C. Alasseur, I. Ben-taher, and A. Matoussi, An Extended Mean Field Game for Storage in Smart Grids, Journal of Optimization Theory and Applications, 2020.
URL : https://hal.archives-ouvertes.fr/hal-01740707

L. Andersen and M. Broadie, Primal-Dual Simulation Algorithm for Pricing Multidimensional American Options, Management Science, vol.50, issue.9, pp.1222-1234, 2004.

F. Antonelli, Backward-Forward Stochastic Differential Equations, The Annals of Applied Probability, vol.3, issue.3, pp.777-793, 1993.

N. Baradel, B. Bouchard, and N. M. Dang, Optimal Control Under Uncertainty and Bayesian Parameters Adjustments, SIAM Journal on Control and Optimization, vol.56, issue.2, pp.1038-1057, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01590606

G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Collection SMAI, 1994.

G. Barles, R. Buckdahn, and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations, Stochastics and Stochastic Reports, vol.60, issue.1-2, pp.57-83, 1997.

P. Barrieu and N. El-karoui, Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs, The Annals of Probability, vol.41, issue.3B, pp.1831-1863, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00843113

E. Bayraktar and R. Kravitz, Stability of exponential utility maximization with respect to market perturbations, Stochastic Processes and their Applications, vol.123, issue.5, pp.1671-1690, 2013.

D. Becherer, Bounded solutions to backward SDEs with jumps for utility optimization and indifference hedging, The Annals of Applied Probability, vol.16, issue.4, pp.2027-2054, 2006.

G. Beer, Wijsman convergence: A survey, Set-Valued Analysis, vol.2, issue.1-2, pp.77-94, 1994.

A. Bensoussan, J. Frehse, and P. Yam, Mean Field Games and Mean Field Type Control Theory, vol.101, 2013.

A. Bensoussan, Y. Li, and S. C. Yam, Backward stochastic dynamics with a subdifferential operator and non-local parabolic variational inequalities, Stochastic Processes and their Applications, vol.128, issue.2, pp.644-688, 2018.

F. E. Benth, K. H. Karlsen, and K. Reikvam, A semilinear Black and Scholes partial differential equation for valuing American options, Finance and Stochastics, vol.7, issue.3, pp.277-298, 2003.

F. E. Benth, K. Karlsen, and K. Reikvam, A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence, Interfaces and Free Boundaries, vol.6, issue.4, pp.379-404, 2004.

J. Bismut, Théorie probabiliste du contrôle des diffusions, Memoirs of the American Mathematical Society, vol.4, issue.167, pp.0-0, 1976.

F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, vol.81, issue.3, pp.637-654, 1973.

B. Bouchard and J. Chassagneux, Delta Hedging in Complete Market, Universitext, pp.127-165, 2016.

B. Bouchard, K. Wai-chau, A. Manai, and A. Sid-ali, Monte-Carlo methods for the pricing of American options: a semilinear BSDE point of view, ESAIM: Proceedings and Surveys, vol.65, pp.294-308x, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01666399

B. Bouchard, X. Tan, X. Warin, and Y. Zou, Numerical approximation of BSDEs using local polynomial drivers and branching processes, Monte Carlo Methods and Applications, vol.23, issue.4, pp.241-263, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01419981

B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Processes and their Applications, vol.111, issue.2, pp.175-206, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00103046

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.
URL : https://hal.archives-ouvertes.fr/hal-00367355

B. Bouchard and X. Warin, Monte-Carlo Valuation of American Options: Facts and New Algorithms to Improve Existing Methods, Springer Proceedings in Mathematics, pp.215-255, 2012.

P. Briand and R. Elie, A simple constructive approach to quadratic BSDEs with or without delay, Stochastic Processes and their Applications, vol.123, issue.8, pp.2921-2939, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00717226

P. Briand and Y. Hu, BSDE with quadratic growth and unbounded terminal value, Probability Theory and Related Fields, vol.136, issue.4, pp.604-618, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00004619

P. Briand and Y. Hu, Quadratic BSDEs with convex generators and unbounded terminal conditions, Probability Theory and Related Fields, vol.141, issue.3-4, pp.543-567, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00136605

R. Buckdahn, B. Djehiche, J. Li, and S. Peng, Mean-field backward stochastic differential equations: A limit approach, The Annals of Probability, vol.37, issue.4, pp.1524-1565, 2009.

