Nizar Touzi

Lecture Notes

Stochastic Calculus in Finance (with Peter Tankov), Ecole Polytechnique, 3ème année, PA Mathématiques Appliquées (pdf file).

Deterministic and Stochastic Control, Application to Finance, Master Probabilité et Finance Ecole Polytechnique – Université Paris 6 (pdf file).

Chaînes de Markov et martingales en temps discret, 2ème année, PA Mathématiques Appliquées (pdf file).

Aléatoire (avec Josselin Garnier & Sylvie Méléard), Ecole Polytechnique, 1ère année, PA Mathématiques Appliquées (pdf file).

Distinctions

Member of Academia Europaea (The Academy of Europe), Section Mathematics, Elected 2024

Minerva Lecture (2020) at the Department of Mathematics, Columbia University, New York.

ICM 2010(Hyderabad) Invited Session Speaker (pdf file)

French Academy of Science – Natixis Foudation, Louis Bachelier Prize 2012

The University of Toronto Dean’s Distinguished Visitor Chair, Fields Institute, April-June 2010

Best Young Researcher in Finance Award 2007 of the Europlace Institute of Finance

Editorial Activities

Applied Mathematics and Optimization, Editor in chief, since January '24

Finance & Stochastics, Co-editor since January '07, and Associate Editor June '00 - December 06

ESAIM: Control, Optimisation and Calculus of Variations, Associate Editor since October '18

Stochastics: an International Journal of Probability and Stochastic Processes, Associate Editor since January ’16

Journal of Optimization Theory and Applications, Associate Editor since January ’14

Mathematical Control and Related Fields, Associate Editor since September ’12

Past editorial activities:

Stochastic Processes and their Applications, Associate Editor since January ’16

Mathematical Finance, Associate Editor, Novembre ’03 - June '20

Advances in Calculus of Variations, Associate Editor, January ’18 — December ‘22

Annals of Applied Probability, Associate Editor, January ’13 - December '18

Electronic Journal of Probability / Electronic Communications in Probability, Associate Editor June ’04 - December ’14

SIAM Journal on Financial Mathematics, Associate Editor January ’09 - December ’12

Journal of Financial Econometrics, Associate Editor January ’01 - December ’06

Applied Mathematics Research eXpress (AMRX), Associate Editor January ’04 - December ’07

Tunisian Journal of Mathematics, Associate Editor since September ’19

Paris-Princeton Lectures in Mathematical Finance, Founder and Co-Editor

Springer Briefs in Mathematical Finance, Associate Editor

PhD Students

Present:

Nathan Sauldubois, Mathieu Lise, Yuxing Huang, Xuyang Lin

Past:

Leila Bassou (June 2024)

Mehdi Talbi (October 2022, Assistant Professor University Paris City)

Bowen Sheng (August 1, 2022)

Heythem Farhat (February 2021, Natixis, Paris)

Kaitong Hu (June 2020, Square Point Capital, London)

Hadrien De March (June 2018, Startup)

Gaoyue Guo (October 2016, Assistant Professor CentraleSupelec)

Zhenjie Ren (October 2015, Professor University Paris Saclay, Evry)

Guillaume Royer (December 2013, Bank of America, London)

Emilie Fabre (February 2012 Société Générale, Paris)

Xiaolu Tan (December 2011, Associate Professor Chinese University of Hong Kong)

Dylan Possamaï (December 2011, Professor ETH Zürich)

Gilles-Edouard Espinosa (June 2010, BNP Paribas)

Arash Fahim (April 2010, Associate Professor Florida State University)

Arnaud Porchet (February 2008, Goldman Sachs, London)

Romuald Elie (December 2006, Professor University Marne la Vallée)

Fabian Astic (March 2007, Moody’s NY)

Imen Ben Tahar (March 2005, Assistant Professor University Paris Dauphine)

Amina Zeghal (January 2005, University Dauphine Tunis)

Moez Mrad (December 2003, Calyon, London)

Bruno Bouchard (January 2001, Professor University Paris Dauphine)

Accepted Papers

[124] L. Bassou, F. M. Djete and N. Touzi, Mean Field Game of Mutual Holding with common noise. SIAM Journal in Financial Mathematics, to appear. ArXiv:2403.16232.

