2021 | Buch

# Time-Inconsistent Control Theory with Finance Applications

verfasst von: Prof. Tomas Björk, Dr. Mariana Khapko, Dr. Agatha Murgoci

Verlag: Springer International Publishing

Buchreihe : Springer Finance

2021 | Buch

verfasst von: Prof. Tomas Björk, Dr. Mariana Khapko, Dr. Agatha Murgoci

Verlag: Springer International Publishing

Buchreihe : Springer Finance

This book is devoted to problems of stochastic control and stopping that are time inconsistent in the sense that they do not admit a Bellman optimality principle. These problems are cast in a game-theoretic framework, with the focus on subgame-perfect Nash equilibrium strategies. The general theory is illustrated with a number of finance applications.

In dynamic choice problems, time inconsistency is the rule rather than the exception. Indeed, as Robert H. Strotz pointed out in his seminal 1955 paper, relaxing the widely used ad hoc assumption of exponential discounting gives rise to time inconsistency. Other famous examples of time inconsistency include mean-variance portfolio choice and prospect theory in a dynamic context. For such models, the very concept of optimality becomes problematic, as the decision maker’s preferences change over time in a temporally inconsistent way. In this book, a time-inconsistent problem is viewed as a non-cooperative game between the agent’s current and future selves, with the objective of finding intrapersonal equilibria in the game-theoretic sense. A range of finance applications are provided, including problems with non-exponential discounting, mean-variance objective, time-inconsistent linear quadratic regulator, probability distortion, and market equilibrium with time-inconsistent preferences.

Time-Inconsistent Control Theory with Finance Applications offers the first comprehensive treatment of time-inconsistent control and stopping problems, in both continuous and discrete time, and in the context of finance applications. Intended for researchers and graduate students in the fields of finance and economics, it includes a review of the standard time-consistent results, bibliographical notes, as well as detailed examples showcasing time inconsistency problems. For the reader unacquainted with standard arbitrage theory, an appendix provides a toolbox of material needed for the book.

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Abstract

In this chapter we introduce the concept of time inconsistency in dynamic choice problems. We start by reviewing the key ideas of dynamic programming and listing the main reasons for the time consistency in a given problem. We then present a number of seemingly simple examples from financial economics in which time consistency fails to hold. To tackle these (and similar problems), we outline the different approaches developed in the literature for handling time inconsistency in a dynamic stochastic control setting. In this book, we take the game-theoretic approach and look for subgame-perfect equilibrium strategies. Additionally, we emphasize that, similar to control problems, a stopping problem can be time inconsistent if it does not admit a Bellman optimality principle.

Abstract

Although our objective is to study time-inconsistent control problems, we will in fact make use of ideas from dynamic programming in our study. In this chapter we therefore give a brief summary of standard discrete-time dynamic programming. We will give the main arguments while going lightly on some of the more technical issues, sweeping measurability and integrability issues under the carpet.

Abstract

In order to illustrate the use of dynamic programming and the Bellman equation we now consider a classical engineering problem: The linear quadratic regulator or LQR. The LQR is a well-known design technique in which a process or machine has its settings optimized by minimizing a quadratic cost function. The cost function is often defined as the sum of deviations for key properties (altitude, temperature, etc.). Applications range from underactuated robotics to nanorobotics, general operation of technical systems such as power or climate control systems, physics, statistics, and econometrics.

Abstract

In this chapter we study a time-consistent equilibrium model, namely a discrete-time simplified version of the Cox–Ingersoll–Ross continuous-time model. We start by analyzing a consumption–investment problem in which the agent considers all prices as exogenously given. We then move to the equilibrium framework, in which the short rate, the stochastic discount factor, and the martingale measure are determined endogenously in equilibrium. In Chap. 10 we will study a time-inconsistent version of this standard problem, and it will be instructive to compare the results.

Abstract

We now go on to study a class of time-inconsistent control problems. Before proceeding to the formal theory, we first recall a part of our discussion on the Bellman optimality principle from the Introduction. We then detail the game-theoretic formulation, whereby we view the problem as a non-cooperative game with one player at each point in time representing different incarnations of decision maker’s preferences. The main result of this chapter is the extension of the standard Bellman equation, in the form of a system of nonlinear equations, for the determination of the equilibrium strategy and the equilibrium value function. We arrive at the extended Bellman system by first studying a couple of simpler problems to highlight the main ideas and then by bringing things together for the general case.

