Time Inconsistent Stochastic Control in Continuous Time: Theory and Examples

58 Pages Posted: 19 Dec 2016 Last revised: 16 Mar 2019

See all articles by Tomas Bjork

Tomas Bjork

Stockholm School of Economics - Swedish House of Finance

Mariana Khapko

University of Toronto - Finance Area; Swedish House of Finance

Agatha Murgoci

Aarhus University - School of Business and Social Sciences

Date Written: December 19, 2016

Abstract

In this paper, which is a continuation of a previous 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.

Keywords: Time Consistency, Time Inconsistency, Time Inconsistent Control, Dynamic Programming, Equilibrium

JEL Classification: C61, C72, C73, G11

Suggested Citation

Bjork, Tomas and Khapko, Mariana and Murgoci, Agatha, Time Inconsistent Stochastic Control in Continuous Time: Theory and Examples (December 19, 2016). Finance Stochastics, Forthcoming, Available at SSRN: https://ssrn.com/abstract=2887328

Tomas Bjork (Contact Author)

Stockholm School of Economics - Swedish House of Finance ( email )

Drottninggatan 98
111 60 Stockholm
Sweden

Mariana Khapko

University of Toronto - Finance Area ( email )

Toronto, Ontario M5S 3E6
Canada

Swedish House of Finance ( email )

Drottninggatan 98
Stockholm
Sweden

Agatha Murgoci

Aarhus University - School of Business and Social Sciences ( email )

Nordre Ringgade 1
Aarhus C, DK-8000
Denmark

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