Skip to main content

Part of the book series: Springer Finance ((FINANCE))

  • 1208 Accesses

Abstract

We now move on to study time-inconsistent control problems in continuous time and, as in Part II, we use a game-theoretic framework, studying subgame-perfect Nash equilibrium strategies. It turns out, however, that the equilibrium concept in continuous time is quite delicate and so requires a detailed investigation. Our main result is an extension of the standard HJB equation to a system of equations: the extended HJB system. We also prove a verification theorem, showing that a solution to the extended HJB system does indeed deliver the equilibrium control and equilibrium value function for our original problem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Basak, S., & Chabakauri, G. (2010). Dynamic mean-variance asset allocation. Review of Financial Studies, 23, 2970–3016.

    Article  Google Scholar 

  • Björk, T., & Murgoci, A. (2014). A theory of Markovian time-inconsistent stochastic control in discrete time. Finance and Stochastics, 18, 545–592.

    Article  MathSciNet  Google Scholar 

  • Cui, X., Li, D., & Shi, Y. (2017). Self-coordination in time inconsistent stochastic decision problems: A planner–doer game framework. Journal of Economic Dynamics and Control, 75, 91–113.

    Article  MathSciNet  Google Scholar 

  • Ekeland, I., & Lazrak, A. (2006). Being serious about non-commitment: Subgame perfect equilibrium in continuous time. Working paper. Available at https://arxiv.org/abs/math/0604264

  • Ekeland, I., & Lazrak, A. (2010). The golden rule when preferences are time inconsistent. Mathematics and Financial Economics, 4, 29–55.

    Article  MathSciNet  Google Scholar 

  • Ekeland, I., & Pirvu, T. (2008). Investment and consumption without commitment. Mathematics and Financial Economics, 2(1), 57–86.

    Article  MathSciNet  Google Scholar 

  • He, X. D., & Jiang, Z. L. (2020b). On the equilibrium strategies for time-inconsistent problems in continuous time. Working paper. Available at https://ssrn.com/abstract=3308274

  • Huang, Y.-J., & Zhou, Z. (2020b). Strong and weak equilibria for time-inconsistent stochastic control in continuous time. Mathematics of Operations Research. Available at https://doi.org/10.1287/moor.2020.1066

  • Lindensjö, K. (2019). A regular equilibrium solves the extended HJB system. Operations Research Letters, 47(5), 427–432.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Björk, T., Khapko, M., Murgoci, A. (2021). Time-Inconsistent Control Theory. In: Time-Inconsistent Control Theory with Finance Applications. Springer Finance. Springer, Cham. https://doi.org/10.1007/978-3-030-81843-2_15

Download citation

Publish with us

Policies and ethics