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

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Authors:
  • Björk, Tomas ;
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    Stockholm School of Economics
  • Murgoci, Agatha
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    Department of Finance, Copenhagen Business School
DOI:
10.1007/s00780-014-0234-y
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.
Type:
Journal article
Language:
English
Published in:
Finance and Stochastics, 2014, Vol 18, Issue 3, p. 545-592
Keywords:
Time consistency; Time inconsistency; Time-inconsistent control; Dynamic programming; Stochastic control; Bellman equation; Hyperbolic discounting; Mean–variance; Mean-variance; GROWTH
Main Research Area:
Social science
Publication Status:
Published
Review type:
Peer Review
Submission year:
2014
Scientific Level:
Scientific
ID:
269206738

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