Mathematics and Financial Economics

, Volume 12, Issue 1, pp 111–134 | Cite as

Backward nonlinear expectation equations

  • Christoph Belak
  • Thomas Seiferling
  • Frank Thomas SeifriedEmail author


Building on an abstract framework for dynamic nonlinear expectations that comprises g-, G- and random G-expectations, we develop a theory of backward nonlinear expectation equations of the form
$$\begin{aligned} X_t = {\mathcal {E}}_t \Bigl [{\textstyle \int _t^T} g(s,X) \mu ({\mathrm {d}}s) + \xi \Bigr ], \quad t \in [0,T]. \end{aligned}$$
We provide existence, uniqueness, and stability results and establish convergence of the associated discrete-time nonlinear aggregations. As an application, we construct continuous-time recursive utilities under ambiguity and identify the corresponding utility processes as limits of discrete-time recursive utilities.


Backward stochastic differential equation Nonlinear expectation Random G-expectation Recursive utility Volatility uncertainty 

Mathematics Subject Classification

60G20 60H30 91B16 

JEL Classification

D81 D91 



We wish to thank Mete Soner, Holger Kraft, Rama Cont, Yuri Kabanov, and Keita Owari for comments and suggestions. Thomas Seiferling gratefully acknowledges financial support from Studienstiftung des Deutschen Volkes.


