUTHOR]

We study a dynamic mean–variance portfolio selection problem with return predictability and trading frictions from price impact. Applying mean-field type control theory, we provide a characterisation of an equilibrium trading strategy for an investor facing stochastic investment opportunities. An explicit equilibrium strategy is derived in terms of the solution to a generalised matrix Riccati differential equation, and a sufficient condition is also provided to ensure the latter's well-posedness. Our solution indicates that the investor should trade gradually towards a target portfolio which accounts for return predictability, price impact and time-consistency. Moreover, an asymptotic analysis around small liquidity costs shows that the investor's target portfolio is an equilibrium portfolio without price impact in the first-order sense, and that her first-order approximated value function does not deteriorate significantly for sufficiently small liquidity costs. Finally, our numerical results demonstrate that the target portfolio is more conservative than an equilibrium portfolio without price impact. [ABSTRACT FROM AUTHOR]

Two experiments examined the effect of risk on waiting preference and temporal preference reversal. Participants made binary choices between smaller–sooner (SS) and larger–later (LL) options following a 2 (time: immediate choices vs. non‐immediate choices) x 4 (risk condition) within‐subjects design. Risk conditions varied in whether the SS and/or the LL was risky or certain. Experiment 1 focused on choices with small magnitudes of payoffs and delays, whereas Experiment 2 focused on large magnitudes. Both experiments showed that making one option risky increased preference for the other option, whereas making both options risky had no impact on waiting preference. Pooling risk conditions together, participants in both experiments demonstrated temporal preference reversal. However, participants showed preference reversal in all 4 risk conditions in Experiment 2, but only in the risky (SS)‐certain (LL) and risky‐risky conditions in Experiment 1. This finding indicates for uncertain outcomes, people were at least equally (if not more) likely to reverse their choices, compared to certain outcomes. [ABSTRACT FROM AUTHOR]

This paper considers time-inconsistent stopping problems in which the inconsistency arises from a class of nonexponential discount functions called weighted discount functions. We show that the smooth pasting (SP) principle, the main approach that is used to construct explicit solutions for classical time-consistent optimal stopping problems, may fail under time-inconsistency. Specifically, a mere change of the discount function from exponential to nonexponential (everything else being the same) will cause the SP approach to fail. In general, we prove that SP solves a time-inconsistent problem, within the intrapersonal game theoretic framework with a general nonlinear cost functional and a geometric Brownian motion, if and only if certain inequalities on the model primitives are satisfied. In the special case of a real options problem, we show that while these inequalities hold trivially for the exponential discount function, they may not hold even for very simple nonexponential discount functions. Moreover, we show that the real options problem actually does not admit any equilibrium whenever the SP approach fails. The negative results in this paper caution one from blindly extending the classical approach for time-consistent stopping problems to their time-inconsistent counterparts. [ABSTRACT FROM AUTHOR]

In a continuous-time setting, the existing notion of equilibrium strategies for timeinconsistent problems in the literature, referred to as weak equilibrium, is not fully aligned with the standard definition of equilibrium in game theory in that the agent may be willing to deviate from a given weak equilibrium strategy. To address this issue, [Y.-J. Huang and Z. Zhou, Math. Oper. Res., 46 (2021), pp. 428-451] propose the notion of strong equilibrium for an infinite-time stochastic control problem in which an agent can control the generator of a time-homogeneous, continuous-time, finite-state Markov chain at each time. We study weak and strong equilibria in a general diffusion framework, provide necessary conditions for a strategy to be a strong equilibrium, and prove that strong equilibrium strategies do not exist for three investment and consumption problems. Finally, we propose a new notion of equilibrium strategies, referred to as regular equilibrium, show that it implies weak equilibrium, provide a sufficient condition under which a weak equilibrium strategy becomes a regular equilibrium, and show that this condition holds for many time-inconsistent problems. [ABSTRACT FROM AUTHOR]

We provide a result on prospect theory decision makers who are naïve about the time inconsistency induced by probability weighting. If a market offers a sufficiently rich set of investment strategies, investors postpone their trading decisions indefinitely due to a strong preference for skewness. We conclude that probability weighting in combination with naïveté leads to unrealistic predictions for a wide range of dynamic setups.

