M. Avellaneda, A. Levy, and A. Paras, Pricing and hedging derivative securities in markets with uncertain volatilities, Applied Mathematical Finance, vol.7, issue.2, pp.73-88, 1995.
DOI : 10.2307/2330793

M. Bardi and I. Capuzzo-dolcetta, Optimal control and viscosity solutions of Hamilton- Jacobi-Belleman equations, 2008.

G. Barles, E. Lesigne, . Sde, and P. Bsde, Backward Stochastic Differential Equations, Pitman Res. Notes Math., Ser, vol.364, pp.47-80, 1997.

G. Barles and P. E. Souganidis, Convergence of Approximation Schemes for Fully Nonlinear Second Order Equation, Asymptotic Anal, pp.271-283, 1991.

G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems, ESAIM: Mathematical Modelling and Numerical Analysis, vol.21, issue.4, pp.557-579, 1987.
DOI : 10.1051/m2an/1987210405571

E. Bayraktar and M. Sirbu, Stochastic Perron???s method and verification without smoothness using viscosity comparison: The linear case, Proc. Amer, pp.3645-3654, 2012.
DOI : 10.1090/S0002-9939-2012-11336-X

E. Bayraktar and M. Sirbu, Stochastic Perron's Method for Hamilton--Jacobi--Bellman Equations, SIAM Journal on Control and Optimization, vol.51, issue.6, pp.4274-4294, 2013.
DOI : 10.1137/12090352X

M. Beiglböck, W. Schachermayer, and B. Veliyev, A short proof of the Doob???Meyer theorem, Stochastic Processes and Applications, pp.1204-1209, 2012.
DOI : 10.1016/j.spa.2011.12.001

H. Berestycki, J. Busca, and I. Florent, Asymptotics and calibration of local volatility models, Quantitative Finance, vol.4, issue.1, pp.61-69, 2002.
DOI : 10.1002/cpa.3160450103

H. Berestycki, J. Busca, and I. Florent, Computing the implied volatility in stochastic volatility models, Communications on Pure and Applied Mathematics, vol.20, issue.140, pp.1352-1373, 2004.
DOI : 10.1002/cpa.20039

V. Bally and A. Matoussi, Weak solutions of Stochastic PDEs and Backward doubly stochastic differential equations, Journal of Theoretical Probability, vol.14, issue.1, pp.125-164, 2001.
DOI : 10.1023/A:1007825232513

J. F. Bonnans, E. Ottenwaelter, and H. Zidani, A fast algorithm for the two dimensional HJB equation of stochastic control, ESAIM: Mathematical Modelling and Numerical Analysis, vol.38, issue.4, pp.723-735, 2004.
DOI : 10.1051/m2an:2004034
URL : https://hal.archives-ouvertes.fr/hal-00988282

B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Process, Appl, vol.111, issue.2, pp.175-206, 2004.

B. Bouchard and N. Touzi, Weak Dynamic Programming Principle for Viscosity Solutions, SIAM Journal on Control and Optimization, vol.49, issue.3, pp.948-962, 2011.
DOI : 10.1137/090752328
URL : https://hal.archives-ouvertes.fr/hal-00367355

L. Caffarelli and X. Cabre, Fully Nonlinear Elliptic Equations, 1995.
DOI : 10.1090/coll/043

P. Cheridito, H. M. Soner, N. Touzi, and N. Victoir, Second order BSDE's and fully nonlinear PDE's, Communications in Pure and Applied Mathematics, pp.60-1081, 2007.

R. Cont and D. Fournie, Functional It?? calculus and stochastic integral representation of martingales, The Annals of Probability, vol.41, issue.1, pp.109-133, 2013.
DOI : 10.1214/11-AOP721

C. Costantini and T. G. Kurtz, Viscosity methods giving uniqueness for martingale problems, Electronic Journal of Probability, vol.20, issue.0
DOI : 10.1214/EJP.v20-3624

M. G. Crandall, P. Lions, and . Lions, Viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, vol.277, issue.1, pp.1-42, 1984.
DOI : 10.1090/S0002-9947-1983-0690039-8

M. G. Crandall, P. Lions, and . Lions, Condition d'unicité pour les solutions généralisées dee équations de Hamilton-Jacobi du premier order, C. R. Acad. Sci. Paris, pp.292-183, 1981.

