No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Equilibrium Strategies in Continuous-Time Time-Inconsistent Problems
... Using cut-off and localization techniques and the properties of characterization operators of transition semigroups, we rigorously establish the associated extended HJB system, which does not only contain equations, but also inequalities as well as a complicated boundary term. Combining the above arguments with parabolic PDE theory, we are able to weaken the conditions in He and Jiang (2019). We emphasize that the derived extended HJB system is an equivalent characterization of equilibriums, which is thus beyond the verification scope. ...
... Constant investment proportion with one-side withdrawal threshold can still be equilibrium in this case, but this is only assured when the dependence on habit is not very "strong" (see Proposition 4.2 for details). This strategy structure is similar to those obtained in existing literature, such as time-consistent stopping control problems Karatzas and Wang (2000), pure stopping problems Christensen and Lindensjö (2018) or pure control problems He and Jiang (2019). However, our numerical experiments show that combining the control and stopping strategies brings novel financial insights. ...
... The value of this radius is related to a universal constant (≈ 8.3419) determined by Bessel functions. This example is interesting in the 1 See Karatzas and Wang (2000), Ekeland and Pirvu (2008), Yong (2012), Björk, Khapko, and Murgoci (2017), Alia, Vives, and Khelfallah (2017) and He and Jiang (2019), among others. aspect of mathematics, and is the first step to further studies on time-inconsistent multi-dimensional stopping control problems. ...
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 multi-dimensional controlled diffusion model and propose a formal definition of their equilibriums. We show that an admissible pair (ˆu, C) of control-stopping policy is equilibrium if and only if the auxiliary function associated to it solves the extended HJB system, providing 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 non-exponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general non constant 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 examplewith hyperbolic discount.
... This can be regarded as a time-inconsistent version of the verification theorem in (classical) stochastic control. The proof of Theorem 1 can be found in Björk et al. (2017) and He and Jiang (2019). Assumption 1 is easy to verify because it involves only the model parameters, i.e., µ, σ, C, F , and G. Assumption 2 imposes some regularity conditions onû, which usually requiresû to be smooth to a certain degree; see He and Jiang (2019) for a sufficient condition for this assumption. ...
... Recently, He and Jiang (2019), Han and Wong (2020), and Hernández and Possamaï (2020) also consider intra-personal equilibria with fixed initial data. Moreover, He and Jiang (2019) propose a formal definition of X 0,x0,û t , calling it the set of reachable states. ...
... In some other works, however, D is set to be the set of constant strategies U; see for instance Ekeland and Lazrak (2006, Björk and Murgoci (2010), and Basak and Chabakauri (2010). He and Jiang (2019) show that the choice of D is irrelevant as long as it at least contains U. Indeed, this can be seen from the observation in Theorem 1 that Γ τ,y,û (t, x; a) = Γ τ,y,û (t, x; a(t, x)) for any a ∈ U. He and Jiang (2019) also show that for strong intra-personal equilibrium, which will be introduced momentarily, the choice of D is relevant. ...
Time inconsistency is prevalent in dynamic choice problems: a plan of actions to be taken in the future that is optimal for an agent today may not be optimal for the same agent in the future. If the agent is aware of this intra-personal conflict but unable to commit herself in the future to following the optimal plan today, the rational strategy for her today is to reconcile with her future selves, namely to correctly anticipate her actions in the future and then act today accordingly. Such a strategy is named intra-personal equilibrium and has been studied since as early as in the 1950s. A rigorous treatment in continuous-time settings, however, had not been available until a decade ago. Since then, the study on intra-personal equilibrium for time-inconsistent problems in continuous time has grown rapidly. In this chapter, we review the classical results and some recent development in this literature.
... Because of time inconsistency, without self-control or the help of commitment devices, the agent cannot commit her future selves to following the dynamic portfolio that maximizes the quantile of her terminal wealth today. Following the literature on time inconsistency, we consider so-called intra-personal equilibrium portfolio strategies; see for instance Strotz (1955Strotz ( -1956, Ekeland and Lazrak (2006), Björk et al. (2017), He and Jiang (2019), and the references therein. More precisely, we assume that the agent has no self control, so we regard her selves at different time to be different players in a game and seek an equilibrium in this game. ...
... The above definition of equilibrium strategies is so-called regular equilibrium, which slightly differs from the notion of weak equilibrium that is used in most studies of continuoustime time-inconsistent problems in the literature. As explained in He and Jiang (2019), the notion of regular equilibrium is preferred to the notion of weak equilibrium because the agent can still be willing to deviate from a weak equilibrium strategy and take a very different alternative. ...
