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Explicit Solution to a Certain Non-ELQG Risk-sensitive Stochastic Control Problem

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Abstract

A risk-sensitive stochastic control problem with finite/infinite horizon is studied with a 1-dimensional controlled process defined by a linear SDE with a linear control-term in the drift. In the criterion function, a non-linear/quadratic term is introduced by using the solution to a Riccati differential equation, and hence, the problem is not ELQG (Exponential Linear Quadratic Gaussian) in general. For the problem, optimal value and control are calculated in explicit forms and the set of admissible risk-sensitive parameters is given in a concrete form. As applications, two types of large deviations control problems, i.e., maximizing an upside large deviations probability and minimizing a downside large deviations probability, are mentioned.

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References

  1. Bensoussan, A.: Stochastic Control of Partially Observable Systems. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  2. Bensoussan, A., Frehse, J., Nagai, H.: Some results on risk-sensitive control with full observation. Appl. Math. Optim. 37(1), 1–40 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bensoussan, A., van Schuppen, J.H.: Optimal control of partially observable stochastic systems with an exponential-of-integral performance index. SIAM J. Control Optim. 23(4), 599–613 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bielecki, T.R., Pliska, S.R.: Risk sensitive dynamic asset management. Appl. Math. Optim. 39, 337–360 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bielecki, T.R., Pliska, S.R.: Risk sensitive Intertemporal CAPM, with application to fixed-income management. IEEE Trans. Automat. Contr. 49(3), 420–432 (2004) (special issue on stochastic control methods in financial engineering)

    Article  MathSciNet  Google Scholar 

  6. Brendle, S.: Portfolio selection under incomplete information. Stoch. Proc. Appl. 116(5), 701–723 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)

    MATH  Google Scholar 

  8. Davis, M.H.A., Lleo, S.: Risk-sensitive benchmarked asset management. Quant. Finance 8(4), 415–426 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fleming, W.H., Sheu, S.J.: Optimal long term growth rate of expected utility of wealth. Ann. Appl. Probab. 9(3), 871–903 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fleming, W.H., Sheu, S.J.: Risk-sensitive control and an optimal investment model. Math. Finance 10(2), 197–213 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fleming, W.H., Sheu, S.J.: Risk-sensitive control and an optimal investment model. II. Ann. Appl. Probab. 12(2), 730–767 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Florens-Landais, D., Pham, P.: Large deviations in estimation of an Ornstein-Uhlenbeck model. J. Appl. Probab. 36, 60–77 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hata, H., Iida, Y.: A risk-sensitive stochastic control approach to an optimal investment problem with partial information. Finance Stoch. 10(3), 395–426 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hata, H., Nagai, H., Sheu, S.J.: Asymptotics of the probability minimizing a “down-side” risk. Ann. Appl. Probab. 20(1), 52–89 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hata, H., Sekine, J.: Solving long term optimal investment problems with Cox-Ingersoll-Ross interest rates. Adv. Math. Econ. 8, 231–255 (2005)

    Article  Google Scholar 

  16. Inoue, A., Nakano, Y.: Optimal long-term investment model with memory. Appl. Math. Optim. 55(1), 93–122 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Jacobson, D.H.: Optimal stochastic linear systems with exponential criteria and their relation to deterministic differential games. IEEE Trans. Automat. Contr. AC-18, 124–131 (1973)

    Article  Google Scholar 

  18. Kac, M.: On some connections between probability theory and differential and integral equations. In: Neyman, J. (ed.) 2nd Berkley Symposium on Mathematical Statistics and Probability, pp. 189–215. University of California Press, Berkeley (1951)

    Google Scholar 

  19. Kaise, H., Sheu, S.J.: Risk sensitive optimal investment: solutions of the dynamical programming equation. In: Mathematics of Finance. Contemp. Math., vol. 351, pp. 217–230. Am. Math. Soc., Providence (2004)

