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

Tomas Björk () and Agatha Murgoci ()

Finance and Stochastics, 2014, vol. 18, issue 3, 545-592

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. Copyright Springer-Verlag Berlin Heidelberg 2014

Keywords: Time consistency; Time inconsistency; Time-inconsistent control; Dynamic programming; Stochastic control; Bellman equation; Hyperbolic discounting; Mean–variance; 49L20; 49L99; 60J05; 60J20; 91A10; 91G10; 91G80; C61; C72; C73; G11 (search for similar items in EconPapers)
Date: 2014
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