Mathematical Methods of Operations Research

, Volume 87, Issue 3, pp 311–346 | Cite as

Non-linear filtering and optimal investment under partial information for stochastic volatility models

  • Dalia IbrahimEmail author
  • Frédéric Abergel
Original Article


This paper studies the question of filtering and maximizing terminal wealth from expected utility in partial information stochastic volatility models. The special feature is that the only information available to the investor is the one generated by the asset prices, and the unobservable processes will be modeled by stochastic differential equations. Using the change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filter processes). Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priori models for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named “Kushner–Stratonovich equations”. Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that, for power and logarithmic utility, the dual value function associated to the martingale approach can be expressed, via the dynamic programming approach, in terms of the solution to a semilinear partial differential equation which depends on the filters estimate and the volatility. We illustrate our results with some examples of stochastic volatility models popular in the financial literature.


Partial information Stochastic volatility Utility maximization Martingale duality method Non-linear filtering Kushner–Stratonovich equations Semilinear partial differential equation 


  1. Bain A, Crisan D (2009) Fundamentals of stochastic filtering. In: Stochastic modelling and applied probability, vol 60. Springer, New YorkGoogle Scholar
  2. Bensoussan A (1992) Stochastic control of partially observable systems. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  3. Ceci C, Colaneri K (2014) The Zakai equation of nonlinear filtering for jump-diffusion observations: existence and uniqueness. Appl Math Optim 69(1):47–82MathSciNetCrossRefzbMATHGoogle Scholar
  4. Crisan D, Lyons T (1999) A particle approximation of the solution of the Kushner–Stratonovitch equation. Probab Theory Relat Fields 115(4):549–578MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cvitanić J, Karatzas I (1992) Convex duality in constrained portfolio optimization. Ann Appl Probab 2(4):767–818MathSciNetCrossRefzbMATHGoogle Scholar
  6. Detemple J (1986) Asset pricing in a production economy with incomplete information. J Finance 41(2):383–391CrossRefGoogle Scholar
  7. Dothan MU, Feldman D (1986) Equilibrium interest rates and multiperiod bonds in a partially observable economy. J Finance 41(2):369–382CrossRefGoogle Scholar
  8. Fleming WH, Soner HM (2006) Controlled Markov processes and viscosity solutions In: Springer Science & Business Media (ed) Stochastic modelling and applied probability, 2nd edn, vol 25. Springer, New YorkGoogle Scholar
  9. Gobet E, Pages G, Pham H, Printemps J (2006) Discretization and simulation of the Zakai equation. SIAM J Numer Anal 44(6):2505–2538MathSciNetCrossRefzbMATHGoogle Scholar
  10. Kallianpur G (1980) Stochastic filtering theory. In: Stochastic modelling and applied probability, vol 13. Springer, New YorkGoogle Scholar
  11. Karatzas I, Lehoczky JP, Shreve SE, Xu GL (1991) Martingale and duality methods for utility maximization in an incomplete market. SIAM J Control Optim 29(3):702–730MathSciNetCrossRefzbMATHGoogle Scholar
  12. Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus. In: Graduate texts in mathematics, vol 113. Springer, New YorkGoogle Scholar
  13. Karatzas I, Zhao X (2001) Bayesian adaptive portfolio optimization. In: Handbooks in mathematical finance: option pricing, interest rates and risk management, 1st edn. Cambridge University Press, Cambridge, pp 632–669Google Scholar
  14. Kurtz TG, Ocone DL (1988) Unique characterization of conditional distributions in nonlinear filtering. Ann Probab 16:80–107MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kuwana Y (1995) Certainty equivalence and logarithmic utilities in consumption/investment problems. Math Finance 5(4):297–309MathSciNetCrossRefzbMATHGoogle Scholar
  16. Lakner P (1995) Utility maximization with partial information. Stoch Process Appl 56(2):247–273MathSciNetCrossRefzbMATHGoogle Scholar
  17. Lakner P (1998) Optimal trading strategy for an investor: the case of partial information. Stoch Process Appl 76(1):77–97MathSciNetCrossRefzbMATHGoogle Scholar
  18. Liptser RS, Shiryaev AN (2001) Statistics of random processes. I, volume 5 of applications of mathematics (New York). Springer, Berlin, expanded edition. General theory, translated from the (1974) Russian original by A. B. Aries, Stochastic modelling and applied probabilityGoogle Scholar
  19. Owen MP (2002) Utility based optimal hedging in incomplete markets. Ann Appl Probab 12(2):691–709MathSciNetCrossRefzbMATHGoogle Scholar
  20. Pardoux E (1991) Filtrage non linéaire et équations aux dérivées partielles stochastiques associées. In: Hennequin PL (ed) École d’Été de Probabilités de Saint-Flour XIX–1989, vol 1464. Lecture Notes in Mathematics. Springer, Berlin, pp 67–163Google Scholar
  21. Pham H (2002) Smooth solutions to optimal investment models with stochastic volatilities and portfolio constraints. Appl Math Optim 46(1):55–78MathSciNetCrossRefzbMATHGoogle Scholar
  22. Pham H, Quenez MC (2001) Optimal portfolio in partially observed stochastic volatility models. Ann Appl Probab 11(1):210–238MathSciNetCrossRefzbMATHGoogle Scholar
  23. Rishel R (1999) Optimal portfolio management with partial observations and power utility function. In: McEneaney WM, Yin GG, Zhang Q (eds) Stochastic analysis, control, optimization and applications, systems control found. Application, Birkhäuser, Boston, pp 605–619CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  1. 1.Laboratoire MICS-Mathématiques et Informatique pour la Complexité et les SystèmesCentraleSupélecGif-sur-YvetteFrance

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