Motivated from time-inconsistent stochastic control problems, we introduce a new type of coupled forward–backward stochastic systems, namely, flows of forward–backward stochastic differential equations. They are coupled systems consisting of a single forward SDE and a continuum of BSDEs, which are defined on different time-intervals. We formulate a notion of equilibrium solutions in a general framework and prove small-time well-posedness of the equations. We also consider discretized flows and show that their equilibrium solutions approximate the original one, together with an estimate of the convergence rate. [ABSTRACT FROM AUTHOR]

This paper studies a class of robust mean–variance portfolio selection problems with state-dependent risk aversion. Model uncertainty, in the sense of considering alternative dominated models, is introduced to the problem to reflect the investor's uncertainty-averse preference. To characterise the robust portfolios, we consider closed-loop equilibrium control and spike variation approaches. Moreover, we show that a closed-loop equilibrium strategy exists and is unique under some technical conditions. This partially addresses open problems left in Björk et al. (Finance Stoch. 21:331–360, 2017) and Pun (Automatica 94:249–257, 2018). By using a necessary and sufficient condition for the equilibrium, we manage to derive the analytical form of the equilibrium strategy via the unique solution to a nonlinear ordinary differential equation system. To validate the proposed closed-loop control framework, we show that when there is no uncertainty, our equilibrium strategy is reduced to the strategy in Björk et al. (Math. Finance 24:1–24, 2014), which cannot be deduced under the open-loop control framework. [ABSTRACT FROM AUTHOR]

This article develops a general framework for preference heterogeneity. This framework includes a discount function, a nonstandard Hamilton–Jacobi–Bellman equation (HJB) and a behavioral equation. When controlling parameters, our discount function and its HJB can reduce to those in Marín‐Solano and Patxot (2012) and among many others. In various fields, our framework can find equilibrium path in the coexistence of present bias, preference heterogeneity, and time‐inconsistency. As an example, the present paper quantifies the impact of preference heterogeneity on household financial decision‐making. It has been proven that our nonstandard HJB yields sophisticated solution (i.e., equilibrium path), while our behavioral equation brings about naive solution and precommitted solution. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study a time-inconsistent consumption-investment problem with random endowments in a possibly incomplete market under general discount functions. We provide a necessary condition and a verification theorem for an open-loop equilibrium consumption-investment pair in terms of a coupled forward-backward stochastic differential equation. Moreover, we prove the uniqueness of the open-loop equilibrium pair by showing that the original time-inconsistent problem is equivalent to an associated time-consistent one. [ABSTRACT FROM AUTHOR]

In both the United States of America and the European Union, Member States are encouraged to prevent young people from starting to smoke by forbidding selling tobacco products to people under a certain age. By contrast, there are in general no legal minimum age requirements for consuming those products. Our aim is to address such discrepancy from a theoretical viewpoint by focusing on the case where people have time-inconsistent preferences. Specifically, we build a three-period (youth, adulthood, old age) model of smoking decision in which individual intertemporal preferences are present-biased. Then, using this model, we show that when agents are naive, that is when they are unaware that their intertemporal preferences are time-inconsistent, it may be worthwhile, from the individual viewpoint, to legally prevent young people from smoking. This conclusion does not always hold, because what is good for an agent in youth can be disputable in adult age (and conversely). When individuals are sophisticated, that is, not naive, a legal smoking age (either for buying, consuming or selling tobacco products) is pointless. This conclusion is also reached if one follows the continuing person approach advocated by Sugden. JEL Classification Numbers : I12, I18, K32, D15 [ABSTRACT FROM AUTHOR]