[123] F. M. Djete, G. Guo, and N. Touzi, Mean field game of mutual holding with defaultable agents, and systemic risk. SIAM Journal on Control and Optimization. To appear.

[122] D. Possamaï and N. Touzi, Is there a Golden Parachute in Sannikov's principal-agent problem ? Mathematics of Operations Research. Published Online:6 May 2024. https://doi.org/10.1287/moor.2022.0305.

[121] R. Aïd, A. Kemper and N. Touzi, A Principal-Agent Model for Optimal Incentives in Renewable Investments International Journal of Theoretical and Applied Finance. Vol. 28, Nos. 1 & 2 (2025) 2550004 (38 pages).

[120] M. Talbi, N. Touzi and J. Zhang, From finite population optimal stopping to mean field optimal stopping. Annals of Applied Probability 2024, Vol. 34, No. 5, 4237-4267.

[119] F. M. Djete and N. Touzi, Mean Field Game of Mutual Holding. Annals of Applied Probability 2024, Vol. 34, No. 6, 4999-5031..

[118] J. Keppo, N. Touzi and Z. Ruiting, Dynamic Contracting in Asset Management under Investor-Partner-Manager Relationship. Operations Research. To appear.

[117] M. Talbi, N. Touzi and J. Zhang, Dynamic programming equation for the mean field optimal stopping problem.SIAM Journal on Control and Optimization, to appear.

[116] M. Talbi, N. Touzi and J. Zhang, Viscosity solutions for obstacle problems on Wasserstein space. SIAM Journal on Control and Optimization, to appear.

[115] Z. Ren, X. Tan, N. Touzi, and J. Yang, Entropic optimal planning for path-dependent mean field games. SIAM Journal on Control and Optimization, Vol. 61, Iss. 3 (2023).

[114] K. Hu, Z. Ren and N. Touzi, On path-dependent multidimensional forward-backward SDEs. Numerical Algebra, Control and Optimization 2023, Volume 13, Issue 3&4: 413-430. Doi: 10.3934/naco.2022010.

[113] Y. Lin, Z. Ren, N. Touzi, J. Yang, Random horizon principal-agent problem. SIAM Journal on Control and Optimization, to appear.

[112] A. Barrasso and N. Touzi, Controlled diffusion Mean Field Games with common noise, and McKean-Vlasov second order backward SDEs. Theory Probab. Appl. Oct 2022 Society for Industrial and Applied Mathematics Vol. 66, No. 4, pp. 613–639.

[111] R. Aïd, D. Possamaï, N. Touzi, Optimal electricity demand response contracting with responsiveness incentives. Mathematics of Operations Research, to appear

[110] Z. Ren, N. Touzi, J. Yang, Nonlinear predictable representation and L1-solutions of second-order backward SDEs. Annales de l'Intitut Henri Poincaré, Probabilité et Statistique, 2022, Vol. 58, No. 2, 639–666.

[109] P. Henry-Labordère, N. Touzi, Branching diffusion representation for nonlinear Cauchy problems and Monte Carlo approximation. Annals of Applied Probability 2021, Vol. 31, No. 5, 2350-2375.

[108] O. El Euch, T. Mastrolia, M. Rosenbaum, N. Touzi, Optimal make-take fees for market making regulation.(This version is a corrigendum of the published one) Mathematical Finance 31, pp109-148 (2021).

[107] Y. Lin, Z. Ren, N. Touzi, J. Yang, Second order backward SDE with random terminal time. Electronic Journal of Probability 2020, Vol. 25, paper no. 99, 1-43.

[106] A. Richard, X. Tan, N. Touzi, On the Root solution to the Skorokhod embedding problem given full marginals. To appear in SIAM Journal on Control and Optimization.

[105] D. Possamaï, N. Touzi, J. Zhang, Zero-sum path-dependent stochastic differential games in weak formulation. To appear in The Annals of Applied Probability.

