Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Risk-Sensitive Stochastic Control

  • Hideo Nagai
Living reference work entry

Latest version View entry history

DOI: https://doi.org/10.1007/978-1-4471-5102-9_233-2

Abstract

Motivated by understanding “robustness” from the viewpoints of stochastic control, the studies of risk-sensitive control have been developed. The idea was applied to portfolio optimization problems in mathematical finance, from which new kinds of problem on stochastic control, named “large deviation control,” have been brought and currently the studies are in progress.

Keywords

Robustness Mathematical finance Large deviation control 
This is a preview of subscription content, log in to check access.

Bibliography

  1. Basar T, Bernhard P (1991) H – optimal control and related minimax design problems. Birkhäuser, BaselCrossRefGoogle Scholar
  2. Bensoussan A (1992) Stochastic control of partially observable systems. Cambridge University Press, Cambridge/New YorkCrossRefGoogle Scholar
  3. Bensoussan A, Nagai H (1997) Min – max characterization of a small noise limit on risk-sensitive control. SIAM J Control Optim 35:1093–1115MathSciNetCrossRefGoogle Scholar
  4. Bensoussan A, Nagai H (2000) Conditions for no breakdown and Bellman equations of risk-sensitive control. Appl Math Optim 42:91–101MathSciNetCrossRefGoogle Scholar
  5. Bensoussan A, Van Schuppen JH (1985) Optimal control of partially observable stochastic systems with an exponential-of-integral performance index. SIAM J Control Optim 23:599–613MathSciNetCrossRefGoogle Scholar
  6. Bensoussan A, Frehse J, Nagai H (1998) Some results on risk-sensitive control with full information. Appl Math Optim 37:1–41MathSciNetCrossRefGoogle Scholar
  7. Bielecki TR, Pliska SR (1999) Risk sensitive dynamic asset management. Appl Math Optim 39:337–360MathSciNetCrossRefGoogle Scholar
  8. Davis M, Lleo S (2008) Risk-sensitive benchmarked asset management. Quant Finan 8:415–426MathSciNetCrossRefGoogle Scholar
  9. Davis M, Lleo S (2015) Risk-sensitive investment management. World Scientific, SingaporezbMATHGoogle Scholar
  10. Elliott RJ, Kalton NJ (1972) The existence of value in differential games. Memoirs Am Math Soc (126)Google Scholar
  11. Fleming WH (1995) Optimal investment models and risk-sensitive stochastic control. IMA Math Appl 65:75–88zbMATHGoogle Scholar
  12. Fleming WH, McEneaney WM (1995) Risk-sensitive control on an infinite horizon. SIAM J Control Optim 33:1881–1915MathSciNetCrossRefGoogle Scholar
  13. Fleming WH, Sheu SJ (1999) Optimal long term growth rate of expected utility of wealth. Ann Appl Probab 9(3):871–903MathSciNetCrossRefGoogle Scholar
  14. Fleming WH, Sheu SJ (2002) Risk-sensitive control and an optimal investment model. II Ann Appl Probab 12(2):730–767MathSciNetCrossRefGoogle Scholar
  15. Hata H, Nagai H, Sheu SJ (2010) Asymptotics of the probability minimizing a “down-side ”risk. Ann Appl Probab 20:52–89MathSciNetCrossRefGoogle Scholar
  16. Jacobson DH (1973) Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games. IEEE Trans Auto Control 18:124–131MathSciNetCrossRefGoogle Scholar
  17. Karatzas I, Shreve SE (2010) Methods of mathematical finance. Springer, New YorkzbMATHGoogle Scholar
  18. Kelly J (1956) A new interpretation of information rate. Bell Syst Tech J 35:917–926MathSciNetCrossRefGoogle Scholar
  19. Kuroda K, Nagai H (2002) Risk sensitive portfolio optimization on infinite time horizon. Stochastics Stochastics Rep 73:309–331MathSciNetCrossRefGoogle Scholar
  20. Merton RC (1998) Continuous time finance. Blackwell, MaldenzbMATHGoogle Scholar
  21. Nagai H (1996) Bellman equations of risk-sensitive control. SIAM J Control Optim 34:74–101MathSciNetCrossRefGoogle Scholar
  22. Nagai H (1999) Risk-sensitive dynamic asset management with partial information. Stochastics in finite and infinite dimensions, a volume in honor of G. Kallianpur. Birkhäuser, Boston, pp 321–340CrossRefGoogle Scholar
  23. Nagai H (2003) Optimal strategies for risk-sensitive portfolio optimization problems for general factor models. SIAM J Control Optim 41:1779–1800MathSciNetCrossRefGoogle Scholar
  24. Nagai H (2011) Asymptotics of the probability minimizing a “down-side” risk under partial information. Quant Finan 11:789–803CrossRefGoogle Scholar
  25. Nagai H (2012) Downside risk minimization via a large deviation approach. Ann Appl Probab 22:608–669MathSciNetCrossRefGoogle Scholar
  26. Nagai H, Peng S (2002) Risk-sensitive dynamic portfolio optimization with partial information on infinite time horizon. Ann Appl Probab 12(1):173–195MathSciNetCrossRefGoogle Scholar
  27. Nagai H, Runggaldier WJ (2008) PDE approach to utility maximization for market models with hidden Markov factors. In: Dalang RC et al (eds) Seminar on stochastic analysis, random fields and applications V, Progress in probability. Birkhäser, Basel, pp 493–506CrossRefGoogle Scholar
  28. Pham H (2003) A large deviations approach to optimal long term investment. Finance Stochast 7:169–195MathSciNetCrossRefGoogle Scholar
  29. Whittle P (1981) Risk-sensitive linear/quadratic/Gaussian control. Adv Appl Probab 13:764–767MathSciNetCrossRefGoogle Scholar
  30. Whittle P (1990) A risk-sensitive maximum principle. Syst Control Lett 15:183–192MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  • Hideo Nagai
    • 1
  1. 1.Department of MathematicsKansai UniversityOsakaJapan

Section editors and affiliations

  • Lei Guo
    • 1
  1. 1.Academy of Mathematics and Systems Science, Chinese Academy of Sciences (CAS)BeijingChina