  1. 1.
    Antonelli, F.: Backward–forward stochastic differential equations. Ann. Appl. Probab. 3(3), 777–793 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Antonelli, F.: Stability of backward stochastic differential equations. Stoch. Process. Appl. 62(1), 103–114 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barles, G., Buckdahn, R., Pardoux, E.: Backward stochastic differential equations and integral–partial differential equations. Stoch. Stoch. Rep. 60(1–2), 57–83 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bismut, J.-M.: Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384–404 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bouchard, B., Elie, R.: Discrete-time approximation of decoupled forward-backward SDE with jumps. Stoch. Process. Appl. 118(1), 53–75 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bouchard, B., Touzi, N.: Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stoch. Process. Appl. 111(2), 175–206 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, Z., Epstein, L.: Ambiguity, risk, and asset returns in continuous time. Econometrica 70(4), 1403–1443 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cheridito, P., Stadje, M.: BS\(\Delta \)Es and BSDEs with non-Lipschitz drivers: comparison, convergence and robustness. Bernoulli 19(3), 1047–1085 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cheridito, P., Soner, H.M., Touzi, N., Victoir, N.: Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Commun. Pure Appl. Math. 60(7), 1081–1110 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Delong, Ł.: Backward Stochastic Differential Equations with Jumps and Their Actuarial and financial applications. European Actuarial Academy (EAA) Series. Springer, London (2013)zbMATHGoogle Scholar
  11. 11.
    Denis, L., Hu, M., Peng, S.: Function spaces and capacity related to a sublinear expectation: application to \(G\)-Brownian motion paths. Potential Anal. 34(2), 139–161 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Duffie, D., Epstein, L.G.: Stochastic differential utility. Econometrica 60(2), 353–394 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dunford, N., Schwartz, J.T.: Linear Operators: I. General Theory. Pure and Applied Mathematics, vol. 7. Interscience Publishers, Inc., New York (1958)zbMATHGoogle Scholar
  14. 14.
    El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Finance 7(1), 1–71 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Epstein, L.G., Ji, S.: Ambiguous volatility and asset pricing in continuous time. Rev. Financ. Stud. 26(7), 1740–1786 (2013)CrossRefGoogle Scholar
  16. 16.
    Epstein, L.G., Ji, S.: Ambiguous volatility, possibility and utility in continuous time. J. Math. Econ. 50, 269–282 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Epstein, L.G., Schneider, M.: Recursive multiple-priors. J. Econ. Theory 113(1), 1–31 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Epstein, L.G., Wang, T.: Intertemporal asset pricing under knightian uncertainty. Econometrica 62(2), 283–322 (1994)CrossRefzbMATHGoogle Scholar
  19. 19.
    Guo, X., Pan, C., Peng, S.: Martingale problem under nonlinear expectations. Preprint (2014). arXiv:1211.2869
  20. 20.
    Hayashi, T.: Intertemporal substitution, risk aversion and ambiguity aversion. Econ. Theory 25(4), 933–956 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hu, M., Ji, S., Peng, S., Song, Y.: Backward stochastic differential equations driven by \(G\)-Brownian motion. Stoch. Process. Appl. 124(1), 759–784 (2014a)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hu, M., Ji, S., Peng, S., Song, Y.: Comparison theorem, Feynman–Kac formula and Girsanov transformation for BSDEs driven by \(G\)-Brownian motion. Stoch. Process. Appl. 124(2), 1170–1195 (2014b)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  24. 24.
    Kraft, H., Seifried, F.T.: Stochastic differential utility as the continuous-time limit of recursive utility. J. Econ. Theory 151, 528–550 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kreps, D.M., Porteus, E.L.: Temporal resolution of uncertainty and dynamic choice theory. Econometrica 46(1), 185–200 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lin, Q., Riedel, F.: Optimal consumption and portfolio choice with ambiguity. Preprint (2014).
  27. 27.
    Nutz, M.: A quasi-sure approach to the control of non-Markovian stochastic differential equations. Electron. J. Probab. 17(23), 1–23 (2012)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Nutz, M.: Random \(G\)-expectations. Ann. Appl. Probab. 23(5), 1755–1777 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Nutz, M., Soner, H.M.: Superhedging and dynamic risk measures under volatility uncertainty. SIAM J. Control Optim. 50(4), 2065–2089 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nutz, M., van Handel, R.: Constructing sublinear expectations on path space. Stoch. Process. Appl. 123(8), 3100–3121 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Peng, S.: Backward SDE and related \(g\)-expectation. In: Backward Stochastic Differential Equations (Paris, 1995–1996), Volume 364 of Pitman Res. Notes Math. Ser., pp. 141–159. Longman, Harlow (1997)Google Scholar
  33. 33.
    Peng, S.: Nonlinear expectations, nonlinear evaluations and risk measures. In: Stochastic Methods in Finance, Volume 1856 of Lecture Notes in Math., pp. 165–253. Springer, Berlin (2004)Google Scholar
  34. 34.
    Peng, S.: Dynamically consistent nonlinear evaluations and expectations. Preprint (2005). arXiv:math/0501415
  35. 35.
    Peng, S.: \(G\)-expectation, \(G\)-Brownian motion and related stochastic calculus of Itô type. In: Stochastic Analysis and Applications, Volume 2 of Abel Symp., pp. 541–567. Springer, Berlin (2007)Google Scholar
  36. 36.
    Peng, S.: Multi-dimensional \(G\)-Brownian motion and related stochastic calculus under \(G\)-expectation. Stoch. Process. Appl. 118(12), 2223–2253 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Peng, S.: Nonlinear expectations and stochastic calculus under uncertainty. Preprint (2010). arXiv:1002.4546
  38. 38.
    Penner, I., Réveillac, A.: Risk measures for processes and BSDEs. Finance Stoch. 19(1), 23–66 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Quenez, M.-C., Sulem, A.: BSDEs with jumps, optimization and applications to dynamic risk measures. Stoch. Process. Appl. 123(8), 3328–3357 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Riedel, F.: Dynamic coherent risk measures. Stoch. Process. Appl. 112(2), 185–200 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Royer, M.: Backward stochastic differential equations with jumps and related non-linear expectations. Stoch. Process. Appl. 116(10), 1358–1376 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Seiferling, T.: Recursive utility and stochastic differential utility: from discrete to continuous time. Ph.D. Thesis, University of Kaiserslautern (2016)Google Scholar
  43. 43.
    Soner, H.M., Touzi, N., Zhang, J.: Martingale representation theorem for the \(G\)-expectation. Stoch. Process. Appl. 121(2), 265–287 (2011a)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Soner, H.M., Touzi, N., Zhang, J.: Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16(67), 1844–1879 (2011b)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Soner, H.M., Touzi, N., Zhang, J.: Wellposedness of second order backward SDEs. Probab. Theory Related Fields 153(1–2), 149–190 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Soner, H.M., Touzi, N., Zhang, J.: Dual formulation of second order target problems. Ann. Appl. Probab. 23(1), 308–347 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Tang, S.J., Li, X.J.: Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32(5), 1447–1475 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Zhang, J.: A numerical scheme for BSDEs. Ann. Appl. Probab. 14(1), 459–488 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Christoph Belak
    • 1
  • Thomas Seiferling
    • 2
  • Frank Thomas Seifried
    • 1
    Email author
  1. 1.Department IV – MathematicsUniversity of TrierTrierGermany
  2. 2.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany

Personalised recommendations