[104] Z. Ren, N. Touzi, and J. Zhang, Comparison of Viscosity Solutions of Semi-linear Path-Dependent PDEs. SIAM J. Control and Optimization 58,1 (2020).

[103] H. De March, N. Touzi, Irreducible convex paving for decomposition of multi-dimensional martingale transport plans. Annals of Probability 2019, Vol. 47, No. 3, 1726-1774.

[102] A.M.G. Cox, J. Obloj and N. Touzi, The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reversal approach. Probability Theory and Related Field 2019, Vol. 173, 211-259.

[101] 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 (B) Probabilités et Statistiques Vol. 55, Number 1 (2019), 184-210.

[100] J. Cvitanic, D. Possamaï and N. Touzi, Dynamic programming approach to Principal-Agent problems. Finance and Stochastics Vol. 22 (2019), 1-37.

[99] Z. Ren, N. Touzi, and J. Zhang, Comparison of viscosity solutions of fully nonlinear degenerate parabolic Path-dependent PDEs. SIAM Journal on Mathematical Analysis, 49(5), 4093-4116 (2017).

[98] A. Cosso, S. Federico, F. Gozzi, M. Rosestolato, and N. Touzi, Path-dependent equations and viscosity solutions in infinite dimension. Annals of Probability, Volume 46, Number 1 (2018), 126-174.

[97] P. Henry-Labordère, X. Tan and N. Touzi, Unbiased simulation of stochastic differential equations. The Annals of Applied Probability Vol. 27, No. 6, 1–37 (2017).

[96] M. Beiglböck, P. Henry-Labordère and N. Touzi, Monotone Martingale Transport Plans and Skorohod Embedding. Stochastic Processes and their Applications Volume 127, Issue 9, Pages 3005-3013 (September 2017).

[95] G. Guo, X. Tan and N. Touzi, On the monotonicity principle of optimal Skorokhod embedding problem. SIAM Journal on Control and Optimization, 54-5 (2016), pp. 2478-2489.

[94] G. Guo, X. Tan and N. Touzi, Tightness and duality of martingale transport on the Skorokhod space. Stochastic Processes and their Applications, 127(3):927-956 (2017).

[93] G. Guo, X. Tan and N. Touzi, Optimal Skorokhod embedding under finitely-many marginal constraints. SIAM Journal on Control and Optimization, 54-4 (2016), pp. 2174-2201.

[92] M. Beiglböck, M. Nutz and N. Touzi, Complete Duality for Martingale Optimal Transport on the Line. Annals of Probability, (2017) Vol. 45, No. 5, 3038–3074. DOI: 10.1214/16-AOP1131.

[91] J. Cvitanic, D. Possamaï and N. Touzi, Moral hazard in dynamic risk management. Management Science, 63(10):3328-3346.

[90] P. Henry-Labordère, X. Tan and N. Touzi, An Explicit Version of the One-dimensional Brenier’s Theorem with Full Marginals Constraint. Stochastic Processes and their Applications, 126(9):2800-2834, 2016.

[89] S. Kallblad, X. Tan and N. Touzi, Optimal Skorokhod embedding given full marginals and Azéma-Yor peacocks. Annals of Applied Probability, 2017, Vol. 27, No. 2, 686-719.

[88] E. Fabre, G. Royer et N. Touzi, Liquidation of an indivisible asset with independent investment. Mathematical Finance, Volume 28, Issue 1 January 2018, pages 153–176.

[87] P. Henry-Labordère and N. Touzi, An Explicit Martingale Version of Brenier's Theorem. Finance and Stochastics, July 2016, Volume 20, Issue 3, pp 635–668.

[86] J. Ma, Z. Ren, N. Touzi and J. Zhang, Large Deviations for Non-Markovian Diffusions and a Path-Dependent Eikonal Equation. Annales de l’IHP: Probabilités et Statistique, 1196-1216 (2016).