A new definition of continuous-time equilibrium controls is introduced. As opposed to the standard definition, which involves a derivative-type operation, the new definition parallels how a discrete-time equilibrium is defined and allows for unambiguous economic interpretation. The terms "strong equilibria" and "weak equilibria" are coined for controls under the new and standard definitions, respectively. When the state process is a time-homogeneous continuous-time Markov chain, a careful asymptotic analysis gives complete characterizations of weak and strong equilibria. Thanks to the Kakutani–Fan fixed-point theorem, the general existence of weak and strong equilibria is also established under an additional compactness assumption. Our theoretic results are applied to a two-state model under nonexponential discounting. In particular, we demonstrate explicitly that there can be incentive to deviate from a weak equilibrium, which justifies the need for strong equilibria. Our analysis also provides new results for the existence and characterization of discrete-time equilibria under infinite horizon. [ABSTRACT FROM AUTHOR]

In this paper, we study the open-loop equilibrium strategy for mean-variance portfolio selection problem under the assumption that the risk tolerance of the investor is a non-negative and non-linear function of his/her wealth. We derive a sufficient and necessary condition for the existence and uniqueness of an open-loop equilibrium strategy via a coupled forward-backward stochastic differential equation. To the best of our knowledge, such an equation appears for the first time in the literature. The well-posedness of this equation is established by merely imposing Lipschitz condition on the risk tolerance. We also present two examples with non-monotone risk tolerances, where some interesting findings are revealed and the equilibrium strategies are obtained explicitly and numerically. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of time-inconsistent stochastic control problems for stochastic differential equations with deterministic coefficients. We study these problems within the game theoretic framework, and look for open-loop Nash equilibrium controls. Under suitable conditions, we derive a verification theorem for equilibrium controls via a flow of forward-backward stochastic partial differential equations. To illustrate our results, we discuss a mean-variance problem with a state-dependent trade-off between the mean and the variance. [ABSTRACT FROM AUTHOR]

This paper solves for the robust time-consistent mean–variance portfolio selection problem on multiple risky assets under a principle component stochastic volatility model. The model uncertainty is introduced to the drifts of the risky assets prices and the stochastic eigenvalues of the covariance matrix of asset returns. Using an extended dynamic programming approach, we manage to derive a semi-closed form solution of the desired portfolio via the solution to a coupled matrix Riccati equation. We provide the conditions, under which we prove the existence and the boundedness of the solution to the coupled matrix Riccati equation and derive the value function of the control problem. Moreover, we conduct numerical and empirical studies to perform sensitivity analyses and examine the losses due to ignoring model uncertainty or volatility information. [ABSTRACT FROM AUTHOR]

Anticipating a bailout in the event of a crisis distorts financial intermediaries' incentives in multiple dimensions. Bailout payments can, for example, lead intermediaries to issue too much short-term debt while simultaneously underinvesting in liquid assets. To correct these distortions, policymakers may choose to regulate the composition of both the assets and liabilities of intermediaries. I examine these regulations in a version of the Diamond and Dybvig [(1983). Bank runs, deposit insurance, and liquidity. Journal of Political Economy, 91(3), 401–419] model with limited commitment. I demonstrate that, contrary to common wisdom, introducing a minimum liquidity requirement can increase intermediaries' susceptibility to a run by their investors. [ABSTRACT FROM AUTHOR]

It is well-known from the work of Kupper and Schachermayer that most law-invariant risk measures are not time-consistent, and thus do not admit dynamic representations as backward stochastic differential equations. In this work we show that in a Brownian filtration the "Optimized Certainty Equivalent" risk measures of Ben-Tal and Teboulle can be computed through PDE techniques, i.e. dynamically. This can be seen as a substitute of sorts whenever they lack time consistency, and covers the cases of conditional value-at-risk and monotone mean-variance. Our method consists of focusing on the convex dual representation, which suggests an expression of the risk measure as the value of a stochastic control problem on an extended the state space. With this we can obtain a dynamic programming principle and use stochastic control techniques, along with the theory of viscosity solutions, which we must adapt to cover the present singular situation. [ABSTRACT FROM AUTHOR]

We present a modified verification theorem for the equilibrium control of a general class of portfolio problems. The general class of portfolio problems studied in this paper, is characterized by an objective where the investor seeks to maximize a functional of two conditional expectations of terminal wealth. The objective functional is allowed to be non-linear in the conditional expectations, and thus the problem class is in general terms time-inconsistent. In addition, we provide a corrected proof of the verification theorem and apply the theorem to a number of quadratic, time-inconsistent portfolio problems and determine their solutions. Some of the quadratic portfolio problems have not previously been solved analytically. [ABSTRACT FROM AUTHOR]

