Optimal investment-consumption-insurance with partial information

  • Hiroaki HataEmail author
Original Paper


We consider an optimal investment, consumption, and life insurance purchase problem for a wage earner. We treat a stochastic factor model that the mean returns of risky assets depend linearly on underlying economic factors formulated as the solutions of linear stochastic differential equations. We discuss the partial information case that the wage earner can not observe the factor process and use only past information of risky assets. Then, our problem is formulated as a stochastic control problem with partial information. Applying the dynamic programming principle, we derive a coupled system of the Hamilton–Jacobi–Bellman (HJB) equation and two backward stochastic differential equations (BSDEs), and obtain the explicit solution. Finally, we strictly prove the verification theorem, and construct the optimal investment-consumption-insurance strategy.


Optimal investment-consumption-insurance HARA utility Stochastic factor model Partial information Hamilton–Jacobi–Bellman equation Backward stochastic differential equation 

Mathematics Subject Classification

49L20 91E28 91E30 93E11 93E20 60H10 60H30 



The authors would like to thank the referees for helpful comments and suggestions.


  1. 1.
    Bensoussan, A.: Stochastic Control of Partially Observable Systems. Cambridge University Press, Cambridge (1992)zbMATHCrossRefGoogle Scholar
  2. 2.
    Bucy, R.S., Joseph, P.D.: Filtering for Stochastic Processes with Applications to Guidance. Chelsea, New York (1987)zbMATHGoogle Scholar
  3. 3.
    Bielecki, T.R., Pliska, S.R.: Risk sensitive dynamic asset management. Appl. Math. Optim. 39, 337–360 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bielecki, T.R., Pliska, S.R.: Risk sensitive Intertemporal CAPM, with application to fixed-income management. IEEE Trans. Auto. Cont. 49(3), 420–432 (2004)zbMATHCrossRefGoogle Scholar
  5. 5.
    Brendle, S.: Portfolio selection under incomplete information. Stoch. Process. Appl. 116, 701–723 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cohen, S.N., Elliott, R.J.: Stochastic Calculus and Applications, 2nd edn. Springer, Cham (2015)zbMATHCrossRefGoogle Scholar
  7. 7.
    Detemple, J.B.: Asset pricing in a production economy with incomplete information. J. Financ. 41(2), 383–391 (1986)CrossRefGoogle Scholar
  8. 8.
    Dokuchaev, N.: Optimal solution of investment problems via linear parabolic equations generated by Kalman filter. SIAM J. Cont. Optim. 44, 1239–1258 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Duarte, I., Pinheiro, D., Pinto, A.A., Pliska, S.R.: Optimal life insurance purchase, consumption and investment on a financial market with multi-dimensional diffusive terms. Optimization 63(11), 1737–1760 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Fleming, W.H., Rishel, R.: Deterministic and Stochastic Optimal Control. Springer, Berlin (1975)zbMATHCrossRefGoogle Scholar
  11. 11.
    Fleming, W.H., Sheu, S.J.: Optimal long term growth rate of expected utility of wealth. Ann. Appl. Probab. 9(3), 871–903 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Fleming, W.H., Sheu, S.J.: Risk-sensitive control and an optimal investment model. Math. Financ. 10(2), 197–213 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fleming, W.H., Sheu, S.J.: Risk-sensitive control and an optimal investment model. II. Ann. Appl. Probab. 12(2), 730–767 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Fouque, J.P., Papanicolaou, G., Sircar, K.R.: Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press (2000)Google Scholar
  15. 15.
    Gennotte, G.: Optimal portfolio choice under incomplete information. J. Financ. 41(3), 733–746 (1986)CrossRefGoogle Scholar
  16. 16.
    Guambe, C., Kufakunesu, R.: A note on optimal investment-consumption-insurance in a Lévy market. Insur. Math. Econ. 65(1), 30–36 (2015)zbMATHCrossRefGoogle Scholar
  17. 17.