[85] Z. Ren, N. Touzi and J. Zhang, An Overview of Viscosity Solutions of Path Dependent PDEs. In Stochastic Analysis and Applications 2014, in Honour of Terry Lyons, Springer Proceedings in Mathematics and Statisctics, Volume 100, pp 397--454.

[84] I. Ekren, N. Touzi and J. Zhang, Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part II. Annals of Probability, Vol. 44, 4 (2016), 2507-2553.

[83] I. Ekren, N. Touzi and J. Zhang, Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I. Annals of Probability, Vol. 44, 2 (2016), 1212-1253.

[82] D. Possamaï, M. Soner and N. Touzi, Homogenization and asymptotics for small transaction costs: the multidimensional case. Communications in Partial Differential Equations, vol. 40 (11), pp 2005-2046, 2015.

[81] J. Obloj, P. Spoida, and N. Touzi, Martingale Inequalities for the Maximum via Pathwise Arguments. Séminaires de Probabilités, XLVII, Lecture Notes in Mathematics 2137, Springer, pp 227--248.

[80] N. Touzi, Martingale Inequalities, Optimal Martingale Transport, and Robust Superhedging. ESAIM: Proceedings and Surveys, Vol. 45 (September 2014), Congrès SMAI 2013.

[79] P. Henry-Labordère, J. Obloj, P. Spoida, and N. Touzi, Maximum Maximum of Martingales given Marginals. Annals of Applied Probability, 2016, Vol. 26, No. 1, 1–44.

[78] D. Possamaï, G. Royer and N. Touzi, On the robust superhedging of measurable claims. Electronic Communications in Probability, 18(95):1-13.

[77] P. Henry-Labordère, X. Tan and N. Touzi, A numerical algorithm for a class of BSDE via branching process. Stochastic Processes and their Applications, 124:1112-1140, 2014.

[76] M. Soner and N. Touzi, Homogenization and asymptotics for small transaction costs. SIAM Journal on Control and Optimization, 51(4), 2893–2921 (2013).

[75] I. Ekren, N. Touzi and J. Zhang, Optimal Stopping under Nonlinear Expectation. Stochastic Processes and Their Applications, (2014), 124, 3277-3311.

[74] A. Galichon, P. Henry-Labordère and N. Touzi, A stochastic control approach to no-arbitrage bounds given marginals, with an application to Lookback options. Annals of Applied Probability, Volume 24, Number 1 (2014), 312-336.

[73] G.-E. Espinosa and N. Touzi, Optimal Investment under Relative Performance Concerns. Mathematical Finance, Volume 25, Issue 2 April 2015, Pages 221–257.

[72] X. Tan and N. Touzi, Optimal Transportation under Controlled Stochastic Dynamics. Annals of Probability, 2013, Vol. 41, No. 5, 3201–324.

[71] I. Ekren, C. Keller, N. Touzi and J. Zhang, On Viscosity Solutions of Path Dependent PDEs. Annals of Probability, Volume 24, Number 1 (2014), 312-336.

[70] G.-E. Espinosa and N. Touzi, Detecting the Maximum of a Mean-Reverting Scalar Diffusion. SIAM Journal on Control and Optimization, Vol. 50, No. 5, pp. 2543–2572 (2012).

[69] R. Carmona, F. Delarue, G.-E. Espinosa and N. Touzi, Singular forward-backward stochastic differential equations and emissions derivatives. Annals of Applied Probability, Volume 23, Number 3 (2013), 1086-1128.

[68] M. Soner, N. Touzi and J. Zhang, Dual Formulation of Second Order Target Problems. Annals of Applied Probability, 23(1), 308-347 (2013).

[67] D. Possamaï, M. Soner, and N. Touzi, Large liquidity expansion of superhedging costs. Asymptotic Analysis, 79, no 1-2, 45-64 (2012).

[66] M. Soner, N. Touzi and J. Zhang, Quasi-sure Stochastic Analysis through Aggregation. Electronic Journal of Probability, 16, 1844-1879 (2011).