... Almost all the existing literature on time inconsistency consider all states including those that are not reachable; seeHe and Jiang (2019) for a discussion and for the relevant references. ...
In this paper, we study a portfolio selection problem in which an agent trades a risk-free asset and multiple risky assets with deterministic mean return rates and volatility and wants to maximize the -quantile of her wealth at some terminal time. Because of the time inconsistency caused by quantiles, we consider intra-personal equilibrium strategies. We find that among the class of time-varying, affine portfolio strategies, the intra-personal equilibrium does not exist when , leads to zero investment in the risky assets when , and is a portfolio insurance strategy when . We then compare the intra-personal equilibrium strategy in the case of , namely under median maximization, to some other strategies and apply it to explain why more wealthy people invest more precentage of wealth in risky assets. Finally, we extend our model to account for multiple terminal time.
... In this paper, we use the game-theoretic approach first adopted by Strotz [58] and then systematically developed by Ekeland and Lazrak [20], Björk et al. [11] and He and Jiang [37] to obtain equilibrium strategy. The basic idea is that the sophisticated individual at each time can implement her strategy in an infinitesimally small, but positive, time period. ...
... 3 We derive the extended HJB equation in the continuous-time setting and obtain a closed-form solution for the logarithmic utility and numerical results for the power utility. Furthermore, we show that the closed-form solution for the logarithmic utility is an equilibrium strategy according to the definition of equilibrium strategy proposed by He and Jiang [37]. By exploring our closed-form solution, we derive a set of predictions that characterize dynamic consumption and investment behaviors. ...
... To derive the time-consistent consumption and portfolio policies, we adopt the game-theoretic approach first proposed by Strotz [58] and then systematically developed by Ekeland and Lazrak [20], Björk et al. [11] and He and Jiang [37]. 15 The game-theoretic approach treats the problem as a non-cooperative game played by the individual and all her future selves. ...
We study the consumption and portfolio decisions by incorporating mortality risk and altruistic factor in the classical model of Merton (Rev Econ Stat 51:247–257, 1969; J Econ Theory 3:373–413, 1971) and Yaari (Rev Econ Stud 32(2):137–150, 1965). We find that besides the present-biased preference, the process of updating mortality information may be another underlying cause of dynamically time-inconsistent consumption behavior. We use the game-theoretic approach to obtain the extended Hamilton–Jacobi–Bellman equation. Furthermore, we obtain the closed-form solution for the logarithmic utility and explore comparative statics and implications for dynamic behavior. We present numerical results for the power utility that shows the sophisticated individual enjoys higher expected discounted utility than the naive. Our analytical solution enables us to generate a set of testable predictions that are consistent with existing empirical evidence. In particular, we show that for a moderate range of expected investment return, individuals will exhibit a “hump-shaped” consumption pattern, as widely documented in the empirical literature.
... Inspired by the notion of equilibrium in [24] and their study of the discrete case in [6], in [9] the authors consider a general Markovian framework with diffusion dynamics for the controlled state process X, and provide a system of PDEs whose solution allows to construct an equilibrium for the problem. Recently, He and Jiang [40] fills in a missing step in [9] by deriving rigorously the PDE system and refining the definition of equilibrium. Nonetheless, so far none of these approaches was able to handle the typical non-Markovian problems that would necessarily arise, for example, in contracting problems involving a principal and time-inconsistent agents (see Cvitanić, Possamaï, and Touzi [16]). ...
... The fist one is that (2.13) holds for P(x)-q.e. x ∈ X . This is in contrast to [6] and [40], where the rewards are respectively compared for all x ∈ X and x in the support of P ν ⋆ ∈ P(x). The approach in [6] is too stringent as it might impose a condition on trajectories that are not reachable by any admissible action. ...
... Furthermore, we could have also considered strict equilibria as those lax equilibria for which (2.13) holds with ε = 0. While this document was being finalised, we were made aware of the works [49] and [40] where this last notion of equilibrium, referred there as regular equilibrium, was studied. See once more Section 3.1 for a discussion on how we compare our definition to this and other notions of equilibria in the literature. ...