M. G. Crandall, H. Ishii, and P. L. Lions, user's guide to viscosity solutions\\ of second order\\ partial differential equations, Bulletin of the American Mathematical Society, vol.27, issue.1, pp.27-28, 1992.
DOI : 10.1090/S0273-0979-1992-00266-5

J. Cvitani? and J. Zhang, Contract Theory in Continuous-Time Models, 2012.
DOI : 10.1007/978-3-642-14200-0

R. W. Darling and E. Pardoux, Backwards SDE with random terminal time and applications to semilinear elliptic PDE, The Annals of Probability, vol.25, issue.3, pp.1135-1159, 1997.
DOI : 10.1214/aop/1024404508

M. H. Davis and G. Burstein, A deterministic approach to stochastic optimal control with application to anticipative control, Stochastics Stochastics Rep, vol.40, pp.3-4, 1992.

K. Debrabant and E. R. Jakobsen, Semi-Lagrangian schemes for linear and fully non-linear diffusion equations, Mathematics of Computation, vol.82, issue.283, pp.1433-1462, 2013.
DOI : 10.1090/S0025-5718-2012-02632-9

C. Dellacherie and P. Meyer, Probabilities and Potential, 1979.

D. Marco, S. Friz, and P. K. , Varadhan's estimates, projected diffusions, and local volatilities, 2013.

L. Denis, M. Hu, and S. Peng, Function Spaces and Capacity Related to a Sublinear Expectation: Application to G-Brownian Motion Paths, Potential Analysis, vol.30, issue.8, pp.139-161, 2011.
DOI : 10.1007/s11118-010-9185-x

L. Denis and C. Martini, A theoretical framework for the pricing of contingent claims in the presence of model uncertainty, The Annals of Applied Probability, vol.16, issue.2, pp.827-852, 2006.
DOI : 10.1214/105051606000000169

J. D. Deuschel, P. K. Friz, V. Jacquier, and S. A. , Marginal Density Expansions for Diffusions and Stochastic Volatility I: Theoretical Foundations, Communications on Pure and Applied Mathematics, vol.20, issue.1, pp.40-82, 2014.
DOI : 10.1002/cpa.21478

J. D. Deuschel, P. K. Friz, V. Jacquier, and S. A. , Marginal Density Expansions for Diffusions and Stochastic Volatility II: Applications, Communications on Pure and Applied Mathematics, vol.20, issue.1, pp.321-350, 2014.
DOI : 10.1002/cpa.21483

J. Diehl, P. Friz, and P. Gassiat, Stochastic control with rough paths, Applied Mathematics & Optimization, vol.6, issue.1, p.2013
DOI : 10.1007/s00245-016-9333-9
URL : https://hal.archives-ouvertes.fr/hal-01419763

Y. Dolinsky, Numerical schemes for G-Expectations, Electronic Journal of Probability, vol.17, issue.0, pp.1-15, 2012.
DOI : 10.1214/EJP.v17-2284

B. Dupire, Functional Itô calculus, papers.ssrn.com, 2009.

I. Ekren, C. Keller, N. Touzi, and J. Zhang, On viscosity solutions of path dependent PDEs, The Annals of Probability, vol.42, issue.1, pp.204-236, 2014.
DOI : 10.1214/12-AOP788

I. Ekren, N. Touzi, and J. Zhang, Optimal stopping under nonlinear expectation, Stochastic Processes and Their Applications, pp.3277-3311, 2014.
DOI : 10.1016/j.spa.2014.04.006
URL : http://arxiv.org/abs/1209.6601

I. Ekren, N. Touzi, and J. Zhang, Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs : Part I, preprint

I. Ekren, N. Touzi, and J. Zhang, Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs : Part II, preprint

E. Karoui and N. , Les Aspects Probabilistes Du Controle Stochastique, Lecture Notes in Mathematics, vol.876, pp.73-238, 1981.
DOI : 10.1007/BFb0097499

E. Karoui, N. Kapoudjian, C. Pardoux, E. Peng, S. Quenez et al., Reflected solutions of backward SDEs, and related obstacle problems for PDEs, Ann. Probab, vol.25, issue.2, pp.702-737, 1997.