[65] Aïd R, G. Chemla, A. Porchet A. and N. Touzi, Hedging and vertical integration in electricity markets. Management Science, Vol. 57, 8, 1438-1452, ISSN:0025-1909.

[64] B. Bouchard and N. Touzi, Weak Dynamic Programming Principle for Viscosity Solutions. SIAM Journal on Control and Optimization, 49, 3, 948-962 (2011).

[63] M. Soner, N. Touzi and J. Zhang, Wellposedness of second order backward SDEs. Probability Theory and Related Fields, 153, 149–190.

[62] M. Soner, N. Touzi and J. Zhang, Martingale Representation Theorem for the G-Expectation. Stochastic Processes and their Applications, 121, 265-287.

[61] R. Aïd, O. Féron, N. Touzi and C. Vialas, An arbitrage-free interest rate model consistent with economic constraints for long-term asset liability management. Bankers, Markets and Investors, Vol. 116 (2012).

[60] Fahim A, N. Touzi. and X. Warin, A Probabilistic Numerical Scheme for Fully Nonlinear PDEs. Annals of Applied Probability, 21, 4, 1322-1364.

[59] I. Ben Tahar, M. Soner and N. Touzi, Merton problem with taxes: characterization, computation and approximation. SIAM Journal on Financial Mathematics, 1, 366-395, (2010).

[58] D. Crisan, K. Manolarakis and N. Touzi, On the Monte Carlo simulation of Backward SDES: an improvement on the Malliavin weights. Stochastic Processes and Their Applications, 120 (2010), no. 7, 1133-1158.

[57] B. Bouchard, R. Elie and N. Touzi, Stochastic Target Problems with Controlled Loss. SIAM Journal on Control and Optimization, 48, 5, pp. 3123-3150.

[56] R. Aïd, L. Campi, A. Nguyen Huu, and N. Touzi, A Structural Risk Neutral Model of Electricity Prices. International Journal of Theoretical and Applied Finance, 12, 7 (2009), 925-947.

[55] M. Soner and N. Touzi, The dynamic programming equation for second order stochastic target problems. SIAM Journal on Control and Optimization, 48, 4, 2344-2365.

[54] U. Cetin, M. Soner and N. Touzi, Option hedging under liquidity costs. Finance and Stochastics, 14, 317-341.

[53] Elie R. and N. Touzi, Optimal lifetime consumption and investment under drawdown constraint. Finance and Stochastics, 12, 3 (July 2008), pp. 299-330.

[52] Porchet A., N. Touzi and X. Warin, Valuation of a powerplant under production constraints and markets incompleteness. Mathematical Methods of Operations research, Volume 70, Issue 1 (2009), Page 47-75.

[51] E. Jouini, W. Schachermayer and N. Touzi, Optimal risk sharing for law invariant monetary utility functions. Mathematical Finance, 18, 2 (2008), pp. 269-292.

[50] I. Ben Tahar, M. Soner and N. Touzi, The dynamic programming equation for the problem of optimal investment under capital gains taxes. SIAM Journal on Control and Optimization, 46, 5 (2007), pp. 1779-1801.

[49] R. Elie, J.-D. Fermanian and N. Touzi, Kernel estimation of Greek weights by parameter randomization. Annals of Applied Probability, 17, 4 (2007), 1399-1423.

[48] G. Carlier, I. Ekeland and N. Touzi, Optimal derivatives design for mean-variance agents under adverse selection. Mathematics and Financial Economics, 1, 1 (April 2007), pp. 57-80.

[47] F. Astic and N. Touzi, No arbitrage conditions and liquidity. Journal of Mathematical Economics, 43 (2007), 692-708.

[46] P. Cheridito, M. Soner, N. Touzi and Nicolas Victoir, Second Order Backward Stochastic Differential Equations and Fully Non-Linear Parabolic PDEs. Communications in Pure and Applied Mathematics, 60 (7): 1081-1110 (2007).

[45] E. Jouini, W. Schachermayer and N. Touzi, Law Invariant Risk Measures have the Fatou Property. Advances in Mathematical Economics, 9, 49-72 (2006).