We develop a theory for continuous-time non-Markovian stochastic control problems which are inherently time-inconsistent. Their distinguishing feature is that the classical Bellman optimality principle no longer holds. Our formulation is cast within the framework of a controlled non-Markovian forward stochastic differential equation, and a general objective functional setting. We adopt a game-theoretic approach to study such problems, meaning that we seek for \emph{sub-game perfect Nash equilibrium} points. As a first novelty of this work, we introduce and motivate a new definition of equilibrium that allows us to establish rigorously an \emph{extended dynamic programming principle}, in the same spirit as in the classical theory. This in turn allows us to introduce a system of backward stochastic differential equations analogous to the classical HJB equation. We prove that this system is fundamental, in the sense that its well-posedness is both necessary and sufficient to characterise the value function and equilibria. As a final step we provide an existence and uniqueness result. Some examples and extensions of our results are also presented.
... Its intuitive and flexible formulation has attracted the attention of numerous researchers, who have sought to strengthen the original framework. Examples include, but are not limited to, Li and Ng (2000); Zhou and Li (2000); Basak and Chabakauri (2010); Czichowsky (2013); ; He and Jiang (2019); Dai et al. (2020). In contrast to expected utility theory (Merton, 1969), MVP suffers from time-inconsistency induced by the variance operator. ...
... With a fixed-point argument, Huang and Zhou (2018) further distinguishes between strong and weak equilibria. Related discussions include He and Jiang (2019) and the references therein. In this paper, we exploit the extended HJB methodology for its wider applications, including MVP. ...
... Roughly speaking, the support contains all possible situations for the paths. We refer to He and Jiang (2019) for the rationale for considering the support rather than the whole space Ω. ...
In this paper, we consider equilibrium strategies under Volterra processes and time-inconsistent preferences embracing mean-variance portfolio selection (MVP). Using a functional Ito calculus approach, we overcome the non-Markovian and non-semimartingale difficulty in Volterra processes. The equilibrium strategy is then characterized by an extended path-dependent Hamilton-Jacobi-Bellman equation system under a game-theoretic framework. A verification theorem is provided. We derive explicit solutions to three problems, including MVP with constant risk aversion, MVP for log returns, and a mean-variance objective with a linear controlled Volterra process. We also thoroughly examine the effect of volatility roughness on equilibrium strategies. Numerical experiments demonstrate that trading rules with rough volatility outperform the classic counterparts.
... As ρ ε S ≥ ε > 0, the condition (1.7) does capture the deviation from stopping to continuing, and is much stronger than (1.3). However, there is still a drawback for (1.7): when the limit is equal to zero, it is possible that for all ε > 0 we have f (x) < E x [δ(ρ ε S )f (X ρ ε S )], and thus there is an incentive to deviate (see [2,Remark 3.5] and [13,1,6] for more details). Roughly speaking, this is similar to a critical point not necessarily being a local maximum in calculus. ...
... This remedies the issue of weak equilibria mention in the above, and captures the economic meaning of "equilibrium" more accurately. Such kind of equilibria is also studied in [13,6] for time inconsistent control. Obviously, a strong equilibrium must be weak. ...
We consider three equilibrium concepts proposed in the literature for time-inconsistent stopping problems, including mild equilibria, weak equilibria and strong equilibria. The discount function is assumed to be log sub-additive and the underlying process is one-dimensional diffusion. We first provide necessary and sufficient conditions for the characterization of weak equilibria. The smooth-fit condition is obtained as a by-product. Next, based on the characterization of weak equilibria, we show that an optimal mild equilibrium is also weak. Then we provide conditions under which a weak equilibrium is strong. We further show that an optimal mild equilibrium is also strong under a certain condition. Finally, we provide several examples including one shows a weak equilibrium may not be strong, and another one shows a strong equilibrium may not be optimal mild.
... Moreover, the key ideas in these proofs provide some novel proof approaches in the literature of time-inconsistent control/stopping. The recent work of [26] discusses notions of equilibrium control based on condition (1.3) in [34]. The focus of their paper is to distinguish between weak and strong (and regular) equilibrium controls. ...
... The notions of mild equilibrium and optimal mild equilibrium only make sense for stopping problems not control problems. Thus the focus of our project is different from [26] and our intuitively unexpected result makes a novel contribution to the literature. ...
