The Annals of Applied Probability

Random $G$-expectations

Marcel Nutz

Full-text: Open access


We construct a time-consistent sublinear expectation in the setting of volatility uncertainty. This mapping extends Peng’s $G$-expectation by allowing the range of the volatility uncertainty to be stochastic. Our construction is purely probabilistic and based on an optimal control formulation with path-dependent control sets.

Article information

Ann. Appl. Probab., Volume 23, Number 5 (2013), 1755-1777.

First available in Project Euclid: 28 August 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 93E20: Optimal stochastic control 91B30: Risk theory, insurance
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)

$G$-expectation volatility uncertainty stochastic domain risk measure time-consistency


Nutz, Marcel. Random $G$-expectations. Ann. Appl. Probab. 23 (2013), no. 5, 1755--1777. doi:10.1214/12-AAP885.

Export citation


  • [1] Avellaneda, M., Levy, A. and Parás, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2 73–88.
  • [2] Bichteler, K. (1981). Stochastic integration and $L^{p}$-theory of semimartingales. Ann. Probab. 9 49–89.
  • [3] Cheridito, P., Soner, H. M., Touzi, N. and Victoir, N. (2007). Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Comm. Pure Appl. Math. 60 1081–1110.
  • [4] Denis, L., Hu, M. and Peng, S. (2011). Function spaces and capacity related to a sublinear expectation: Application to $G$-Brownian motion paths. Potential Anal. 34 139–161.
  • [5] Denis, L. and Martini, C. (2006). A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16 827–852.
  • [6] Lyons, T. J. (1995). Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2 117–133.
  • [7] Mandelkern, M. (1990). On the uniform continuity of Tietze extensions. Arch. Math. (Basel) 55 387–388.
  • [8] Nutz, M. and Soner, H. M. (2012). Superhedging and dynamic risk measures under volatility uncertainty. SIAM J. Control Optim. 50 2065–2089.
  • [9] Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
  • [10] Peng, S. (1997). Backward SDE and related $g$-expectation. In Backward Stochastic Differential Equations (Paris, 19951996). Pitman Research Notes in Mathematics Series 364 141–159. Longman, Harlow.
  • [11] Peng, S. (2004). Filtration consistent nonlinear expectations and evaluations of contingent claims. Acta Math. Appl. Sin. Engl. Ser. 20 191–214.
  • [12] Peng, S. (2005). Nonlinear expectations and nonlinear Markov chains. Chin. Ann. Math. Ser. B 26 159–184.
  • [13] Peng, S. (2007). $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type. In Stochastic Analysis and Applications. Abel Symp. 2 541–567. Springer, Berlin.
  • [14] Peng, S. (2008). Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation. Stochastic Process. Appl. 118 2223–2253.
  • [15] Peng, S. (2010). Nonlinear expectations and stochastic calculus under uncertainty. Preprint. Available at arXiv:1002.4546v1.
  • [16] Peng, S. (2010). Backward stochastic differential equation, nonlinear expectation and their applications. In Proceedings of the International Congress of Mathematicians. Volume I 393–432. Hindustan Book Agency, New Delhi.
  • [17] Soner, H. M., Touzi, N. and Zhang, J. (2013). Dual formulation of second order target problems. Ann. Appl. Probab. 23 308–347.
  • [18] Soner, H. M., Touzi, N. and Zhang, J. (2012). Wellposedness of second order backward SDEs. Probab. Theory Related Fields 153 149–190.
  • [19] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233. Springer, Berlin.