The Annals of Applied Probability

Risk-sensitive control and an optimal investment model II

W. H. Fleming and S. J. Sheu

Full-text: Open access

Abstract

We consider an optimal investment problem proposed by Bielecki and Pliska. The goal of the investment problem is to optimize the long-term growth of expected utility of wealth. We consider HARA utility functions with exponent $-\infty< \gamma< 1$. The problem can be reformulated as an infinite time horizon risk-sensitive control problem. Some useful ideas and results from the theory of risk-sensitive control can be used in the analysis. Especially, we analyze the associated dynamical programming equation. Then an optimal (or approximately optimal) Markovian investment policy can be derived.

Article information

Source
Ann. Appl. Probab. Volume 12, Number 2 (2002), 730-767.

Dates
First available: 17 July 2002

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1026915623

Digital Object Identifier
doi:10.1214/aoap/1026915623

Mathematical Reviews number (MathSciNet)
MR1910647

Zentralblatt MATH identifier
01906173

Subjects
Primary: 90A09 93E20: Optimal stochastic control
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
Risk-sensitive stochastic control optimal investment model long-term growth rate dynamical programming equation Ricatti equation

Citation

Fleming, W. H.; Sheu, S. J. Risk-sensitive control and an optimal investment model II. The Annals of Applied Probability 12 (2002), no. 2, 730--767. doi:10.1214/aoap/1026915623. http://projecteuclid.org/euclid.aoap/1026915623.


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References

  • BENSOUSSAN, A. and FREHSE, J. (1992). On Bellman equation of ergodic control in Rn. J. Reine Angew. Math. 429 125-160.
  • BIELECKI, T. R. and PLISKA, S. R. (1999). Risk sensitive dy namic asset management. Appl. Math. Optim. 39 337-360.
  • BIELECKI, T. R. and PLISKA, S. R. (2000). Risk sensitive intertemporal CAPM, with application to fixed income management. Preprint.
  • BIELECKI, T. R., PLISKA, S. R. and SHERRIS, M. (2000). Risk sensitive asset allocation. J. Econom. Dy nam. Control 24 1145-1177.
  • BRENNAN, M. J., SCHWARTZ, E. S. and LAGNADO, R. (1997). Strategic asset allocation. J. Econom. Dy namics Control 21 1377-1403.
  • BROCKETT, R. W. (1970). Finite Dimensional Linear Sy stems. Wiley, New York.
  • CVITANIC, J. and KARATZAS, I. (1995). On portfolio optimization under "drawdown" constraints. IMA Vol. Math. Appl. 65 35-45.
  • FLEMING, W. H. (1995). Optimal investment models and risk-sensitive stochastic control. IMA Vol. Math. Appl. 65 75-88.
  • FLEMING, W. H., GROSSMAN, S. G., VILA, J.-L. and ZARIPHOPOULOU, T. (1990). Optimal portfolio rebalancing with transaction costs. Preprint.
  • FLEMING, W. H. and JAMES, M. R. (1995). The risk-sensitive index and the H2 and H norms for nonlinear sy stems. Math. Control Signals Sy stems 8 199-221.
  • FLEMING, W. H. and MCENEANEY, W. M. (1995). Risk-sensitive control on an infinite time horizon. SIAM J. Control Optim. 33 1881-1915.
  • FLEMING, W. H. and RISHEL, R. W. (1975). Deterministic and Stochastic Optimal Control. Springer, New York.
  • FLEMING, W. H. and SHEU, S. J. (1997). Asy mptotics for the principal eigenvalue and eigenfunction for a nearly first-order operator with large potential. Ann. Probab. 25 1953-1994.
  • FLEMING, W. H. and SHEU, S. J. (1999). Optimal long term growth rate of expected utility of wealth. Ann. Appl. Probab. 9 871-903.
  • FLEMING, W. H. and SHEU, S. J. (2000). Risk sensitive control and an optimal investment model. Math. Finance 10 197-213.
  • FLEMING, W. H. and SONER, H. M. (1992). Controlled Markov Processes and Viscosity Solutions. Springer, New York.
  • KHASMINSKII, R. Z. (1980). Stochastic Stability of Differential Equations. Sijhoff and Noordhoff, Alphen aan der Rijn.
  • KONNO, H., PLISKA, S. R. and SUZUKI, R. I. (1993). Optimal portfolios with asy mptotic criteria. Ann. Oper. Res. 45 187-204.
  • KUCERA, V. (1972). A contribution to matrix quadratic equation. IEEE Trans. Automat. Control 17 344-347.
  • KURODA, K. and NAGAI, H. (2000). Risk-sensitive portfolio optimization on infinite-time horizon. Preprint.
  • LIPTSER, R. S. and SHIRy AYEV, A. N. (1977). Statistics of Random Processes I. Springer, New York.
  • MCENEANEY, W. M. (1993). Connections between risk-sensitive stochastic control, differential games and H-infinite control: The nonlinear case. Ph.D. dissertation, Brown Univ.
  • MCENEANEY, W. M. (1995). Uniqueness for viscosity solutions of nonstationary Hamilton-Jacobi- Bellman equations under some a priori conditions (with application). SIAM J. Control Optim. 33 1560-1576.
  • NAGAI, H. (1996). Bellman equations of risk sensitive control. SIAM J. Control Optim. 34 74-101.
  • PLATEN, E. and REBOLLEDO, R. (1996). Principles for modelling financial markets. J. Appl. Probab. 33 601-603.
  • WHITTLE, P. (1990). Risk-Sensitive Optimal Control. Wiley, New York.
  • WILLEMS, J. C. (1971). Least squares stationary optimal control and the algebraic Riccati equation. IEEE Trans. Automat. Control 16 621-635.
  • WONHAM, W. M. (1968). On a matrix Riccati equation of stochastic control. SIAM J. Control Optim. 6 681-697.
  • PROVIDENCE, RI 02912 E-MAIL: whf@cfm.brown.edu INSTITUTE OF MATHEMATICS ACADEMIA SINICA
  • NANKANG, TAIPEI TAIWAN REPUBLIC OF CHINA E-MAIL: sheusj@math.sinica.edu.tw