Abstract

In this chapter we provide some extensions of the basic theory. In particular, we allow for a more general term in the nonlinear G part of the objective functional, study the infinite-horizon case, and consider a generalized expected utility model. Furthermore, we discuss questions of existence and uniqueness, present a useful scaling result, and provide an equivalence result between time-inconsistent and time-consistent problems.

Abstract

Problems with non-exponential discounting constitute an important subclass of the family of time-inconsistent problems. To see how the general theory works in a more concrete case, we now consider a fairly general model class with non-exponential discounting. As a special case, we will then study the case of hyperbolic discounting, and we will obtain an analytic solution for an example with logarithmic utility. We also discuss a generalization of the Euler equation for hyperbolic consumers, the so-called hyperbolic Euler equation of Harris and Laibson (Econometrica 69:935–957, 2001).

Abstract

In this chapter we study a multi-period version of the classical mean-variance problem. The mean-variance problem is a cornerstone of modern portfolio theory. An agent faces the trade-off between higher returns and higher risk, the latter measured as variance. The formulation is both simple and intuitive, and it is widely used both in financial industry and academia. Initially formulated as a static problem, it has been widely studied in a pre-committed setup or for myopic agents that optimize only for the next period, step by step. When applying the theory developed above, we can study the intertemporal hedging that arises in a fully dynamic setting.

Abstract

In this chapter we study two time-inconsistent versions of the classical linear quadratic regulator. In the first version, time inconsistency enters through a quadratic function of the expected value. In the second version, time inconsistency enters through an explicit dependence of the initial state in the quadratic term. As time passes, the final target point changes.

Abstract

In this chapter we consider the equilibrium model from Chap. 4, which is a discrete-time simplified version of the Cox–Ingersoll–Ross type of economy. We now assume, however, that the representative agent has time-inconsistent preferences. The time inconsistency enters the problem via time- and state-dependence of the agent’s utility function. Loosely speaking, when standing today and making decisions about the future, the preference ordering of the decision maker is allowed to depend on the current state and the current time. In this setting we study both the agent’s problem (intrapersonal equilibrium) and the pricing implications (market equilibrium).

Abstract

In this chapter we review the theory of dynamic programming in continuous time. This can be done within the framework of a general controlled Markov process, but in order to keep the theory reasonably concrete we restrict ourselves to the case of a controlled stochastic differential equation (SDE) driven by a finite-dimensional Wiener process. The extension to an arbitrary controlled Markov process is rather obvious and we will comment upon it later. As the reader will see, the arguments in the continuous-time case will be almost exactly the same as for the discrete-time case.

Abstract

In this chapter, we apply the dynamic programming theory to the continuous-time linear quadratic regulator problem (LQR). The LQR is a classical engineering problem and design technique in which a process has its settings optimized by minimizing a quadratic cost function. The cost function is often defined as the sum of deviations for key properties (altitude, temperature, etc.).

Abstract

In this chapter we consider a standard consumption–investment problem. In this problem, an economic agent, taking prices as given, makes decisions about how much to consume and how much to save, as well as how to allocate their savings between the available assets. (For the reader without previous experience from economic theory, Appendix A provides the necessary background on arbitrage and portfolio theory.)

Abstract

We now go on to analyze the simplest possible equilibrium model. Unlike the previous chapter, where we studied individual consumption and portfolio choices while taking prices as given, in this chapter our goal will be to solve for the equilibrium prices of assets in the economy. In particular, we will be able to derive the equilibrium risk-free interest rate, the equilibrium Girsanov kernel, and the equilibrium stochastic discount factor. See Appendix A for the necessary background in arbitrage theory.

Abstract

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.

Abstract

Here we present a couple of important special cases and comment on possible extensions of the continuous-time results that were developed in the previous chapter. In particular, we show how the extended HJB system simplifies in the special case when we have time inconsistency entering the problem only through state-dependency or only through a nonlinear function of the expected value. Results are extended to allow for infinite horizon and point process dynamics. The chapter also establishes a connection between the original time-inconsistent problem and an equivalent time-consistent problem that yields the same solution.

Abstract

We now illustrate the theory developed in Chap. 15, and the first example we consider is a fairly general case of a continuous-time control problem with non-exponential discounting. As mentioned before, non-exponential discounting captures preferences under which delayed rewards are not discounted at a constant rate. As delay discounting arises naturally in economics, understanding deviations from exponential discounting has important implications for a wide range of problems, from finance and pension economics to studies of climate change and natural resource allocation.

