Skip to main content
SpringerLink
Log in

Optimal investment under partial information

  • Published:
Mathematical Methods of Operations Research Aims and scope Submit manuscript

Abstract

We consider the problem of maximizing terminal utility in a model where asset prices are driven by Wiener processes, but where the various rates of returns are allowed to be arbitrary semimartingales. The only information available to the investor is the one generated by the asset prices and, in particular, the return processes cannot be observed directly. This leads to an optimal control problem under partial information and for the cases of power, log, and exponential utility we manage to provide a surprisingly explicit representation of the optimal terminal wealth as well as of the optimal portfolio strategy. This is done without any assumptions about the dynamical structure of the return processes. We also show how various explicit results in the existing literature are derived as special cases of the general theory.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bäuerle N, Rieder U (2004) Portfolio optimization with Markov-modulated stock prices and interest rates. IEEE Trans Automat Control 49(3): 442–447

    Article  MathSciNet  Google Scholar 

  • Bäuerle N, Rieder U (2005) Portfolio optimization with unobservable Markov-modulated drift process. J Appl Probab 42(2): 362–378

    Article  MATH  MathSciNet  Google Scholar 

  • Bäuerle N, Rieder U (2007) Portfolio optimization with jumps and unobservable intensity. Math Financ 17(2): 205–224

    Article  MATH  Google Scholar 

  • Bensoussan A (2004) Stochastic control of partially-observable systems, 2nd edn. Cambridge University Press, Cambridge

    Google Scholar 

  • Brendle S (2004) Portfolio selection under partial observation and constant absolute risk aversion, Working paper, Princeton University

  • Brendle S (2006) Portfolio selection under incomplete information. Stoch Process Appl 116(5): 701–723

    Article  MATH  MathSciNet  Google Scholar 

  • Brennan M (1998) The role of learning in dynamic portfolio decisions. Eur Financ Rev 1:295–306

    Article  Google Scholar 

  • Brennan M, Xia Y (2001) Assessing asset pricing anomalies. Rev Financ Stud 14(4): 905–942

    Article  Google Scholar 

  • Callegaro G, Di Masi G, Runggaldier W (2006) Portfolio optimization in discontinuous markets under incomplete ionformation. Asia-Pac Financ Mark 13: 373–394

    Article  MATH  Google Scholar 

  • Cvitanic J, Lazrak A, Martinelli L, Zapatero F (2006) Dynamic portfolio choice with parameter uncertainty and the economic value of analysts’ recommendations. Rev Financ Stud 19: 1113–1156

    Article  Google Scholar 

  • Da Prato G, Zabzcyk J (1996) Ergodicity for infinite dimensional systems. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Dana R, Jeanblanc M (1992) Financial markets in continuous time. Cambridge University Press, Cambridge

    Google Scholar 

  • Dothan M, Feldman D (1986) Equilibrium interest rates and multiperiod bonds in a partially observable economy. J Financ 41(2): 369–382

    Article  Google Scholar 

  • Feldman D (1989) The term structure of interest rates in a partially observed economy. J Financ 44: 789–812

    Article  Google Scholar 

  • Feldman D (1992) Logarithmic preferences, myopic decisions, and incomplete information. J Financ Quant Anal 27: 619–629

    Article  Google Scholar 

  • Feldman D (2003) Production and the real rate of interest: a sample path equilibrium. Rev Financ 7: 247–275

    Google Scholar 

  • Feldman D (2007) Incomplete information equilibria: separation theorems and other myths. Ann Oper Res 151: 119–149

    Article  MATH  MathSciNet  Google Scholar 

  • Fujisaki M, Kallianpur G, Kunita H (1972) Stochastic differential equations for the non linear filtering problem. Osaka J Math 9: 19–40

    MATH  MathSciNet  Google Scholar 

  • Gennotte G (1986) Optimal portfolio choice under incomplete information. J Financ 41: 733–749

    Article  Google Scholar 

  • Haussmann UG, Sass J (2004a) Optimal terminal wealth under partial information for HMM stock returns. In: Mathematics of finance (Contemp. Math. 351). AMS

  • Haussmann UG, Sass J (2004) Optimizing the terminal wealth under partial information: the drift process as a continuous time Markov chain. Financ Stoch 8: 553–577

    MATH  MathSciNet  Google Scholar 

  • Honda T (2003) Optimal portfolio choice for unobservable and regime-switching mean returns. J Econ Dyn Control 28: 45–78

    Article  MATH  MathSciNet  Google Scholar 

  • Karatzas I, Lehoczky J, Shreve S (1987) Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM J Control Optim 25: 1557–1586

    Article  MATH  MathSciNet  Google Scholar 

  • Karatzas I, Shreve S (1998) Methods of mathematical finance. Springer, New York

    MATH  Google Scholar 

  • Lakner P (1995) Utility maximization with partial information. Stoch Process Appl 56: 247–249

    Article  MATH  MathSciNet  Google Scholar 

  • Lakner P (1998) Optimal trading strategy for an investor: the case of partial information. Stoch Process Appl 76: 77–97

    Article  MATH  MathSciNet  Google Scholar 

  • Liptser R, Shiryayev A (2004) Statistics of random processes vol. I, 2nd edn. Springer, New York

    Google Scholar 

  • Nagai H, Runggaldier W (2006) PDE approach to utility maximization for market models with hidden Markov factors. In: 5th Seminar on stochastic analysis, random fields and applications. Birkhauser Verlag

  • Rogers L, Williams D (1987) Diffusions, Markov processes and Martingales vol. 2. Wiley, New York

    Google Scholar 

  • Sass J (2007) Utility maximization with convex constraints and partial information. Acta Appl Math 97: 221–238

    Article  MATH  MathSciNet  Google Scholar 

  • Sass J, Wunderlich R (2009) Optimal portfolio policies under bounded expected loss and partial information, Working paper

  • Xia Y (2001) Learning about predictability: the effects of parameter uncertainty on dynamic asset allocation. J Financ 56: 205–246

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Finance, Stockholm School of Economics, Box 6501, 113 83, Stockholm, Sweden

    Tomas Björk

  2. Department of Mathematics, Imperial College, SW7 2AZ, London, England

    Mark H. A. Davis

  3. Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden

    Camilla Landén

Authors
  1. Tomas Björk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark H. A. Davis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Camilla Landén
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tomas Björk.

Additional information

Support from the Tom Hedelius and Jan Wallander Foundation is gratefully acknowledged. The authors are very grateful to the Associate editor and two anonymous referees for a number of very helpful comments.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Björk, T., Davis, M.H.A. & Landén, C. Optimal investment under partial information. Math Meth Oper Res 71, 371–399 (2010). https://doi.org/10.1007/s00186-010-0301-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-010-0301-x

Keywords

Mathematics Subject Classification (2000)

Search

Navigation