R. Buckdahn, J. Li, and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Processes and their Applications, vol.119, issue.10, pp.3133-3154, 2009.

P. Cardaliaguet, Mean field games: the master equation and the mean field limit, Séminaire Laurent Schwartz ? EDP et applications, pp.1-10, 2015.

R. Carmona and F. Delarue, Probabilistic Analysis of Mean-Field Games, SIAM Journal on Control and Optimization, vol.51, issue.4, pp.2705-2734, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00714589

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications II, Probabilistic Theory of Mean Field Games with Applications I-II, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01868145

R. Carmona, F. Delarue, and A. Lachapelle, Control of McKean?Vlasov dynamics versus mean field games, Mathematics and Financial Economics, vol.7, issue.2, pp.131-166, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00655842

P. Carr, R. Jarrow, and R. Myneni, ALTERNATIVE CHARACTERIZATIONS OF AMERICAN PUT OPTIONS, Mathematical Finance, vol.2, issue.2, pp.87-106, 1992.

M. G. Crandall, H. Ishii, and P. Lions, user's guide to viscosity solutions\\ of second order\\ partial differential equations, Bulletin of the American Mathematical Society, vol.27, issue.1, pp.1-68, 1992.

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.

F. Delarue, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic Processes and their Applications, vol.99, issue.2, pp.209-286, 2002.

F. Delbaen, P. Grandits, T. Rheinlander, D. Samperi, M. Schweizer et al., Exponential Hedging and Entropic Penalties, Mathematical Finance, vol.12, issue.2, pp.99-123, 2002.

F. Delbaen, Y. Hu, and X. Bao, Backward SDEs with superquadratic growth, Probability Theory and Related Fields, vol.150, issue.1-2, pp.145-192, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00362685

B. Djehiche and S. Hamadène, Mean-field backward-forward stochastic differential equations and nonzero sum stochastic differential games, 2019.

D. Duffie, Dynamic asset pricing theory, 2010.

N. E. Karoui, S. Hamadène, and A. Matoussi, Chapter Eight. BSDEs And Applications, Indifference Pricing, pp.267-320, 2008.

N. E. Karoui, A. Matoussi, and A. Ngoupeyou, Quadratic exponential semimartingales and application to BSDEs with jumps, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01740692

N. El-karoui, E. Pardoux, and M. C. Quenez, Reflected Backward SDEs and American Options, Numerical Methods in Finance, vol.13, pp.215-231, 1997.

N. El-karoui, S. Peng, and M. C. Quenez, Backward Stochastic Differential Equations in Finance, Mathematical Finance, vol.7, issue.1, pp.1-71, 1997.

W. H. Fleming and P. E. Souganidis, On the existence of value functions of twoplayer, zero-sum stochastic differential games, Indiana University Mathematics Journal, vol.38, issue.2, pp.293-314, 1989.

C. Frei, Convergence results for the indifference value based on the stability of BSDEs, Stochastics, vol.85, issue.3, pp.464-488, 2012.

C. Frei, Convergence results for the indifference value based on the stability of BSDEs, Stochastics, vol.85, issue.3, pp.464-488, 2012.

D. A. Gomes and J. Saúde, Mean Field Games Models?A Brief Survey, Dynamic Games and Applications, vol.4, issue.2, pp.110-154, 2013.

S. Hamadène, Backward?forward SDE?s and stochastic differential games, Stochastic Processes and their Applications, vol.77, issue.1, pp.1-15, 1998.

S. Hamadene and J. P. Lepeltier, Zero-sum stochastic differential games and backward equations, Systems & Control Letters, vol.24, issue.4, pp.259-263, 1995.

J. M. Harrison and S. R. Pliska, A stochastic calculus model of continuous trading: Complete markets, Stochastic Processes and their Applications, vol.15, issue.3, pp.313-316, 1983.

