Stochastic Partial Differential Equations and Portfolio Choice


We introduce a stochastic partial differential equation which describes the evolution of the investment performance process in portfolio choice models. The equation is derived for two formulations of the investment problem, namely, the traditional one (based on maximal expected utility of terminal wealth) and the recently developed forward formulation. The novel element in the forward case is the volatility process which is up to the investor to choose. We provide various examples for both cases and discuss the differences and similarities between the different forms of the equation as well as the associated solutions and optimal processes.


Optimal Portfolio Stochastic Volatility Incomplete Market Stochastic Partial Differential Equation Portfolio Choice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ait-Sahalia, Y., Brandt, M.: Variable selection for portfolio choice. J. Finance 56, 1297–1351 (2001) CrossRefGoogle Scholar
  2. 2.
    Bates, D.S.: Post-87 crash fears and S&P futures options. J. Econom. 94, 181–238 (2000) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Becherer, D.: The numeraire portfolio for unbounded semimartingales. Finance Stoch. 5, 327–344 (2001) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Berrier, F., Rogers, L.C., Tehranchi, M.: A characterization of forward utility functions. Preprint (2009) Google Scholar
  5. 5.
    Borkar, V.S.: Optimal control of diffusion processes. Pitman Res. Notes Math. Ser., 203 (1983) Google Scholar
  6. 6.
    Bouchard, B., Touzi, N.: Weak dynamic programming principle for viscosity solutions. Preprint (2009) Google Scholar
  7. 7.
    Brandt, M.: Estimating portfolio and consumption choice: A conditional Euler equation approach. J. Finance 54, 1609–1645 (1999) CrossRefGoogle Scholar
  8. 8.
    Chacko, G., Viceira, L.M.: Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets. Rev. Financ. Stud. 18, 1369–1402 (2005) CrossRefGoogle Scholar
  9. 9.
    Cvitanić, J., Schachermayer, W., Wang, H.: Utility maximization in incomplete markets with random endowment. Finance Stoch. 5, 259–272 (2001) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    El Karoui, N., Nguyen, D.H., Jeanblanc, M.: Compactification methods in the control of degenerate diffusions: Existence of an optimal control. Stochastics 20, 169–220 (1987) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Finance 7(1), 1–71 (1997) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Fama, W.E., Schwert, G.W.: Asset returns and inflation. J. Financ. Econ. 5, 115–146 (1977) CrossRefGoogle Scholar
  13. 13.
    Ferson, W.E., Harvey, C.R.: The risk and predictability of international equity returns. Rev. Financ. Stud. 6, 527–566 (1993) Google Scholar
  14. 14.
    Fleming, W.H., Soner, M.H.: Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, New York (2005) Google Scholar
  15. 15.
    French, K.R., Schwert, G.W., Stambaugh, R.F.: Expected stock returns and volatility. J. Financ. Econ. 19, 3–29 (1987) CrossRefGoogle Scholar
  16. 16.
    Glosten, L.R., Jagannathan, R., Runkle, D.E.: On the relation between the expected value and the volatility of the nominal excess return of stocks. J. Finance 48, 1779–1801 (1993) CrossRefGoogle Scholar
  17. 17.
    Goll, T., Kallsen, J.: Optimal portfolios for logarithmic utility. Stoch. Process. Appl. 89, 31–48 (2000) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Goll, T., Kallsen, J.: A complete explicit solution to the log-optimal portfolio problem. Ann. Appl. Probab. 12(2), 774–799 (2003) MathSciNetGoogle Scholar
  19. 19.
    Heaton, J., Lucas, D.: Market frictions, savings behavior and portfolio choice. Macroecon. Dyn. 1, 76–101 (1997) MATHCrossRefGoogle Scholar
  20. 20.
    Hugonnier, J., Kramkov, D.: Optimal investment with random endowments in incomplete markets. Ann. Appl. Probab. 14, 845–864 (2004) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Karatzas, I., Kardaras, C.: The numeraire portfolio in semimartingale financial models. Finance Stoch. 11, 447–493 (2007) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Karatzas, I., Zitković, G.: Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Appl. Probab. 31(4), 1821–1858 (2003) MATHGoogle Scholar
  23. 23.
    Kim, T.S., Omberg, E.: Dynamic nonmyopic portfolio behavior. Rev. Financ. Stud. 9, 141–161 (1996) CrossRefGoogle Scholar
