# 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

**References:** View references in EconPapers View complete reference list from CitEc

**Citations:** View citations in EconPapers (16) Track citations by RSS feed

**Downloads:** (external link)

http://hdl.handle.net/10.1007/s00780-014-0234-y (text/html)

Access to full text is restricted to subscribers.

**Related works:**

This item may be available elsewhere in EconPapers: Search for items with the same title.

**Export reference:** BibTeX
RIS (EndNote, ProCite, RefMan)
HTML/Text

**Persistent link:** https://EconPapers.repec.org/RePEc:spr:finsto:v:18:y:2014:i:3:p:545-592

**Ordering information:** This journal article can be ordered from

http://www.springer. ... ance/journal/780/PS2

Access Statistics for this article

Finance and Stochastics is currently edited by *M. Schweizer*

More articles in Finance and Stochastics from Springer

Bibliographic data for series maintained by Sonal Shukla ().