Department of Economics and Business Economics

A theory of Markovian time-inconsistent stochastic control in discrete time

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  • Tomas Bjork, Stockholm Sch Econ, Stockholm School of Economics, Dept Finance
  • ,
  • Agatha Murgoci, Copenhagen Business School, Denmark

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
Issue3
Pages (from-to)545-592
Number of pages48
ISSN0949-2984
DOIs
Publication statusPublished - Jul 2014

    Research areas

  • Time consistency, Time inconsistency, Time-inconsistent control, Dynamic programming, Stochastic control, Bellman equation, Hyperbolic discounting, Mean-variance, GROWTH

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