[44] M. Mrad, N. Touzi, and A. Zeghal, Monte Carlo estimation of a joint density using Malliavin calculus. Computational Economics, 27, 4, 497-531 (2006).

[43] M. Soner and N. Touzi, Hedging under gamma constraints by optimal stopping and face-lifting. Mathematical Finance, 17, 1, 59-80 (2007).

[42] R. Carmona and N. Touzi, Optimal multiple stopping and valuation of swing options. Mathematical Finance, 18, 2 (April 2008) pp. 239-268.

[41] B. Bouchard, N. El Karoui and N. Touzi, Maturity randomisation for stochastic control problems. Annals of Applied Probability, 15, 4, 2575-2605 (2005).

[40] P. Cheridito, M. Soner and N. Touzi, Small time path behavior of double stochastic integrals and applications to stochastic control. Annals of Applied Probability, 15, 4, 2472-2495 (2005).

[39] P. Cheridito, M. Soner and N. Touzi, The multi-dimensional super-replication problem under Gamma constraints. Annales de l’Institut Henri Poincaré, Série C: Analyse Non-Linéaire, 22, 633-666 (2005).

[38] A. Bensoussan, N. Touzi and J.-L. Menaldi, Penalty approximation and analytical characterization of the problem of super-replication under portfolio constraints. Asymptotic Analysis, 41, 311-330 (2005).

[37] B. Bouchard and N. Touzi, Discrete-time approximation and Monte Carlo simulation of backward stochastic differential equations. Stochastic Processes and their Applications, 111, 175-206 (2004).

[36] B. Bouchard, I. Ekeland and N. Touzi, On the Malliavin approach to Monte Carlo approximation of conditional expectations. Finance and Stochastics, 8, 45-71 (2004).

[35] E. Jouini, M. Meddeb and N. Touzi, Vector-valued measure of risk. Finance and Stochastics, 8, 531-552.

[34] B. Bouchard, N. Touzi and A. Zeghal, Dual Formulation of the Utility Maximization Problem : the case of Nonsmooth Utility. Annals of Applied Probability, 14, 678-717 (2004).

[33] H.M. Soner and N. Touzi, The problem of super-replication under constraints. [33] H.M. Soner and N. Touzi, The problem of super-replication under constraints, to appear in Paris-Princeton Lectures in Mathematical Finance, Lecture Notes in Mathematics, Springer-Verlag, Vol. 1814, 133-172.

[32] H.M. Soner and N. Touzi, Stochastic representation of mean curvature type geometric flows. Annals of Probability, 31, 1145-1165 (2003).

[31] H.M. Soner and N. Touzi, Level set characterization of stochastic target problems. Communications in PDE’s, 27, 2031-2053.

[30] B. Bouchard, Y. Kabanov and N. Touzi, Option pricing by large risk aversion utility under transaction costs. Decision in Economics and Finance, 24, 127-136 (2001).

[29] H.M. Soner and N. Touzi, Dynamic programming for stochastic target problems and geometric flows. Journal of the European Mathematical Society, 4, 201-236 (2002).

[28] G. Deelstra, H. Pham and N. Touzi, Dual formulation of the utility maximization problem under transaction costs. Annals of Applied Probability, 11 (4), 1353-1383 (2002).

[27] N. Touzi and N. Vieille, Continuous-time Dynkin games with mixed strategies. SIAM Journal on Control and Optimization, 41, 1073-1088 (2002).

[26] H.M. Soner and N. Touzi, Stochastic target problems, dynamic programming and viscosity solutions. SIAM Journal on Control and Optimization, 41, 404-424 (2002).

[25] N. Touzi, Super-replication under proportional transaction costs: from discrete to continuous-time models. Mathematical Methods of Operations Research, 50, 297-320 (1999).

[24] B. Bouchard and N. Touzi, Explicit solution of the multivariate super-replication problem under transaction costs. Annals of Applied Probability, 10, 685-708 (2000).

[23] N. Touzi, Direct characterization of the value of super-replication under stochastic volatility and portfolio constraints. Stochastic Processes and their Applications, 88, 305-328 (2000).