  24. 24.
    Korn, R., Korn, E.: Option Pricing and Portfolio Optimization – Modern Methods of Financial Mathematics. Am. Math. Soc., Providence (2001) MATHGoogle Scholar
  25. 25.
    Kraft, H.: Optimal portfolios and Heston’s stochastic volatility model. Quant. Finance 5, 303–313 (2005) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9(3), 904–950 (1999) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Kramkov, D., Schachermayer, W.: Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 13(4), 1504–1516 (2003) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Kramkov, D., Sirbu, M.: On the two times differentiability of the value function in the problem of optimal investment in incomplete market. Ann. Appl. Probab. 16(3), 1352–1384 (2006) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Krylov, N.: Controlled Diffusion Processes. Springer, New York (1980) MATHGoogle Scholar
  30. 30.
    Liu, J.: Portfolio selection in stochastic environments. Rev. Financ. Stud. 20(1), 1–39 (2007) MATHCrossRefGoogle Scholar
  31. 31.
    Merton, R.: Lifetime portfolio selection under uncertainty: The continuous-time case. Rev. Econ. Stat. 51, 247–257 (1969) CrossRefGoogle Scholar
  32. 32.
    Mnif, M.: Portfolio optimization with stochastic volatilities and constraints: An application in high dimension. Appl. Math. Optim. 56, 243–264 (2007) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Musiela, M., Zariphopoulou, T.: The backward and forward dynamic utilities and the associated pricing systems: The case study of the binomial model. Preprint (2003) Google Scholar
  34. 34.
    Musiela, M., Zariphopoulou, T.: Investment and valuation under backward and forward dynamic exponential utilities in a stochastic factor model. Advances in Mathematical Finance. Applied and Numerical Harmonic Analysis Series, 303–334 (2007) Google Scholar
  35. 35.
    Musiela, M., Zariphopoulou, T.: Optimal asset allocation under forward exponential performance criteria (2006). In: Ethier, S., Feng, J., Stockbridge, R. (eds.) Markov Processes and related topics. A Festschrift for T.G. Kurtz. Lecture Notes-Monograph Series, 285–300. Institute of Mathematical Statistics, Beachwood (2008) CrossRefGoogle Scholar
  36. 36.
    Musiela, M., Zariphopoulou, T.: Portfolio choice under space-time monotone performance criteria. SIAM Journal on Financial Mathematics 1, 326–365 (2010) MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Musiela, M., Zariphopoulou, T.: The single period binomial model. In: Carmona R. (ed.) Indifference Pricing. Princeton University Press, Princeton, 3–41 (2009) Google Scholar
  38. 38.
    Musiela, M., Zariphopoulou, T.: Portfolio choice under dynamic investment performance criteria. Quant. Finance 9(2), 161–170 (2009) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Pham, H.: Smooth solutions to optimal investment models with stochastic volatilities and portfolio constraints. Appl. Math. Optim. 46, 1–55 (2002) MathSciNetCrossRefGoogle Scholar
  40. 40.
    Schachermayer, W.: A super-martingale property of the optimal portfolio process. Finance Stoch. 7(4), 433–456 (2003) MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Wachter, J.: Portfolio and consumption decisions under mean-reverting returns: An exact solution for complete markets. J. Financ. Quant. Anal. 37, 63–91 (2002) CrossRefGoogle Scholar
  42. 42.
    Yong, J., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999) MATHGoogle Scholar
  43. 43.
    Zariphopoulou, T.: A solution approach to valuation with unhedgeable risks. Finance Stoch. 5, 61–82 (2001) MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Zariphopoulou, T.: Optimal asset allocation in a stochastic factor model – An overview and open problems. In: Hansjorg, A., Runggaldier, W., Schachermayer, W. (eds.) Advanced Financial Modeling. RADON Series on Computational and Applied Mathematics, 8, 427–453 (2009) Google Scholar
  45. 45.
    Žitković, G.: A dual characterization of self-generation and exponential forward performances. Ann. Appl. Probab. 19(6), 2176–2210 (2009) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.BNP ParibasLondonUK
  2. 2.Oxford-Man Institute and Mathematical InstituteUniversity of OxfordOxfordUK
  3. 3.Departments of Mathematics and IROM, McCombs School of BusinessThe University of Texas at AustinAustinUSA

Personalised recommendations