A Theory of Markovian Time-inconsistent Stochastic Control in Discrete Time

Tomas Björk, Agatha Murgoci

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

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.
Original languageEnglish
JournalFinance and Stochastics
Volume18
Issue number3
Pages (from-to)545-592
ISSN0949-2984
DOIs
Publication statusPublished - 2014

Cite this

Björk, Tomas ; Murgoci, Agatha. / A Theory of Markovian Time-inconsistent Stochastic Control in Discrete Time. In: Finance and Stochastics. 2014 ; Vol. 18, No. 3. pp. 545-592.
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A Theory of Markovian Time-inconsistent Stochastic Control in Discrete Time. / Björk, Tomas; Murgoci, Agatha.

In: Finance and Stochastics, Vol. 18, No. 3, 2014, p. 545-592.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - A Theory of Markovian Time-inconsistent Stochastic Control in Discrete Time

AU - Björk, Tomas

AU - Murgoci, Agatha

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Time consistency

KW - Time inconsistency

KW - Time-inconsistent control

KW - Dynamic programming

KW - Stochastic control

KW - Bellman equation

KW - Hyperbolic discounting

KW - Mean–variance

U2 - 10.1007/s00780-014-0234-y

DO - 10.1007/s00780-014-0234-y

M3 - Journal article

VL - 18

SP - 545

EP - 592

JO - Finance and Stochastics

JF - Finance and Stochastics

SN - 0949-2984

IS - 3

ER -