Applied Mathematics & Optimization

, Volume 75, Issue 1, pp 117–147 | Cite as

Dynamic Robust Duality in Utility Maximization

  • Bernt Øksendal
  • Agnès Sulem


A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities:
  1. (i)

    The optimal terminal wealth \(X^*(T) : = X_{\varphi ^*}(T)\) of the problem to maximize the expected U-utility of the terminal wealth \(X_{\varphi }(T)\) generated by admissible portfolios \(\varphi (t); 0 \le t \le T\) in a market with the risky asset price process modeled as a semimartingale;

  2. (ii)

    The optimal scenario \(\frac{dQ^*}{dP}\) of the dual problem to minimize the expected V-value of \(\frac{dQ}{dP}\) over a family of equivalent local martingale measures Q, where V is the convex conjugate function of the concave function U.

In this paper we consider markets modeled by Itô-Lévy processes. In the first part we use the maximum principle in stochastic control theory to extend the above relation to a dynamic relation, valid for all \(t \in [0,T]\). We prove in particular that the optimal adjoint process for the primal problem coincides with the optimal density process, and that the optimal adjoint process for the dual problem coincides with the optimal wealth process; \(0 \le t \le T\). In the terminal time case \(t=T\) we recover the classical duality connection above. We get moreover an explicit relation between the optimal portfolio \(\varphi ^*\) and the optimal measure \(Q^*\). We also obtain that the existence of an optimal scenario is equivalent to the replicability of a related T-claim. In the second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a similar dynamic relation between them. In particular, we show how to get from the solution of one of the problems to the other. We illustrate the results with explicit examples.


Robust portfolio optimization Robust duality Dynamic duality method Stochastic maximum principle Backward stochastic differential equation Itô-Lévy market 

Mathematics Subject Classification

Primary 60H10 93E20 Secondary 91B70 46N10 



The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no [228087].


  1. 1.
    Bordigoni, G., Matoussi, A., Schweizer, M.: A stochastic control approach to a robust utility maximization problem. In: Benth, F.E., et al. (eds.) Stochastic Analysis and Applications. The Abel Symposium 2005. Springer, Berlin (2007)Google Scholar
  2. 2.
    El Karoui, N., Quenez, M.-C.: Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33, 29–66 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    El Karoui, N., Peng, S., Quenez, M.-C.: Backward stochastic differential equations in finance. Math. Financ. 7, 1–71 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Föllmer, H., Schie, A., Weber, S.: Robust preferences and robust portfolio choice. In: Ciarlet, P., Bensoussan, A., Zhang, Q. (eds.) Mathematical Modelling and Numerical Methods in Finance. Handbook of Numerical Analysis 15, pp. 29–88. Springer, New York (2009)Google Scholar
  5. 5.
    Fontana, C., Øksendal, B., Sulem, A.: Viability and martingale measures in jump diffusion markets under partial information. Methodol. Comput. Appl. Probab. 1(2), 209–222 (2014). doi: 10.1007/s11009-014-9397-4 Google Scholar
  6. 6.
    Gushkin, A.: Dual characterization of the value function in the robust utility maximization problem. Theory Probab. Appl. 55, 611–630 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Jeanblanc, M., Matoussi, A., Ngoupeyou, A.: Robust utility maximization in a discontinuous filtration, arXiv 1201.2690 v3 (2013)Google Scholar
  8. 8.
    Kramkov, D., Schachermayer, W.: Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Annal. Appl. Probab. 13, 1504–1516 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kreps, D.: Arbitrage and equilibrium in economics with infinitely many commodities. J. Math. Econ. 8, 15–35 (1981)CrossRefzbMATHGoogle Scholar
  10. 10.
    Lim, T., Quenez, M.-C.: Exponential utility maximization and indifference price in an incomplete market with defaults. Electron. J. Probab. 16, 1434–1464 (2011)CrossRefzbMATHGoogle Scholar
  11. 11.
    Loewenstein, M., Willard, G.: Local martingales, arbitrage, and viability. Econ. Theory 16, 135–161 (2000)CrossRefzbMATHGoogle Scholar
  12. 12.
    Maenhout, P.: Robust portfolio rules and asset pricing. Rev. Financ. Stud. 17, 951–983 (2004)CrossRefGoogle Scholar
  13. 13.
    Øksendal, B., Sulem, A.: Applied Stochastic Control of Jump Diffusions, 2nd edn. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
  14. 14.
    Øksendal, B., Sulem, A.: Forward-backward stochastic differential games and stochastic control under model uncertainty. J. Optim. Theory Appl. 2014(161), 22–55 (2012). doi: 10.1007/S10957-012-0166-7 MathSciNetzbMATHGoogle Scholar
  15. 15.
    Øksendal, B., Sulem, A.: Risk minimization in financial markets modeled by Itô-Lévy processes. Afrika Math. 26, 939–979 (2015). doi: 10.1007/s13370-014-0248-9 CrossRefzbMATHGoogle Scholar
  16. 16.
    Quenez, M.-C.: Optimal portfolio in a multiple-priors model. In: Dalang, R.C., Dozzi, M., Russo, F. (eds.) Seminar on Stochastic Analysis, pp. 291–321. Random Fields and Applications IV, Birkauser (2004)Google Scholar
  17. 17.
    Quenez, M.-C., Sulem, A.: BSDEs with jumps, optimization and applications to dynamic risk measures. Stoch. Process. Appl. 123, 3328–3357 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Royer, M.: Backward stochastic differential equations with jumps and related non-linear expectations. Stoch. Process. Appl. 116, 1358–1376 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  20. 20.
    Tang, S.H., Li, X.: Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32, 1447–1475 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OsloOsloNorway
  2. 2.INRIA Paris, 2 Rue Simone Iff75589 Paris Cedex 12France
  3. 3.Université Paris-EstMarne-la-ValléeFrance

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