This thesis consists of two problems on time inconsistency and one problem on mean field games, all featuring the study of equilibrium and applications in economics and finance. In Chapter II, we deal with time inconsistency in the infinite horizon mean- variance stopping problem under discrete time setting. In order to determine a proper time-consistent plan, we investigate subgame perfect Nash equilibria among three different types of strategies, pure stopping times, randomized stopping times and liquidation strategies. We show that equilibria among pure stopping times or randomized stopping times may not exist, while an equilibrium liquidation strategy always exists. Furthermore, we argue that the mean-standard deviation variant of this problem makes more sense for this type of strategies in terms of time consistency. The existence and uniqueness of optimal equilibrium liquidation strategies are also analyzed. In Chapter III, we delve into equilibrium concepts for time inconsistent stopping problems in continuous time. We point out that the two existing notions of equi- librium in the literature, which we call mild equilibrium and weak equilibrium, are inadequate to capture the idea of subgame perfect Nash equilibrium. To characterize it more accurately, we introduce a new notion, strong equilibrium. It is proved that an optimal mild equilibrium is always a strong equilibrium. Moreover, we provide a new iteration method that can directly construct an optimal mild equilibrium and thus also guarantees its existence. xi In Chapter IV, we adopt a mean field game (MFG) approach to analyze a costly job search model with incomplete credit and insurance markets. The MFG approach enables us to quantify the impact of a class of countercyclical unemployment benefit policies on labor supply in general equilibrium. Our model provides two interesting predictions. First, the difference between unemployment rates under a countercyclical policy and an acyclical policy is positive and increases rapidly with the size of the aggregate shock. Second, compared with a baseline policy without means test, a means-tested policy which is targeted to provide more generous benefits to liquidity constrained individuals turns out to provide improved consumption insurance to all individuals as well as results in a lower equilibrium unemployment rate relative to a comparable non-targeted policy.
... Another question that interests researchers is the support of an SDE and the set of states that the SDE can reach at certain future time. For example, in the study of time-inconsistent stochastic control problems, the set of reachable states of an SDE is an important constituent of the definition of equilibrium strategies in He and Jiang (2019). ...
... In other words, the X t is the union of int(S X(t) ) and the smallest relatively closed subset A of ∂S X(t) such that P(X(t) ∈ A) = P(X(t) ∈ ∂S X(t) ). For an application of the set of reachable sets, see He and Jiang (2019). ...
In this paper, we prove a sufficient and necessary condition for the transition probability distribution of a general, time-inhomogeneous linear SDE to possess a density function and study the differentiability of the density function and the transition quantile function of the SDE. Moreover, we completely characterize the support of the marginal distribution of this SDE.
... Recently, Lindensjö [27] derived rigourously the extended HJB equations without using arguments from the discrete-time case. He and Jiang [20] and Hernández and Possamaï [21] generalized [5] by refining the definition of the closed-loop equilibrium concept. For equilibrium stopping times in time-inconsistent Markovian Problems, see e.g. ...
... Recently, Lindensjö [24] derived rigourously the extended HJB equations without using arguments from the discrete-time case. He and Jiang [19] and Hernández and Possamaï [20] generalized [6] by refining the definition of the closed-loop equilibrium concept. For equilibrium stopping times in time-inconsistent Markovian Problems, see e.g. ...
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 an illustration of the general theory, we discuss a mean-variance portfolio selection problem under a jump-diffusion model.
... Using the observation in [4, Remark 3.5] as a starting point [23] introduces -in a time-inconsistent stochastic control framework -the notion of strong equilibrium, which adapted to the problem of the present paper corresponds to the condition that there should exist a fixedh > 0 such that for each h ∈ [0,h] holds that the numerator of (2.2) is non-negative. The notion of strong equilibrium for time-inconsistent control is also studied in [18,19]. ...
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 choose the intensity function of a Cox process is introduced. A subgame perfect Nash equilibrium is defined. The equilibrium is characterized and other results with different necessary and sufficient conditions for equilibrium are proven. This includes a smooth fit result. A mean-variance problem and a variance problem are studied as examples. The state process is a general one-dimensional It\^{o} diffusion.
We provide a unified approach to find equilibrium solutions for time-inconsistent problems with distribution dependent rewards, which are important to the study of behavioral finance and economics. Our approach is based on equilibrium master equation, a non-local partial differential equation on Wasserstein space. We refine the classical notion of derivatives with respect to distribution, and establish Itô's formula in the sense of such refined derivatives. Our approach is inspired by theories of Mckean-Vlasov stochastic control and mean field games, but is significantly different from both in that: we prohibit marginal distribution of state to be an input of closed loop control; we solve the best reaction to individual selves in an intra-person game instead of the best reaction to large populations as in mean field games. As applications, we reexamine the dynamic portfolio choice problem with rank dependent utility based on the proposed novel approach. We also recover the celebrated extended HJB equation when reward of the problem has a nonlinear function of expectation , while reformulate and weaken the assumptions needed. Most importantly, we provide a procedure to find equilibrium solution of a dynamic mean-ES portfolio choice problem, which is completely new to the literature.