E. Karoui, N. Jeanblanc-picqué, M. Shreve, and S. E. , Robustness of the Black and Scholes Formula, Mathematical Finance, vol.8, issue.2, pp.93-126, 1998.
DOI : 10.1111/1467-9965.00047

E. Karoui, N. Peng, S. , M. Quenez, and M. , Backward Stochastic Differential Equations in Finance, Mathematical Finance, vol.7, issue.1, pp.1-71, 1997.
DOI : 10.1111/1467-9965.00022

N. Karoui and X. Tan, Capacities, measurable selection and dynamic programming Part I : abstract framework, preprint

N. Karoui and X. Tan, Capacities, measurable selection and dynamic programming Part II : application in stochastic control problems, preprint

L. C. Evans and H. Ishii, A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.2, issue.1, pp.1-20, 1985.
DOI : 10.1016/S0294-1449(16)30409-7

L. C. Evans, P. E. Souganidis, G. Fournier, and M. Willem, A PDE approach to certain large deviation problems for systems of parabolic equations, pp.229-258, 1989.

A. Fahim, N. Touzi, and X. Warin, A probabilistic numerical method for fully nonlinear parabolic PDEs, The Annals of Applied Probability, vol.21, issue.4, pp.1322-1364, 2011.
DOI : 10.1214/10-AAP723
URL : https://hal.archives-ouvertes.fr/hal-00367103

J. Feng and T. G. Kurtz, Large Deviations for Stochastic Processes, Mathematical Surveys and Monographs, vol.131

M. Fisher and G. Nappo, On the Moments of the Modulus of Continuity of It?? Processes, Stochastic Analysis and Applications, vol.36, issue.1, pp.103-122, 2010.
DOI : 10.1007/978-1-4612-0949-2

W. H. Fleming, Exit probabilities and optimal stochastic control, Applied Mathematics & Optimization, vol.17, issue.1, pp.329-346, 1978.
DOI : 10.1007/BF01442148

W. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2006.

W. H. Fleming and E. Souganidis, PDE-viscosity solution approach to some problems of large deviations, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4 e série, pp.171-192, 1986.

M. Ford and A. Jacquier, SMALL-TIME ASYMPTOTICS FOR IMPLIED VOLATILITY UNDER THE HESTON MODEL, International Journal of Theoretical and Applied Finance, vol.12, issue.06, pp.861-876, 2009.
DOI : 10.1142/S021902490900549X

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, 1984.

F. Gao and J. Liu, LARGE DEVIATIONS FOR SMALL PERTURBATIONS OF SDES WITH NON-MARKOVIAN COEFFICIENTS AND THEIR APPLICATIONS, Stochastics and Dynamics, vol.06, issue.04, pp.487-520, 2006.
DOI : 10.1142/S0219493706001840

J. Gatheral, L. Hsu, E. P. Ouyang, C. , W. et al., ASYMPTOTICS OF IMPLIED VOLATILITY IN LOCAL VOLATILITY MODELS, Mathematical Finance, vol.8, issue.1, pp.591-620
DOI : 10.1111/j.1467-9965.2010.00472.x

E. Gobet, J. P. Lemor, and X. Warin, A regression-based Monte Carlo method to solve backward stochastic differential equations, The Annals of Applied Probability, vol.15, issue.3, pp.2172-2202, 2005.
DOI : 10.1214/105051605000000412

W. Guo, J. Zhang, and J. Zhuo, A monotone scheme for high-dimensional fully nonlinear PDEs, The Annals of Applied Probability, vol.25, issue.3
DOI : 10.1214/14-AAP1030

J. Guyon and P. Henry-labordère, Uncertain Volatility Model : A Monte-Carlo Approach, Journal of Computational Finance, vol.14, issue.3, 2011.

I. Gyöngy, Mimicking the one-dimensional marginal distributions of processes having an ito differential, Probability Theory and Related Fields, vol.4, issue.4, pp.501-516, 1986.
DOI : 10.1007/BF00699039

P. Hagan, A. Lesniewski, and D. Woodward, Probability distribution in the SABR model of stochastic volatility, Working Paper, 2004.

S. Hamadène, Reflected BSDE's with discontinuous barrier and application, Stochastics An International Journal of Probability and Stochastic Processes, vol.74, issue.3, pp.571-596, 2002.
DOI : 10.1080/1045112021000036545

P. Henry-labordère, Analysis, Geometry, and Modeling in Finance, Chapman & Hall/CRC, Financial Mathematics Series, vol.20085168, 2008.
DOI : 10.1201/9781420087000

P. Henry-labordère, C. Litterer, and Z. Ren, A dual algorithm for stochastic control problem : applications to uncertain volatility models and CVA

P. Henry-labordere, X. Tan, and N. Touzi, A numerical algorithm for a class of BSDEs via the branching process. Stochastic Process, Appl, vol.124, issue.2, pp.1112-1140, 2014.