    Hata, H., Nagai, H., Sheu, S.J.: An optimal consumption problem for general factor models. SIAM J. Cont. Optim. 56(5), 3149–3183 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Hata, H., Nagai, H., Sheu, S.J.: Asymptotics of the probability minimizing a “Down-side” risk. Ann. Appl. Probab. 20(1), 52–89 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Hata, H., Sekine, J.: Solving long term optimal investment problems with Cox-Ingersoll-Ross interest rates. Adv. Math. Econ. 8, 231–255 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Hata, H., Sekine, J.: Explicit solution to a certain Non-ELQG risk-sensitive stochastic control problem. Appl. Math. Optim. 62, 341–380 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hata, H., Sheu, S.J.: On the Hamilton–Jacobi–Bellman equation for an optimal consumption problem: II. Verification Theorem. SIAM J. Cont. Optim. 50(4), 2401–2430 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Hata, H., Sheu, S.J.: An optimal consumption and investment problem with partial information. Adv. Appl. Probab. 50(1), 131–153 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Huang, H., Milevsky, M.A.: Portfolio choice and mortality-contingent claims: the general HARA case. J. Bank. Financ. 32, 2444–2452 (2008)CrossRefGoogle Scholar
  24. 24.
    Kuroda, K., Nagai, H.: Risk sensitive portfolio optimization infinite time horizon. Stoch. Stoch. Rep. 73, 309–331 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Liu, J.: Portfolio selection in stochastic environments. Rev. Financ. Stud. 20, 1–39 (2007)CrossRefGoogle Scholar
  26. 26.
    Lakner, P.: Utility maximization with partial information. Stoch. Process. Appl. 56, 247–273 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Lakner, P.: Optimal trading strategy for an investor: the case of partial information. Stoch. Process. Appl. 76, 77–97 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Liptser, R.S., Shiryaev, A.N.: Statistics of Random Processes, I. General Theory, 2nd edn. Springer, New York (2001)zbMATHCrossRefGoogle Scholar
  29. 29.
    Merton, R.C.: Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory 3, 373–413 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Nagai, H.: Risk-sensitive dynamic asset management with partial information. In: Hida, T., Karandikar, R.L., Kunita, H., Rajput, B.S., Watanabe, S. (eds.) Stochastics in Finite and Infinite Dimensions, in Honor of Gopinath Kallianpur, pp. 321–339. Birkháuser, Boston (2001)CrossRefGoogle Scholar
  31. 31.
    Nagai, H., Peng, S.: Risk-sensitive dynamic portfolio optimization with partial information on infinite time horizon. Ann. Appl. Probab. 12(1), 173–195 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Putschögl, W., Sass, J.: Optimal consumption and investment under partial information. Decis. Econ. Financ. 31, 137–170 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Pliska, S., Ye, J.: Optimal life insurance purchase and consumption/investment under uncertain lifetime. J. Bank. Financ. 31(5), 1307–1319 (2007)CrossRefGoogle Scholar
  34. 34.
    Richard, S.: Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model. J. Financ. Ecom. 2, 187–203 (1975)CrossRefGoogle Scholar
  35. 35.
    Rishel, R.: Optimal portfolio management with partial observation and power utility function. In: Stochastic Analysis, Control, Optimization and Applications, pp. 605–619 . Birkhäuser, Basel (1999)zbMATHCrossRefGoogle Scholar
  36. 36.
    Shen, Y., Wei, J.: Optimal investment-consumption-insurance with random parameters. Scand. Actuar. J. 2018(1), 37–62 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Wachter, J.A.: Portfolio and consumption decisions under mean-reverting returns: an exact solution for complete markets. J. Financ. Quant. Anal. 37, 63–91 (2002)CrossRefGoogle Scholar
  38. 38.
    Zhang, J.: Backward stochastic differential equations. From linear to fully nonlinear theory. Probability Theory and Stochastic Modelling, vol. 86. Springer, New York (2017)CrossRefGoogle Scholar
  39. 39.
    Zohar, G.: A generalized Cameron–Martin formula with applications to partial observed dynamic portfolio optimization. Math. Financ. 11, 475–494 (2015)zbMATHCrossRefGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of EducationShizuoka UniversityShizuokaJapan

Personalised recommendations