M. B. Haugh and L. Kogan, Pricing American Options: A Duality Approach, Operations Research, vol.52, issue.2, pp.258-270, 2004.

P. Henry-labordère, Deep Primal-Dual Algorithm for BSDEs: Applications of Machine Learning to CVA and IM, SSRN Electronic Journal, vol.25, issue.7, p.67, 2017.

P. Henry-labordère, N. Oudjane, X. Tan, N. Touzi, and X. Warin, Branching diffusion representation of semilinear PDEs and Monte Carlo approximation, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, vol.55, issue.1, pp.184-210, 2019.

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

U. Horst, Y. Hu, P. Imkeller, A. Réveillac, and J. Zhang, Forward?backward systems for expected utility maximization, Stochastic Processes and their Applications, vol.124, issue.5, pp.1813-1848, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00631727

Y. Hu, P. Imkeller, and M. Müller, Utility maximization in incomplete markets, The Annals of Applied Probability, vol.15, issue.3, pp.1691-1712, 2005.

Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations, Probability Theory and Related Fields, vol.103, issue.2, pp.273-283, 1995.

S. D. Jacka, Optimal Stopping and the American Put, Mathematical Finance, vol.1, issue.2, pp.1-14, 1991.

P. Jaillet, D. Lamberton, and B. Lapeyre, Variational inequalities and the pricing of American options, Acta Applicandae Mathematicae, vol.21, issue.3, pp.263-289, 1990.
URL : https://hal.archives-ouvertes.fr/hal-01667008

M. Jeanblanc, A. Matoussi, and A. Ngoupeyou, Preprint repository arXiv achieves milestone million uploads, Physics Today, 2014.

E. Jouini and C. Napp, Convergence of utility functions and convergence of optimal strategies, Finance and Stochastics, vol.8, issue.1, pp.133-144, 2004.
URL : https://hal.archives-ouvertes.fr/halshs-00151579

M. Kac, Foundations of kinetic theory, Proceedings of The third Berkeley symposium on mathematical statistics and probability, vol.3, pp.171-197, 1956.

I. Karatzas, J. P. Lehoczky, and S. E. Shreve, Optimal Portfolio and Consumption Decisions for a ?Small Investor? on a Finite Horizon, SIAM Journal on Control and Optimization, vol.25, issue.6, pp.1557-1586, 1987.

I. Karatzas, J. P. Lehoczky, S. E. Shreve, and G. Xu, Martingale and Duality Methods for Utility Maximization in an Incomplete Market, SIAM Journal on Control and Optimization, vol.29, issue.3, pp.702-730, 1991.

I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, vol.39, 1998.

C. Kardaras, The continuous behavior of the numéraire portfolio under small changes in information structure, probabilistic views and investment constraints, Stochastic Processes and their Applications, vol.120, issue.3, pp.331-347, 2010.

C. Kardaras and G. ?itkovi?, STABILITY OF THE UTILITY MAXIMIZATION PROBLEM WITH RANDOM ENDOWMENT IN INCOMPLETE MARKETS, Mathematical Finance, vol.21, issue.2, pp.no-no, 2010.

N. Kazamaki, BMO-martingales, Lecture Notes in Mathematics, pp.25-52, 1994.

I. J. Kim, The Analytic Valuation of American Options, Review of Financial Studies, vol.3, issue.4, pp.547-572, 1990.

M. Kobylanski, differential equations with quadratic growth, The Annals of Probability, vol.28, issue.2, pp.558-602, 2000.

D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, The Annals of Applied Probability, vol.9, issue.3, pp.904-950, 1999.

K. Larsen, CONTINUITY OF UTILITY-MAXIMIZATION WITH RESPECT TO PREFERENCES, Mathematical Finance, vol.19, issue.2, pp.237-250, 2009.

K. Larsen and G. ?itkovi?, Stability of utility-maximization in incomplete markets, Stochastic Processes and their Applications, vol.117, issue.11, pp.1642-1662, 2007.

J. Lasry and P. Lions, Jeux à champ moyen. I ? Le cas stationnaire, Comptes Rendus Mathematique, vol.343, issue.9, pp.619-625, 2006.