M. Hu, S. Ji, and S. Peng, Backward Stochastic Differential Equation Driven by Fractional Brownian Motion, Stochastic Processes and their Applications, 2014.
DOI : 10.1137/070709451

R. Karandikar, On pathwise stochastic integration, Stochastic Processes and Their Applications, pp.11-18, 1995.

I. Kharroubi, N. Langrené, and H. Pham, Abstract., Monte Carlo Methods and Applications, vol.20, issue.2, pp.145-165
DOI : 10.1515/mcma-2013-0024
URL : https://hal.archives-ouvertes.fr/hal-00395363

I. Kharroubi, N. Langrené, and H. Pham, Discrete time approximation of fully nonlinear HJB equations via BSDEs with nonpositive jumps, The Annals of Applied Probability, vol.25, issue.4
DOI : 10.1214/14-AAP1049
URL : https://hal.archives-ouvertes.fr/hal-01172286

M. Kobylanski, differential equations with quadratic growth, The Annals of Probability, vol.28, issue.2, pp.558-602, 2000.
DOI : 10.1214/aop/1019160253

M. Kobylanski and M. Quenez, Optimal stopping in a general framework, IMS) : OAJ, pp.1-28, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00519457

N. V. Krylov, On the rate of convergence of finite-difference approximations for Bellmans equations with variable coefficients. Probability Theory and Related Fields, pp.1-16, 2000.

N. V. Krylov, Approximating Value Functions for Controlled Degenerate Diffusion Processes by Using Piece-Wise Constant Policies, Electronic Journal of Probability, vol.4, issue.0, pp.1-19, 1999.
DOI : 10.1214/EJP.v4-39

H. J. Kushner and P. Dupuis, Numerical methods for stochastic control problems in continuous time, Applications of Mathematics, vol.24, 1992.

P. Lions, Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part I: the case of bounded stochastic evolutions, Acta Mathematica, vol.161, issue.0, pp.3-4, 1988.
DOI : 10.1007/BF02392299

P. Lions, Viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions. II. Optimal control of Zakai's equation. Stochastic partial differential equations and applications, II (Trento, Lecture Notes in Math, pp.147-170, 1390.

P. Lions, Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. III. Uniqueness of viscosity solutions for general second-order equations, Journal of Functional Analysis, vol.86, issue.1, pp.1-18, 1989.
DOI : 10.1016/0022-1236(89)90062-1

N. Lukoyanov and . Yu, On viscosity solution of functional Hamilton-Jacobi type equations for hereditary systems, Proceedings of the Steklov Institute of Mathematics Issue 2 Supplement, pp.190-200, 2007.
DOI : 10.1134/S0081543807060132

T. J. Lyons, Uncertain volatility and the risk-free synthesis of derivatives, Applied Mathematical Finance, vol.20, issue.2, pp.117-133, 1995.
DOI : 10.1002/cpa.3160450103

J. Ma, Z. Ren, N. Touzi, and J. Zhang, Large deviations for non-Markovian diffusions and a path-dependent Eikonal equation, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.52, issue.3, pp.1407-5314
DOI : 10.1214/15-AIHP678
URL : https://hal.archives-ouvertes.fr/hal-01026115

A. Matoussi, D. Possamaï, and C. , Zhou Robust utility maximization in non-dominated models with 2BSDE : the uncertain volatility model, Mathematical Finance

A. Neufeld and M. Nutz, Superreplication under volatility uncertainty for measurable claims, Electronic Journal of Probability, vol.18, issue.0, pp.1-14, 2013.
DOI : 10.1214/EJP.v18-2358

M. Nutz and G. Random, Random $G$-expectations, The Annals of Applied Probability, vol.23, issue.5, pp.1755-1777, 2013.
DOI : 10.1214/12-AAP885

M. Nutz and R. Van-handel, Constructing sublinear expectations on path space, Stochastic Processes and their Applications, pp.3100-3121, 2013.
DOI : 10.1016/j.spa.2013.03.022

E. Pardoux and S. Peng, Adapted solutions of backward stochastic differential equations, System and Control Letters, pp.55-61, 1990.