We now move on to study time-inconsistent control problems in continuous time and, as in Part II, we use a game-theoretic framework, studying subgame-perfect Nash equilibrium strategies. It turns out, however, that the equilibrium concept in continuous time is quite delicate and so requires a detailed investigation. Our main result is an extension of the standard HJB equation to a system of equations: the extended HJB system. We also prove a verification theorem, showing that a solution to the extended HJB system does indeed deliver the equilibrium control and equilibrium value function for our original problem.
We study portfolio selection in a complete continuous-time market where the preference is dictated by the rank-dependent utility. As such a model is inherently time inconsistent due to the underlying probability weighting, we study the investment behavior of a sophisticated consistent planners who seek (subgame perfect) intra-personal equilibrium strategies. We provide sufficient conditions under which an equilibrium strategy is a replicating portfolio of a final wealth. We derive this final wealth profile explicitly, which turns out to be in the same form as in the classical Merton model with the market price of risk process properly scaled by a deterministic function in time. We present this scaling function explicitly through the solution to a highly nonlinear and singular ordinary differential equation, whose existence of solutions is established. Finally, we give a necessary and sufficient condition for the scaling function to be smaller than one corresponding to an effective reduction in risk premium due to probability weighting.
A new notion of equilibrium, which we call strong equilibrium, is introduced for time‐inconsistent stopping problems in continuous time. Compared to the existing notions introduced in Huang, Y.‐J., & Nguyen‐Huu, A. (2018, Jan 01). Time‐consistent stopping under decreasing impatience. Finance and Stochastics, 22(1), 69–95 and Christensen, S., & Lindensjö, K. (2018). On finding equilibrium stopping times for time‐inconsistent markovian problems. SIAM Journal on Control and Optimization, 56(6), 4228–4255, which in this paper are called mild equilibrium and weak equilibrium, respectively, a strong equilibrium captures the idea of subgame perfect Nash equilibrium more accurately. When the state process is a continuous‐time Markov chain and the discount function is log subadditive, we show that an optimal mild equilibrium is always a strong equilibrium. Moreover, we provide a new iteration method that can directly construct an optimal mild equilibrium and thus also prove its existence.
We extend a recent result of Trybuła and Zawisza (2019), who investigate a continuous-time portfolio optimization problem under monotone mean–variance preferences. Their main finding is that the optimal strategies for monotone and classical mean–variance preferences coincide in a stochastic factor model for the financial market. We generalize this result to any model for the financial market where asset prices are continuous.
Under non-exponential discounting, we develop a dynamic theory for stopping problems in continuous time. Our framework covers discount functions that induce decreasing impatience. Due to the inherent time inconsistency, we look for equilibrium stopping policies, formulated as fixed points of an operator. Under appropriate conditions, fixed-point iterations converge to equilibrium stopping policies. This iterative approach corresponds to the hierarchy of strategic reasoning in game theory and provides “agent-specific” results: it assigns one specific equilibrium stopping policy to each agent according to her initial behavior. In particular, it leads to a precise mathematical connection between the naive behavior and the sophisticated one. Our theory is illustrated in a real options model.
Standard Markovian optimal stopping problems are consistent in the sense that the first entrance time into the stopping set is optimal for each initial state of the process. Clearly, the usual concept of optimality cannot in a straightforward way be applied to non-standard stopping problems without this time-consistent structure. This paper is devoted to the solution of time-inconsistent stopping problems with the reward depending on the initial state using an adaptation of Strotz's consistent planning. More precisely, we give a precise equilibrium definition --- of the type subgame perfect Nash equilibrium based on pure Markov strategies. In general, such equilibria do not always exist and if they exist they are in general not unique. We, however, develop an iterative approach to finding equilibrium stopping times for a general class of problems and apply this approach to one-sided stopping problems on the real line. We furthermore prove a verification theorem based on a set of variational inequalities which also allows us to find equilibria. In the case of a standard optimal stopping problem, we investigate the connection between the notion of an optimal and an equilibrium stopping time. As an application of the developed theory we study a selling strategy problem under exponential utility and endogenous habit formation.
