The Annals of Probability

Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II

Ibrahim Ekren, Nizar Touzi, and Jianfeng Zhang

Full-text: Open access


In our previous paper [Ekren, Touzi and Zhang (2015)], we introduced a notion of viscosity solutions for fully nonlinear path-dependent PDEs, extending the semilinear case of Ekren et al. [Ann. Probab. 42 (2014) 204–236], which satisfies a partial comparison result under standard Lipshitz-type assumptions. The main result of this paper provides a full, well-posedness result under an additional assumption, formulated on some partial differential equation, defined locally by freezing the path. Namely, assuming further that such path-frozen standard PDEs satisfy the comparison principle and the Perron approach for existence, we prove that the nonlinear path-dependent PDE has a unique viscosity solution. Uniqueness is implied by a comparison result.

Article information

Ann. Probab., Volume 44, Number 4 (2016), 2507-2553.

Received: May 2013
Revised: September 2014
First available in Project Euclid: 2 August 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35D40: Viscosity solutions 35K10: Second-order parabolic equations 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.)

Path dependent PDEs nonlinear expectation viscosity solutions comparison principle Perron’s approach


Ekren, Ibrahim; Touzi, Nizar; Zhang, Jianfeng. Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II. Ann. Probab. 44 (2016), no. 4, 2507--2553. doi:10.1214/15-AOP1027.

Export citation


  • [1] Bayraktar, E. and Sîrbu, M. (2013). Stochastic Perron’s method for Hamilton–Jacobi–Bellman equations. SIAM J. Control Optim. 51 4274–4294.
  • [2] Buckdahn, R., Ma, J. and Zhang, J. (2015). Pathwise Taylor expansions for random fields on multiple dimensional paths. Stochastic Process. Appl. 125 2820–2855.
  • [3] Cont, R. and Fournié, D.-A. (2013). Functional Itô calculus and stochastic integral representation of martingales. Ann. Probab. 41 109–133.
  • [4] Crandall, M. G., Ishii, H. and Lions, P.-L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 1–67.
  • [5] Dupire, B. (2009). Functional Itô calculus. Available at
  • [6] Ekren, I., Keller, C., Touzi, N. and Zhang, J. (2014). On viscosity solutions of path dependent PDEs. Ann. Probab. 42 204–236.
  • [7] Ekren, I., Touzi, N. and Zhang, J. (2014). Optimal stopping under nonlinear expectation. Stochastic Process. Appl. 124 3277–3311.
  • [8] Ekren, I., Touzi, N. and Zhang, J. (2016). Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I. Ann. Probab. 44 1212–1253.
  • [9] Fleming, W. H. and Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions, 2nd ed. Stochastic Modelling and Applied Probability 25. Springer, New York.
  • [10] Fleming, W. H. and Vermes, D. (1988). Generalized solutions in the optimal control of diffusions. In Stochastic Differential Systems, Stochastic Control Theory and Applications (Minneapolis, Minn., 1986). IMA Vol. Math. Appl. 10 119–127. Springer, New York.
  • [11] Fleming, W. H. and Vermes, D. (1989). Convex duality approach to the optimal control of diffusions. SIAM J. Control Optim. 27 1136–1155.
  • [12] Henry-Labordère, P., Tan, X. and Touzi, N. (2014). A numerical algorithm for a class of BSDEs via the branching process. Stochastic Process. Appl. 124 1112–1140.
  • [13] Ishii, H. (1987). Perron’s method for Hamilton–Jacobi equations. Duke Math. J. 55 369–384.
  • [14] Krylov, N. V. (2000). On the rate of convergence of finite-difference approximations for Bellman’s equations with variable coefficients. Probab. Theory Related Fields 117 1–16.
  • [15] Lieberman, G. M. (1996). Second Order Parabolic Differential Equations. World Scientific, River Edge, NJ.
  • [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] Pham, T. and Zhang, J. (2014). Two person zero-sum game in weak formulation and path dependent Bellman–Isaacs equation. SIAM J. Control Optim. 52 2090–2121.
  • [18] Zhang, J. and Zhuo, J. (2014). Monotone schemes for fully nonlinear parabolic path dependent PDEs. Journal of Financial Engineering 1 1450005.