A game-theoretic framework for time-inconsistent stopping problems where the time-inconsistency is due to the consideration of a non-linear function of an expected reward is developed. A class of mixed strategy stopping times that allows the agents in the game to jointly choose the intensity function of a Cox process is introduced and motivated. A subgame perfect Nash equilibrium is defined. The equilibrium is characterized and other necessary and sufficient equilibrium conditions including a smooth fit result are proved. Existence and uniqueness are investigated. A mean–variance and a variance problem are studied. The state process is a general one-dimensional Itô diffusion. [ABSTRACT FROM AUTHOR]

• A benevolent planner motivates a time-inconsistent agent who takes private actions by only committing to provide feedback on outcomes. • A revelation principle specific to this environment–where private actions are taken after the mechanism is chosen–is established. • The characterisation provides conditions when the optimal mechanism takes the simple form of a cutoff. • The key normative result is that incentives through feedback always makes the time-inconsistent agent unambiguously better off. I model a benevolent planner who motivates a time-inconsistent agent by only committing to provide feedback. The optimal feedback mechanism always takes the simple form of recommending an action. I also provide conditions for when the optimal feedback mechanism takes the simple form of a cutoff. I show that providing incentives through feedback always makes the time-inconsistent agent unambiguously better off, a property that does not necessarily hold when monetary instruments are used. [ABSTRACT FROM AUTHOR]

In a dynamic setting, decision makers update their reference point as a function of previous decision and outcomes. In "Failing to Foresee the Updating of the Reference Point Leads to Time-Inconsistent Investment," Strub and Li investigate the influence of reference point updating on decision making and in particular, address whether a decision maker foresees the updating of the reference point in the context of discrete time portfolio optimization. By deriving and comparing optimal trading strategies under various frameworks and reference point updating rules and then, simulating how typical investment behavior would look like in each setting, they come to the following conclusion: a loss-averse decision maker with a time-varying reference point exhibiting realistic investment behavior fails to foresee the future updating of the reference point, and this failure leads to time-inconsistent investment. The current literature on behavioral portfolio optimization with reference point updating assumes that the decision maker foresees how the reference point will evolve and thus solves a time-consistent problem formulation. Empirical findings, however, suggest that decision makers often fail to foresee the updating of the reference point and consequently make time-inconsistent decisions. We analyze and compare the optimal investment strategies for a discrete time behavioral portfolio optimization problem with loss-aversion and time-varying reference points under both the time-consistent and time-inconsistent framework and for different updating rules for the reference point. There is only one framework predicting realistic investment behavior: the decision maker fails to foresee the updating of the reference point and thus faces a time-inconsistent problem, solves for a dynamically optimal strategy, and updates the reference point in a nonrecursive manner. [ABSTRACT FROM AUTHOR]

We model adjustment costs in a general equilibrium setting using a “transport sector”. This sector provides services needed to re-allocate a factor of production across wo other sectors. A market imperfection in the transport sector causes adjustment to occur too slowly in the absence of government intervention. The government has a restricted menu of second best policies to remedy this imperfection. Given this restricted menu, the optimal policy choice depends on the government’s ability to make commitments. The key to these results is our replacement of the black box of adjustment costs with an explicit model of these costs.

In this study, we examine dynamic allocation of resources and consumption over an infinite time horizon in an overlapping-generations model with two-sided altruism. Each generation cares not only about its own consumption and the utility of its successor, but also about the utility of its parental generation. The consumption decisions of different generations could display time-inconsistency in a model of two-sided altruism. Consequently, the outcome solutions could differ depending on whether members of the current generation are naïve and falsely believe that the future generation will comply with their, or are sophisticated and know exactly how the future generation will choose to behave. We assume that the young generation is naïve and behaves as a central planner in allocating capital resources. The capital stocks inherited by each generation from the parental generation are actually realized and used in the joint production of capital stocks and consumption goods. We refer to a sequence of such capital stocks as a two-sided altruistic path. This study examines a more general production function than earlier models of two-sided altruism, and our results prove that any two-sided altruistic path will eventually converge to the two-sided altruistic stationary path. [ABSTRACT FROM AUTHOR]