In this paper, which is a continuation of the previously published discrete time paper we develop a theory for continuous time stochastic control problems which, in various ways, are time inconsistent in the sense that they do not admit a Bellman optimality principle. We study these problems within a game theoretic framework, and we look for Nash subgame perfect equilibrium points. For a general controlled continuous time Markov process and a fairly general objective functional we derive an extension of the standard Hamilton-Jacobi-Bellman equation, in the form of a system of non-linear equations, for the determination for the equilibrium strategy as well as the equilibrium value function. As applications of the general theory we study non exponential discounting, various types of mean variance problems, a point process example, as well as a time inconsistent linear quadratic regulator. We also present a study of time inconsistency within the framework of a general equilibrium production economy of Cox-Ingersoll-Ross type.
The real options framework has been used extensively to analyze the timing of invest-ment under uncertainty. While standard real options models assume that agents possess a constant rate of time preference, there is substantial evidence that agents are very im-patient about choices in the short-term, but are quite patient when choosing between long-term alternatives. We extend the real options framework to model the investment timing decisions of entrepreneurs with such time-inconsistent preferences. Two opposing forces determine investment timing: while evolving uncertainty induces entrepreneurs to defer investment in order to take advantage of the option to wait, their time-inconsistent preferences motivate them to invest earlier in order to avoid the time-inconsistent be-havior they will display in the future. We find that the precise trade-off between these two forces depends on such factors as whether entrepreneurs are sophisticated or naive in their expectations regarding their future time-inconsistent behavior, as well as whether the payoff from investment occurs all at once or over time. We extend the model to consider equilibrium investment behavior for an industry comprised of time-inconsistent entrepreneurs. Such an equilibrium involves the dual problem of entrepreneurs playing dynamic games against competitors as well as against their own future selves., the anonymous referee, and seminar participants at Columbia and Wisconsin (Madison) for helpful comments.
We investigate the classical Ramsey problem of economic growth when the planner uses non-constant discounting. It is well-known
that this leads to time inconsistency, so that optimal strategies are no longer implementable. We then define equilibrium
strategies to be such that unilateral deviations occurring during a small time interval are penalized. Non-equilibrium strategies
are not implementable, so only equilibrium strategies should be considered by a rational planner. We show that there exists
such strategies which are (a) smooth, and (b) lead to stationary growth, as in the classical Ramsey model. Finally, we prove
an existence and multiplicity result: for logarithmic utility and quasi-exponential discount, there is an interval I such that, for every k in I, there is an equilibrium strategy converging to k. We conclude by giving an example where the planner is led to non-constant discount rates by considerations of intergenerational
equity.
KeywordsTime inconsistency–Markov strategies–Ramsey models–Nash equilibria–Intergenerational equity–Implicit differential equation
In this paper, we formulate a general time-inconsistent stochastic
linear--quadratic (LQ) control problem. The time-inconsistency arises from the
presence of a quadratic term of the expected state as well as a state-dependent
term in the objective functional. We define an equilibrium, instead of optimal,
solution within the class of open-loop controls, and derive a sufficient
condition for equilibrium controls via a flow of forward--backward stochastic
differential equations. When the state is one dimensional and the coefficients
in the problem are all deterministic, we find an explicit equilibrium control.
As an application, we then consider a mean-variance portfolio selection model
in a complete financial market where the risk-free rate is a deterministic
function of time but all the other market parameters are possibly stochastic
processes. Applying the general sufficient condition, we obtain explicit
equilibrium strategies when the risk premium is both deterministic and
stochastic.
This paper considers the portfolio management problem of optimal investment,
consumption and life insurance. We are concerned with time inconsistency of
optimal strategies. Natural assumptions, like different discount rates for
consumption and life insurance, or a time varying aggregation rate lead to time
inconsistency. As a consequence, the optimal strategies are not implementable.
We focus on hyperbolic discounting, which has received much attention lately,
especially in the area of behavioural finance. Following [10], we consider the
resulting problem as a leader-follower game between successive selves, each of
whom can commit for an infinitesimally small amount of time. We then define
policies as subgame perfect equilibrium strategies. Policies are characterized
by an integral equation which is shown to have a solution. Although we work on
CRRA preference paradigm, our results can be extended for more general
preferences as long as the equations admit solutions. Numerical simulations
reveal that for the Merton problem with hyperbolic discounting, the consumption
increases up to a certain time, after which it decreases; this pattern does not
occur in the case of exponential discounting, and is therefore known in the
litterature as the "consumption puzzle". Other numerical experiments explore
the effect of time varying aggregation rate on the insurance premium.