E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in Control and Inform. Sci, vol.176, pp.200-217, 1991.
DOI : 10.1007/BFb0007334

S. Peng, Backward Stochastic Differential Equation, Nonlinear Expectation and Their Applications, Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), 2010.
DOI : 10.1142/9789814324359_0019

S. Peng, Backward SDE and related g-expectation, Backward stochastic differential equations, pp.141-159, 1995.

S. Peng and G. , Brownian motion and dynamic risk measure under volatility uncertainty, preprint

S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyers type, Probability Theory and Related Fields, vol.113, issue.4, pp.473-499, 1999.
DOI : 10.1007/s004400050214

S. Peng and M. Xu, The smallest g-supermartingale and reflected BSDE with single and double L2L2 obstacles, Annales de l'Institut Henri Poincare (B) Probability and Statistics, vol.41, issue.3, pp.605-630, 2005.
DOI : 10.1016/j.anihpb.2004.12.002

T. Pham and J. Zhang, Two Person Zero-Sum Game in Weak Formulation and Path Dependent Bellman--Isaacs Equation, SIAM Journal on Control and Optimization, vol.52, issue.4, pp.2090-2121, 2014.
DOI : 10.1137/120894907

D. Possamai and X. Tan, Weak approximation of second-order BSDEs, The Annals of Applied Probability, vol.25, issue.5
DOI : 10.1214/14-AAP1055
URL : https://hal.archives-ouvertes.fr/hal-01102789

Z. Ren, Perron's method for viscosity solutions of semilinear path dependent PDEs, pre- print

Z. Ren, Viscosity solutions of fully nonlinear elliptic path dependent partial differential equations, The Annals of Applied Probability, vol.26, issue.6
DOI : 10.1214/16-AAP1178

Z. Ren and X. Tan, On the convergence of monotone schemes for path-dependent PDEs, Stochastic Processes and their Applications
DOI : 10.1016/j.spa.2016.10.002

Z. Ren, N. Touzi, and J. Zhang, An overview on Viscosity Solutions of Path-Dependent PDEs, Stochastic Analysis and Applications 2014 -In Honour of Terry Lyons, Proceedings in Mathematics and Statistics, 2014.

Z. Ren, N. Touzi, and J. Zhang, Comparison of viscosity solutions of semilinear pathdependent partial differential equations

L. C. Rogers, Pathwise Stochastic Optimal Control, SIAM Journal on Control and Optimization, vol.46, issue.3, pp.1116-113, 2007.
DOI : 10.1137/050642885

W. Rudin, Real and Complex Analysis second edition, International Series in Pure and Applied Mathematics, 1991.

A. I. Sakhanenko, A new way to obtain estimates in the invariance principle, High Dimensional Probability II, pp.221-243, 2000.

H. M. Soner, N. Touzi, and J. Zhang, Quasi-sure Stochastic Analysis through Aggregation, Electronic Journal of Probability, vol.16, issue.0, pp.1844-1879, 2011.
DOI : 10.1214/EJP.v16-950

H. M. Soner, N. Touzi, and J. Zhang, Well-posedness of second order backward SDEs, Probability Theory and Related Fields, pp.149-190, 2012.

A. Swiech, Unbounded second order partial differential equations in infinite-dimensional Hilbert spaces, Comm. Partial Differential Equations, vol.19, pp.1999-2036, 1994.

X. Tan, A splitting method for fully nonlinear degenerate parabolic PDEs, Electron, J. Probab, vol.18, issue.15, pp.1-24, 2013.

X. Tan, Discrete-time probabilistic approximation of path-dependent stochastic control problems, The Annals of Applied Probability, vol.24, issue.5, pp.1803-1834, 2014.
DOI : 10.1214/13-AAP963
URL : https://hal.archives-ouvertes.fr/hal-01246998

L. Wang, On the regularity theory of fully nonlinear parabolic equations: II, Communications on Pure and Applied Mathematics, vol.45, issue.2, pp.141-178, 1992.
DOI : 10.1002/cpa.3160450202

J. Zhang, A numerical scheme for BSDEs, The Annals of Applied Probability, vol.14, issue.1, pp.459-488, 2004.
DOI : 10.1214/aoap/1075828058

J. Zhang and J. Zhuo, Monotone schemes for fully nonlinear parabolic path dependent PDEs, Journal of Financial Engineering, vol.01, issue.01, pp.1450005-1450015, 2014.
DOI : 10.1142/S2345768614500056