In this paper, we study general time-inconsistent stochastic control models which are driven by a stochastic differential equation with random jumps. Specifically, the time-inconsistency arises from the presence of a non-exponential discount function in the objective functional. We consider equilibrium, instead of optimal, solution within the class of open-loop controls. We prove an equivalence relationship between our time-inconsistent problem and a time-consistent problem such that the equilibrium controls for the time-consistent problem coincide with the equilibrium controls for the time-inconsistent problem. We establish two general results which characterize the open-loop equilibrium controls. As special cases, a generalized Merton's portfolio problem and a linear-quadratic problem are discussed. [ABSTRACT FROM AUTHOR]

• We analyze the determinants of the incentives that a social planner should provide to time-inconsistent consumers. • Consumption can be reduced by exerting a non-observable self-control effort. • We focus on the case in which incentives can only be conditioned on future behavior. • Differently from sin taxes theory, optimal incentives are socially costly even when consumers are homogeneous. • Optimal incentives are non-monotonic in the level of time-inconsistency and naivety. We analyze the determinants of the incentives that a paternalistic social planner should provide to improve the behavior of time-inconsistent consumers. We focus on the specific case in which consumers can reduce their future consumption by exerting self-control effort in non-observable and non-contractible activities. For example, they can reduce their future healthcare costs by practicing physical exercise. Differently from the results obtained by the sin taxes theory, we find that incentives that address the under-exertion of effort by present-biased consumers are always socially costly even when consumers are homogeneous. Moreover, we show that incentives are first increasing and then decreasing in the level of time-inconsistency, due to the ineffectiveness of incentive policies. Incentives are also affected by the degree of naivety in a non-monotonic way. [ABSTRACT FROM AUTHOR]

• Formulate the time-consistent mean-variance hedging for an illiquid asset using a cointegrated liquid asset. • Solve for the explicit closed-form equilibrium hedging strategy. • Demonstrate improvement in hedging effectiveness of the novel hedging solution with real empirical data. When a financial institution has a difficulty to close a position in an illiquid asset, a practical solution is to hedge the risk of the position with a related liquid asset. This may frequently occur in insurance institutions who diversify asset allocation by illiquid investment and face high transaction costs when try to close the illiquid asset position. A conventional approach constructs optimal hedging based on the correlation between the liquid and illiquid assets. We show, however, that ignoring the serial dependence through cointegration significantly reduces the hedging effectiveness. This paper investigates the time-consistent mean-variance optimal hedging using cointegration for the risk of a non-traded portfolio. Specifically, only the investment amount in the liquid asset is the control variable. The mathematical challenge stems on the derivation of the optimal equilibrium strategy when we have to deal with cointegration, illiquidity and time-inconsistency simultaneously. Using a stochastic differential game approach, we characterize the equilibrium hedging strategy from a system of partial differential equations that involves an HJB equation for the time-consistent problem with cointegration. Novel solutions to the equilibrium strategy and the efficient frontier are obtained in explicit closed-form formulas. The significance of cointegration in the corresponding hedging problem is evidenced with real empirical data. [ABSTRACT FROM AUTHOR]

We investigate an exponential utility maximization problem for an insurer who faces a stream of non-hedgeable claims. The insurer's risk aversion coefficient changes in time and depends on the current insurer's net asset value (the excess of assets over liabilities). We use the notion of an equilibrium strategy and derive the HJB equation for our time-inconsistent optimization problem. We assume that the insurer's risk aversion coefficient consists of a constant risk aversion and a small amount of a wealth-dependent risk aversion. Using perturbation theory, the equilibrium value function, which solves the HJB equation, is expanded on the parameter controlling the degree of risk aversion depending on wealth. We find the first-order approximations to the equilibrium value function and the equilibrium investment strategy. Some new results for exponential utility maximization problem with constant risk aversion are derived in order to approximate the solution to our exponential utility maximization problem with wealth-dependent risk aversion. [ABSTRACT FROM AUTHOR]