Optimal Control

  • Ernst Eberlein
  • Jan Kallsen
Part of the Springer Finance book series (FINANCE)


Dynamic stochastic optimisation problems play an important role in Mathematical Finance and other applications. In this chapter we provide basic tools for their mathematical treatment in continuous time.


  1. 14.
    B. Bassan, C. Ceci, Regularity of the value function and viscosity solutions in optimal stopping problems for general Markov processes. Stochastics Stochastics Rep. 74(3–4), 633–649 (2002)MathSciNetCrossRefGoogle Scholar
  2. 28.
    J.-M. Bismut, An introductory approach to duality in optimal stochastic control. SIAM Rev. 20(1), 62–78 (1978)MathSciNetCrossRefGoogle Scholar
  3. 76.
    C. Dellacherie, P.-A. Meyer, Probabilities and Potential B: Theory of Martingales (North-Holland, Amsterdam, 1982)zbMATHGoogle Scholar
  4. 96.
    N. El Karoui, Les aspects probabilistes du contrôle stochastique, in Ninth Saint Flour Probability Summer School—1979 (Saint Flour, 1979), volume 876 of Lecture Notes in Math. (Springer, Berlin, 1981), pp. 73–238Google Scholar
  5. 112.
    W. Fleming, M. Soner, Controlled Markov Processes and Viscosity Solutions, 2nd edn. (Springer, New York, 2006)zbMATHGoogle Scholar
  6. 115.
    H. Föllmer, P. Protter, Local martingales and filtration shrinkage. ESAIM Probab. Stat. 15(In honor of Marc Yor, suppl.), S25–S38 (2011)Google Scholar
  7. 126.
    T. Goll, J. Kallsen, A complete explicit solution to the log-optimal portfolio problem. Ann. Appl. Probab. 13(2), 774–799 (2003)MathSciNetCrossRefGoogle Scholar
  8. 135.
    M. Haugh, L. Kogan, Pricing American options: a duality approach. Oper. Res. 52(2), 258–270 (2004)MathSciNetCrossRefGoogle Scholar
  9. 154.
    J. Jacod, A. Shiryaev, Limit Theorems for Stochastic Processes, 2nd edn. (Springer, Berlin, 2003)CrossRefGoogle Scholar
  10. 206.
    M. Lenga, Representable Options. PhD thesis, Kiel University, 2017Google Scholar
  11. 217.
    R. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case. Rev. Econ. Stat. 51(3), 247–257 (1969)CrossRefGoogle Scholar
  12. 218.
    R. Merton, Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3(4), 373–413 (1971)MathSciNetCrossRefGoogle Scholar
  13. 219.
    R. Merton, Theory of rational option pricing. Bell J. Econom. Manag. Sci. 4, 141–183 (1973)MathSciNetCrossRefGoogle Scholar
  14. 226.
    B. Øksendal, Stochastic Differential Equations, 6th edn. (Springer, Berlin, 2003)CrossRefGoogle Scholar
  15. 227.
    B. Øksendal, A. Sulem, Applied Stochastic Control of Jump Diffusions, 2nd edn. (Springer, Berlin, 2007)CrossRefGoogle Scholar
  16. 232.
    G. Peskir, A. Shiryaev, Optimal Stopping and Free-Boundary Problems (Birkhäuser, Basel, 2006)Google Scholar
  17. 233.
    H. Pham, On quadratic hedging in continuous time. Math. Methods Oper. Res. 51(2), 315–339 (2000)MathSciNetCrossRefGoogle Scholar
  18. 234.
    H. Pham, Continuous-Time Stochastic Control and Optimization with Financial Applications (Springer, Berlin, 2009)CrossRefGoogle Scholar
  19. 249.
    C. Rogers, Monte Carlo valuation of American options. Math. Finance 12(3), 271–286 (2002)MathSciNetCrossRefGoogle Scholar
  20. 260.
    W. Schachermayer, M. Sîrbu, E. Taflin, In which financial markets do mutual fund theorems hold true? Finance Stochast. 13(1), 49–77 (2009)MathSciNetCrossRefGoogle Scholar
  21. 270.
    M. Schweizer, A guided tour through quadratic hedging approaches. Option Pricing, Interest Rates and Risk Management (Cambridge Univ. Press, Cambridge, 2001), pp. 538–574Google Scholar
  22. 271.
    A. Seierstad, Stochastic Control in Discrete and Continuous Time (Springer, New York, 2009)CrossRefGoogle Scholar
  23. 287.
    N. Touzi, Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE (Springer, New York, 2013)CrossRefGoogle Scholar
  24. 295.
    J. Yong, X. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations (Springer, New York, 1999)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ernst Eberlein
    • 1
  • Jan Kallsen
    • 2
  1. 1.Department of Mathematical StochasticsUniversity of FreiburgFreiburg im BreisgauGermany
  2. 2.Department of MathematicsKiel UniversityKielGermany

Personalised recommendations