J. Lasry and P. Lions, Jeux à champ moyen. II ? Horizon fini et contrôle optimal, Comptes Rendus Mathematique, vol.343, issue.10, pp.679-684, 2006.

J. Lasry and P. Lions, Mean field games, Japanese Journal of Mathematics, vol.2, issue.1, pp.229-260, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00667356

J. P. Lepeltier and J. San-martin, Backward stochastic differential equations with continuous coefficient, Statistics & Probability Letters, vol.32, issue.4, pp.425-430, 1997.

J. Li, Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integral-PDEs, Stochastic Processes and their Applications, vol.128, issue.9, pp.3118-3180, 2018.

P. Lions and J. Lasry, Large Investor Trading Impacts on Volatility, Paris-Princeton Lectures on Mathematical Finance 2004, pp.173-190, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00667394

J. Liu and J. Ren, Comparison theorem for solutions of backward stochastic differential equations with continuous coefficient, Statistics & Probability Letters, vol.56, issue.1, pp.93-100, 2002.

Y. Liu and J. Ma, Optimal reinsurance/investment problems for general insurance models, The Annals of Applied Probability, vol.19, issue.4, pp.1495-1528, 2009.

J. Ma, J. Morel, and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, p.132, 1999.

J. Ma, P. Protter, and J. Yong, Solving forward-backward stochastic differential equations explicitly ? a four step scheme, Probability Theory and Related Fields, vol.98, issue.3, pp.339-359, 1994.

A. Manai and A. Matoussi, Stability of the utility maximization problem in non-equivalent markets: A BSDE point of view, 2019.

M. Mania and M. Schweizer, Dynamic exponential utility indifference valuation, The Annals of Applied Probability, vol.15, issue.3, pp.2113-2143, 2005.

X. Mao, Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients, Stochastic Processes and their Applications, vol.58, issue.2, pp.281-292, 1995.

A. Matoussi, A. Manai, and R. Salhi, Preprint repository arXiv achieves milestone million uploads, Physics Today, 2014.

H. P. Mckean, Application of brownian motion to the equation of kolmogorov-petrovskii-piskunov, Communications on Pure and Applied Mathematics, vol.28, issue.3, pp.323-331, 1975.

H. Mckean, A free boundary problem for the heat equation arising from a problem in mathematical economics, Industrial Management Review, vol.6, pp.32-39, 1965.

H. P. Mckean, A CLASS OF MARKOV PROCESSES ASSOCIATED WITH NONLINEAR PARABOLIC EQUATIONS, Proceedings of the National Academy of Sciences, vol.56, issue.6, pp.1907-1911, 1966.

R. C. Merton, Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case, The Review of Economics and Statistics, vol.51, issue.3, p.247, 1969.

R. C. Merton, Optimum Consumption and Portfolio Rules in a Continuous-Time Model**I would like to thank P. A. Samuelson, R. M. Solow, P. A. Diamond, J. A. Mirrlees, J. A. Flemming, and D. T. Scheffman for their helpful discussions. Of course, all errors are mine. Aid from the National Science Foundation is gratefully acknowledged. An earlier version of the paper was presented at the second World Congress of the Econometric Society, Cambridge, England., Stochastic Optimization Models in Finance, pp.621-661, 1975.

H. Min, Y. Peng, and Y. Qin, Fully Coupled Mean-Field Forward-Backward Stochastic Differential Equations and Stochastic Maximum Principle, Abstract and Applied Analysis, vol.2014, pp.1-15, 2014.

M. Mocha and N. Westray, Quadratic Semimartingale BSDEs Under an Exponential Moments Condition, Lecture Notes in Mathematics, pp.105-139, 2012.

M. Mocha and N. Westray, The Stability of the Constrained Utility Maximization Problem: A BSDE Approach, SIAM Journal on Financial Mathematics, vol.4, issue.1, pp.117-150, 2013.

O. Morgenstern and J. V. Neumann, Theory of games and economic behavior, 1953.

M. Morlais, Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem, Finance and Stochastics, vol.13, issue.1, pp.121-150, 2008.