    Google Scholar 

  20. Kaise, H., Sheu, S.J.: On the structure of solutions of ergodic type Bellman equations related to risk-sensitive control. Ann. Probab. 34(1), 284–320 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Karatzas, I.: On a stochastic representation for the principal eigenvalue of a second-order differential equation. Stochastics 3, 305–321 (1980)

    MATH  MathSciNet  Google Scholar 

  22. Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin (1991)

    MATH  Google Scholar 

  23. Kumar, P.R., van Schuppen, J.H.: On the optimal control of stochastic systems with an exponential-of-integral performance index. J. Math. Anal. Appl. 11, 312–332 (1981)

    Article  Google Scholar 

  24. Kuroda, K., Nagai, H.: Risk sensitive portfolio optimization on infinite time horizon. Stoch. Stoch. Rep. 73, 309–331 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Nagai, H.: Bellman equations of risk-sensitive control. SIAM J. Control. Optim. 37, 74–101 (1996)

    Article  MathSciNet  Google Scholar 

  26. Nagai, H.: Optimal strategies for risk-sensitive portfolio optimization problems for general factor models. SIAM J. Control. Optim. 41, 1779–1800 (2003)

    Article  MathSciNet  Google Scholar 

  27. Nagai, H.: Asymptotics of probability minimizing a “down-side” risk under partial information. Quant. Finance (2008, to appear). doi:10.1080/14697680903341814

  28. Nagai, H.: Down-side risk minimization as large deviation control. Preprint (2008)

  29. Nagai, H., Peng, S.: Risk-sensitive dynamic portfolio optimization with partial information on infinite time horizon. Ann. Appl. Probab. 12(1), 173–195 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  30. Ocone, D.: Asymptotic stability of Beneš filters. Stoch. Anal. Appl. 17, 1053–1074 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  31. Pham, H.: A large deviations approach to optimal long term investment. Finance Stoch. 7, 169–195 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  32. Pham, H.: A risk-sensitive control dual approach to a large deviations control problem. Syst. Control Lett. 49, 295–309 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Reed, M., Simon, B.: Methods of Modern Mathematics Physics IV. Academic Press, San Diego (1978)

    Google Scholar 

  34. Rishel, R.: Optimal portfolio management with partial observation and power utility function. Stoch. Anal. Control Optim. Appl. 605–619 (1999), a volume in honor of W.H. Fleming

  35. Sekine, J.: A note on long-term optimal portfolios under drawdown constraints. Adv. Appl. Probab. 38, 673–692 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. Sekine, J.: On a large deviations control for a linear-quadratic model: the complete dual solution. In: Proceedings of 4th JSIAM-SIMAI Meeting. Gakuto International Series, Mathematica Sciences and Application, vol. 28, pp. 322–333. Gakkatosho, Tokyo (2008)

    Google Scholar 

  37. Speyer, J.L., Deyst, J., Jacobson, D.H.: Optimisation of stochastic linear systems with additive measurement and process noise using exponential performance criteria. IEEE Trans. Automat. Contr. 19, 358–366 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  38. Whittle, P.: Risk-sensitive linear/quadratic/Gaussian control. Adv. Appl. Probab. 13, 764–777 (1981)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Academia Sinica, No. 1, Section 4, Roosevelt Road, Taipei, 106-17, Taiwan

    Hiroaki Hata

  2. Institute of Economic Research, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, 606-8501, Japan

    Jun Sekine

Authors
  1. Hiroaki Hata
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  2. Jun Sekine
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Correspondence to Jun Sekine.

Additional information

Communicating Editor: Hideo Nagai.

Jun Sekine’s research is supported by Grant-in-Aid for Scientific Research (C), No. 20540115, MEXT.

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Hata, H., Sekine, J. Explicit Solution to a Certain Non-ELQG Risk-sensitive Stochastic Control Problem. Appl Math Optim 62, 341–380 (2010). https://doi.org/10.1007/s00245-010-9106-9

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