Hyperbolic discount functions induce dynamically inconsistent preferences, implying a motive for consumers to constrain their
own future choices. This paper analyzes the decisions of a hyperbolic consumer who has access to an imperfect commitment technology:
an illiquid asset whose sale must be initiated one period before the sale proceeds are received. The model predicts that consumption
tracks income, and the model explains why consumers have asset-specific marginal propensities to consume. The model suggests
that financial innovation may have caused the ongoing decline in U. S. savings rates, since financial innovation increases
liquidity, eliminating commitment opportunities. Finally, the model implies that financial market innovation may reduce welfare
by providing “too much” liquidity.
Extending Barro (1999) and Luttmer and Mariotti (2003), we introduce a new model of time preferences: the instantaneous-gratification model. This model applies tractably to a much wider range of settings than existing models. It applies to both complete- and incomplete-market settings and it works with generic utility functions. It works in settings with linear policy rules and in settings in which equilibrium cannot be supported by linear rules. The instantaneous-gratification model also generates a unique equilibrium, even in infinite-horizon applications, thereby resolving the multiplicity problem hitherto associated with dynamically inconsistent models. Finally, it simultaneously features a single welfare criterion and a behavioral tendency towards overconsumption. © The Author(s) 2012. Published by Oxford University Press, on behalf of President and Fellows of Harvard College. All rights reserved.
In this paper, we investigate the Merton portfolio management problem in the context of non-exponential discounting. This gives rise to time-inconsistency of the decision-maker. If the decision-maker at time t=0 can commit his/her successors, he/she can choose the policy that is optimal from his/her point of view, and constrain the others to abide by it, although they do not see it as optimal for them. If there is no commitment mechanism, one must seek a subgame-perfect equilibrium strategy between the successive decision-makers. In the line of the earlier work by Ekeland and Lazrak we give a precise definition of equilibrium strategies in the context of the portfolio management problem, with finite horizon, we characterize it by a system of partial differential equations, and we show existence in the case when the utility is CRRA and the terminal time T is small. We also investigate the infinite-horizon case and we give two different explicit solutions in the case when the utility is CRRA (in contrast with the case of exponential discount, where there is only one). Some of our results are proved under the assumption that the discount function h(t) is a linear combination of two exponentials, or is the product of an exponential by a linear function.
For an infinite-horizon continuous-time optimal stopping problem under non-exponential discounting, we look for an optimal equilibrium, which generates larger values than any other equilibrium does on the entire state space. When the discount function is log sub-additive and the state process is one-dimensional, an optimal equilibrium is constructed in a specific form, under appropriate regularity and integrability conditions. While there may exist other optimal equilibria, we show that they can differ from the constructed one in very limited ways. This in particular leads to a sufficient condition for the uniqueness of optimal equilibria, up to some closedness condition. To illustrate our theoretic results, comprehensive analysis is carried out for three specific stopping problems. For each one of them, an optimal equilibrium is characterized through an explicit formula.
Many households display inertia in investment management over their life cycles. Our calibrated dynamic life cycle portfolio choice model can account for such an apparently 'irrational' outcome, by incorporating the fact that investors must forgo acquiring job-specific skills when they spend time managing their money, and their efficiency in financial decision making varies with age. Resulting inertia patterns mesh well with findings from prior studies and our own empirical results from Panel Study of Income Dynamics (PSID) data. We also analyze how people optimally choose between actively managing their assets versus delegating the task to financial advisors. Delegation proves valuable to both the young and the old. Our calibrated model quantifies welfare gains from including investment time and money costs as well as delegation in a life cycle setting.
We study an infinite-horizon discrete-time optimal stopping problem under non-exponential discounting. A new method, which we call the iterative approach, is developed to find subgame perfect Nash equilibriums. When the discount function induces decreasing impatience, we establish the existence of an equilibrium through fixed-point iterations. Moreover, we show that there exists a unique optimal equilibrium, which generates larger value than any other equilibrium does at all times. To the best of our knowledge, this is the first time a dominating subgame perfect Nash equilibrium is shown to exist in the literature of time-inconsistency.
In this paper, we continue our study on a general time-inconsistent stochastic linear-quadratic control problem originally formulated in [Y. Hu, H. Jin, and X. Y. Zhou, SIAM J. Control. Optim., 50 (2012), pp. 1548-1572]. We derive a necessary and sufficient condition for equilibrium controls via a ow of forward-backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we prove that the explicit equilibrium control constructed in [Y. Hu, H. Jin, and X. Y. Zhou, SIAM J. Control. Optim., 50 (2012), pp. 1548-1572] is indeed unique. Our proof is based on the derived equivalent condition for equilibria as well as a stochastic version of the Lebesgue differentiation theorem. Finally, we show that the equilibrium strategy is unique for a mean-variance portfolio selection model in a complete financial market where the risk-free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes.