M. Morlais, Utility maximization in a jump market model, Stochastics, vol.81, issue.1, pp.1-27, 2009.
URL : https://hal.archives-ouvertes.fr/hal-01835198

M. Morlais, A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem, Stochastic Processes and their Applications, vol.120, issue.10, pp.1966-1995, 2010.

R. Myneni, The pricing of the American option, The Annals of Applied Probability, pp.1-23, 1992.

A. B. Ngoupeyou, Optimisation des portefeuilles d'actifs soumis au risque de défaut, 2010.

E. Pardoux, Generalized discontinuous backward stochastic differential equations. Pitman Research Notes in Mathematics Series, pp.207-219, 1997.

É. Pardoux, Backward Stochastic Differential Equations and Viscosity Solutions of Systems of Semilinear Parabolic and Elliptic PDEs of Second Order, Stochastic Analysis and Related Topics VI, pp.79-127, 1998.

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems & Control Letters, vol.14, issue.1, pp.55-61, 1990.

E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probability Theory and Related Fields, vol.114, issue.2, pp.123-150, 1999.

S. Peng, A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation, Stochastics: An International Journal of Probability and Stochastic Processes, vol.38, pp.119-134, 1992.

S. Peng and Z. Wu, Fully Coupled Forward-Backward Stochastic Differential Equations and Applications to Optimal Control, SIAM Journal on Control and Optimization, vol.37, issue.3, pp.825-843, 1999.

H. Pham, Optimal stopping, free boundary, and American option in a jump-diffusion model, Applied Mathematics and Optimization, vol.35, issue.2, pp.145-164, 1997.

H. Pham, Optimal stopping of controlled jump diffusion processes: a viscosity solution approach, In Journal of Mathematical Systems, Estimation and Control, 1998.

D. Possamai, N. Kazi-tani, and C. Zhou, Quadratic BSDEs with jumps: a fixed-point approach, Electronic Journal of Probability, vol.20, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01245360

L. C. Rogers, Monte Carlo valuation of American options, Mathematical Finance, vol.12, issue.3, pp.271-286, 2002.

R. Rouge and N. El-karoui, Pricing Via Utility Maximization and Entropy, Mathematical Finance, vol.10, issue.2, pp.259-276, 2000.

M. Royer, Backward stochastic differential equations with jumps and related non-linear expectations. Stochastic processes and their applications, vol.116, pp.1358-1376, 2006.

P. A. Samuelson, Rational theory of warrant pricing, vol.6, p.13, 1965.

W. Schachermayer, Optimal investment in incomplete markets when wealth may become negative, The Annals of Applied Probability, vol.11, issue.3, pp.694-734, 2001.

S. Tang and X. Li, Necessary Conditions for Optimal Control of Stochastic Systems with Random Jumps, SIAM Journal on Control and Optimization, vol.32, issue.5, pp.1447-1475, 1994.

R. Tevzadze, Solvability of backward stochastic differential equations with quadratic growth, Stochastic Processes and their Applications, vol.118, issue.3, pp.503-515, 2008.

K. Weston, Stability of utility maximization in nonequivalent markets, Finance and Stochastics, vol.20, issue.2, pp.511-541, 2016.

Z. Wu, Fully coupled FBSDE with Brownian motion and Poisson process in stopping time duration, Journal of the Australian Mathematical Society, vol.74, issue.2, pp.249-266, 2003.

H. Xing, Stability of the exponential utility maximization problem with respect to preferences, Mathematical Finance, vol.27, issue.1, pp.38-67, 2017.

J. Yong, Finding adapted solutions of forward-backward stochastic differential equations: method of continuation. Probability Theory and Related Fields, vol.107, pp.537-572, 1997.

J. Yong, Forward-backward stochastic differential equations with mixed initial-terminal conditions, Transactions of the American Mathematical Society, vol.362, issue.2, pp.1047-1096, 2010.

J. Zhang, A numerical scheme for BSDEs, The Annals of Applied Probability, vol.14, issue.1, pp.459-488, 2004.

W. Zhen, Forward-backward stochastic differential equations with Brownian motion and poisson process, Acta Mathematicae Applicatae Sinica, vol.15, issue.4, pp.433-443, 1999.