Abstract
In this paper, which is a continuation of the discrete-time paper (Björk and Murgoci in Finance Stoch. 18:545–592, 2004), we study a class of 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 nonlinear equations, for the determination of the equilibrium strategy as well as the equilibrium value function. The main theoretical result is a verification theorem. As an application of the general theory, we study 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 (Cox et al. in Econometrica 53:363–384, 1985).
Highlights
The purpose of this paper is to study a class of stochastic control problems in continuous time which have the property of being time-inconsistent in the sense that they do not allow a Bellman optimality principle
We attack a fairly general class of timeinconsistent problems by using a game-theoretic approach; so instead of searching for optimal strategies, we search for subgame perfect Nash equilibrium strategies
– In Sect. 6, the results of Sect. 5 are extended to a more general reward functional. – Section 7 treats the infinite-horizon case. – In Sect. 8, we study a time-inconsistent version of the linear-quadratic regulator to illustrate how the theory works in a concrete case. – Section 9 is devoted to a rather detailed study of a general equilibrium model for a production economy with time-inconsistent preferences. – In Sect. 10, we review some remaining open problems
Summary
The purpose of this paper is to study a class of stochastic control problems in continuous time which have the property of being time-inconsistent in the sense that they do not allow a Bellman optimality principle. The very concept of optimality becomes problematic, since a strategy which is optimal given a specific starting point in time and space may be non-optimal when viewed from a later date and a different state. We attack a fairly general class of timeinconsistent problems by using a game-theoretic approach; so instead of searching for optimal strategies, we search for subgame perfect Nash equilibrium strategies. The paper presents a continuous-time version of the discrete-time theory developed in our previous paper [5]. Since we build heavily on the discrete-time paper, the reader is referred to that for motivating examples and more detailed discussions on conceptual issues
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