Abstract

In this chapter we will consider dynamic mean-variance optimization. This is a continuous-time version of a standard Markowitz investment problem, where we penalize the risk undertaken by the conditional variance. In Sect. 18.1 we first consider the simplest possible case of a Wiener driven single risky asset and re-derive the corresponding results of Basak and Chabakauri (Rev. Financ. Stud. 23:2970–3016, 2010). We then extend the model in Sect. 18.2 and study the case when the risky asset is driven by a point process as well as by a Wiener process. In Sect. 18.3, we study mean-variance portfolio choice with wealth-dependent preferences.

Abstract

In this chapter we study a simple time-inconsistent version of the linear quadratic regulator in continuous time. Time inconsistency enters through an explicit dependence on the initial state for the final quadratic term. Loosely speaking, we want to control a system such that the final state is close to the initial point while at the same time keeping the control energy small. As the starting point changes, the final target point also changes as time passes.

Abstract

In this chapter we apply the previously developed theory to a rather detailed study of a general equilibrium model for a production economy with time-inconsistent preferences. The model under consideration is a time-inconsistent version of the classic Cox–Ingersoll–Ross model in Cox et al. (Econometrica, 53:363–384, 1985). Our main objective is to investigate the structure of the equilibrium short rate, the equilibrium Girsanov kernel, and the equilibrium stochastic discount factor. See Appendix A for the necessary background from arbitrage theory.

Abstract

Optimal stopping theory studies problems that involve determining the best time to intervene and stop a process in order to maximize expected rewards or minimize expected costs. Applications of optimal stopping theory are plentiful and include asset trading (e.g., the best time to sell an asset), derivative pricing (e.g., American options), real options theory (e.g., the best time to invest in a project), economics of gambling (e.g., when to stop gambling in a casino), and search and matching (e.g., when to stop searching and accept a job). In this chapter we briefly summarize standard optimal stopping theory in discrete time.

Abstract

In this chapter we provide a brief summary of optimal stopping in continuous time. Our objective here is simply to present some of the main ideas and arguments. We will often sweep technical problems (mostly concerning regularity) under the carpet. The continuous-time case is technically a much more complicated subject than the discrete-time theory discussed in Chap. 21. We refer the reader to the literature referenced in the Notes at the end of this chapter for technical details and proofs, as well as more precise formulations.

Abstract

We now go on to study a class of time-inconsistent stopping problems in discrete time. We start by defining the concepts of Markovian stopping strategies and subgame-perfect Nash equilibrium stopping strategies. Following similar steps to those in the control case, we then proceed to derive an extension of the standard Wald–Bellman equation to a non-standard extended system that allows for the determination of the equilibrium value function and the equilibrium stopping strategy. Examples studied at the end of the chapter include a time-inconsistent version of a simple secretary problem and a procrastination problem for a time-inconsistent agent who decides when to complete a task.

Abstract

We now turn to the case of time-inconsistent stopping problems in continuous time. We start by formulating the problem and formally defining the continuous-time equilibrium concept. In order to derive the relevant extension of variational inequalities for the time-inconsistent problem, we discretize (to some extent) the continuous-time problem, use our previously derived results in discrete time, and then take the limit. We present a sketch of a verification argument establishing the connection between the extended dynamic programming system and our continuous-time equilibrium concept. Finally, we exemplify the theory with two examples of an asset-selling problem with non-exponential discounting and the mean-variance objective.

Abstract

In this chapter we study stopping strategies in the presence of distorted probabilities, in both discrete and continuous time. Probability distortion is a salient ingredient for a number of important models in behavioral economics, including cumulative prospect theory (Kahneman and Tversky (Econometrica 47:263–291, 1979), Tversky and Kahneman (Journal of Risk and Uncertainty 5:297–323, 1992)) and rank-dependent utility (Quiggin (Journal of Economic Behavior & Organization 3:323–343, 1982), Schmeidler (Econometrica 57:571–587, 1989)). Contrary to the expected utility theory, in the prospect theory model, economic agents do not weight outcomes by their objective probabilities but rather by transformed probabilities. These transformed probabilities (or decision weights) allow the model to capture economic behavior observed in experimental settings showing that people tend to overweight small probabilities and underweight large probabilities. Similarly, rank-dependent expected utility overweighs unlikely extreme outcomes. Importantly, in a dynamic context probability weighting makes the decision maker’s problem inherently time inconsistent. Mathematically, the reward functional with probability distortion involves the so-called Choquet integral (Choquet (Annales de l’Institut Fourier, 5:131–295, 1954)), instead of the conventional expectation.