[22] H.M. Soner and N. Touzi, Super-replication under Gamma constraint. Journal on Control and Optimization, 39, 73-96 (2000).

[21] P.-F. Koehl, H. Pham and N. Touzi, On super-replication under Transaction costs in general discrete-time models. Theory of Probability and its Applications, 45, 783-788 (1999).

[20] L. Carassus, H. Pham and N. Touzi, Arbitrage and domination cost in a discrete-time model with convex portfolio constraints. Mathematical Finance, 11, 3, 315-329 (2001).

[19] E. Jouini, P.-F. Koehl and N. Touzi, Optimal investment with taxes: an existence result. Journal of Mathematical Economics, 33, 373-388.

[18] E. Jouini, P.-F. Koehl and N. Touzi, Optimal investment with taxes~: an optimal control problem with endogeneous delay. Nonlinear Analysis~: Theory, Methods and Applications, 37, 31-56 (1999).

[17] N. Touzi, Optimal insurance demand under marked point processes shocks. Annals of Applied Probability, 10, 283-312 (2000).

[16] J. Cvitanic, H. Pham and N. Touzi, Super-replication in stochastic volatility models with portfolio constraints. Journal of Applied Probability, 36, 523-545 (1999).

[15] J. Cvitanic, H. Pham and N. Touzi, A closed-form solution for the problem of super-replication under transaction costs. Finance and Stochastics, 3, 35-54 (1999).

[14] E. Fournié, J.M. Lasry and N. Touzi, Monte Carlo Methods in Stochastic Volatility Models. in Numerical Methods in Financial Mathematics, edited by C. Rogers et D. Talay, Cambridge University Press (1997).

[13] E. Fournié, J.M. Lasry, J. Lebuchoux, P.-L. Lions and N. Touzi, Some applications of Malliavin calculus to Monte Carlo methods in finance. Finance and Stochastics, 3, 391-412 (1999).

[12] P.-F. Koehl, H. Pham and N. Touzi, Hedging in discrete-time under Transaction costs and continuous-time limit. Journal of Applied Probability, 36, 163-178 (1999).

[11] H. Pham and N. Touzi, The fundamental theorem of asset pricing with cone constraints. Journal of Mathematical Economics, 31(2), 265-279 (March 1999).

[10] N. Touzi, American options exercise boundary when the volatility changes randomly. Applied Mathematics and Optimization 39, 411-422 (1999).

[9] E. Fournié, J. Lebuchoux and N. Touzi, Small Noise Expansion and Importance Sampling. Asymptotic Analysis 14, 361-376 (1997).

[8] E. Jouini, P.-F. Koehl and N. Touzi, Incomplete Markets. Transaction costs and Liquidity effects, European Journal of Finance 4, 325-348.

[7] L.P. Hansen, J.A. Scheinkman and N. Touzi, Spectral Methods for Identifying Scalar Diffusions. Journal of Econometrics 86, 1-32.

[6] J.-P. Florens, E. Renault and N. Touzi, Testing Embeddability by Stationary Reversible Continuous-Time Markov Processes. Econometric Theory 14.

[5] C. Gouriéroux, E. Renault and N. Touzi, Calibration by Simulation for Small Sample Bias Correction. in Simulation Based Inference in Econometrics Methods edited by J. Geweke and R. Mariano.

[4] S. Pastorello, E. Renault and N. Touzi, Statistical Inference for Random Variance Option Pricing. Journal of Business and Economic Statistics 2000, vol. 18, no 3, p. 358-367.

[3] M. Romano and N. Touzi, Contingent Claims and Market Completeness in a Stochastic Volatility Model. Mathematical Finance 7, 399-412 (1997).

[2] H. Pham and N. Touzi, Equilibrium State Prices in a Stochastic Volatility Model. Mathematical Finance 6, 215-236 (1996).

[1] E. Renault and N. Touzi, Option Hedging and Implicit Volatilities. Mathematical Finance 6, 279-302 (1996).