A time-inconsistent stochastic optimal control problem with a recursive cost func- tional is studied. Equilibrium strategy is introduced, which is time-consistent and locally approxi- mately optimal. By means of multiperson hierarchical differential games associated with partitions of the time interval, a family of approximate equilibrium strategy is constructed, and by sending the mesh size of the time interval partition to zero, an equilibrium Hamilton{Jacobi{Bellman (HJB) equation is derived through which the equilibrium value function can be identified and the equilib- rium strategy can be obtained. Moreover, a well-posedness result of the equilibrium HJB equation is established under certain conditions, and a verification theorem is proved. Finally, an illustrative example is presented, and some comparisons of different definitions of equilibrium strategy are put in order.
We develop a theory for a general class of discrete-time stochastic control problems that, in various ways, are time-inconsistent in the sense that they do not admit a Bellman optimality principle. We attack these problems by viewing them within a game theoretic framework, and we look for subgame perfect Nash equilibrium points. For a general controlled Markov process and a fairly general objective functional, we derive an extension of the standard Bellman equation, in the form of a system of nonlinear equations, for the determination of the equilibrium strategy as well as the equilibrium value function. Most known examples of time-inconsistent stochastic control problems in the literature are easily seen to be special cases of the present theory. We also prove that for every time-inconsistent problem, there exists an associated time-consistent problem such that the optimal control and the optimal value function for the consistent problem coincide with the equilibrium control and value function, respectively for the time-inconsistent problem. To exemplify the theory, we study some concrete examples, such as hyperbolic discounting and mean-variance control.
This paper presents a problem which I believe has not heretofore been analysed2 and provides a theory to explain, under different circumstances, three related phenomena: (1) spendthriftiness; (2) the deliberate regimenting of one’s future economic behaviour— even at a cost; and (3) thrift. The senses in which we deal with these topics can probably not be very well understood, however, until after the paper has been read; but a few sentences at this point may shed some light on what we are up to.
The object of this paper is to study the mean–variance portfolio op-timization 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 [2] where the authors assumed a con-stant risk aversion parameter. This assumption leads to an equilibrium control where the dollar amount invested in the risky asset is indepen-dent 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 is allowed to depend dy-namically 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 [4], we provide a fairly detalied anaysis on the general case. We also study the particular case when the risk aversion is inversely proportional to wealth, and for this case we provide an analytic solution where the equilibrium dollar amount invested in the risky asset is proportional to current wealth. The equi-librium for this model thus appears more reasonable than the one for the model with constant risk aversion.
We develop a theory for stochastic control problems which, in various ways, are time inconsistent in the sense that they do not admit a Bellman optimality principle. We attach these problems by viewing them within a game theoretic framework, and we look for Nash subgame perfect equilibrium points. For a general controlled Markov process and a fairly general objective functional we derive an extension of the standard Hamilton-Jacobi-Bellman equation, in the form of a system of on-linear equations, for the determination for the equilibrium strategy as well as the equilibrium value function. All known examples of time inconsistency in the literature are easily seen to be special cases of the present theory. We also prove that for every time inconsistent problem, there exists an associated time consistent problem such that the optimal control and the optimal value function for the consistent problem coincides with the equilibrium control and value function respectively for the time inconsistent problem. We also study some concrete examples.
A general time-inconsistent optimal control problem is considered for
stochastic differential equations with deterministic coefficients. Under
suitable conditions, a Hamilton-Jacobi-Bellman type equation is derived for the
equilibrium value function of the problem. Well-posedness and some properties
of such an equation is studied, and time-consistent equilibrium strategies are
constructed. As special cases, the linear-quadratic problem and a generalized
Merton's portfolio problem are investigated.
The authors examine self-control problems--modeled as time-inconsistent, present-biased preferences--in a model where a person must do an activity exactly once. They emphasize two distinctions: do activities involve immediate costs or immediate rewards, and are people sophisticated or naive about future self-control problems? Naive people procrastinate immediate-cost activities and preproperate--do too soon--immediate-reward activities. Sophistication mitigates procrastination but exacerbates preproperation. Moreover, with immediate costs, a small present bias can severely harm only naive people, whereas with immediate rewards it can severely harm only sophisticated people. Lessons for savings, addiction, and elsewhere are discussed.
A model of casino gambling
- N Barberis
Dynamic mean-variance asset allocation
- S Basak
Portfolio inertia and stock market fluctuations
- Y Bilias