Z. Ci-(d, U. , Z. , Y. Upper, and C. I. , , vol.1

U. , Z. Lower, C. , U. , Z. et al., Table 5.2: Lower and upper estimators for a call option and different degrees, vol.1

, 3: Lower and upper estimators for a butterfly option and different degrees Bibliography

F. Abergel and R. Tachet, A non-linear partial integro-differential equation from mathematical finance, Discrete Contin. Dyn. Syst, issue.3, pp.907-917, 2010.

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables, Courier Corporation, vol.55, 1964.

Y. Achdou and O. Pironneau, Computational Methods for Option Pricing. SIAM series, Frontiers in Applied Mathematics, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00019751

A. Agarwal, S. Marco, E. Gobet, J. G. López-salas, F. Noubiagain et al., Numerical approximations of McKean Anticipative Backward Stochastic Differential Equations arising in Initial Margin requirements, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01686952

J. M. Albin, A continuous non-Brownian motion martingale with Brownian motion marginal distributions, Statistics and Probability Letters, vol.78, issue.6, p.682, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00612014

A. Alfonsi, C. Labart, and J. Lelong, Stochastic local intensity loss models with interacting particle systems, Mathematical Finance, vol.26, issue.2, pp.366-394, 2016.
URL : https://hal.archives-ouvertes.fr/hal-00786239

D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc, vol.73, issue.6, pp.890-896, 1967.

P. Artzner, F. Delbaen, J. M. Eber, and D. Heath, Coherent measures of risk, Math. Finance, vol.9, issue.3, pp.203-228, 1999.

R. Avikainen, On irregular functionals of SDEs and the Euler scheme, Finance and Stochastics, vol.13, issue.3, pp.381-401, 2009.

D. Baker, C. Donati-martin, and M. Yor, A Sequence of Albin Type Continuous Martingales with Brownian Marginals and Scaling, pp.441-449, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00471169

V. Bally and D. Talay, The law of the euler scheme for stochastic differential equations. Probability Theory and Related Fields, vol.104, pp.43-60, 1996.
URL : https://hal.archives-ouvertes.fr/inria-00074427

, Margin requirements for non-centrally-cleared derivatives, 2015.

V. I. Bogachev, M. Röckner, and S. V. Shaposhnikov, Positive densities of transition probabilities of diffusion processes, Theory of Probability & Its Applications, vol.53, pp.194-215, 2009.

M. Bossy and J. Jabir, Particle approximation for lagrangian stochastic models with specular boundary condition, Electron. Commun. Probab, vol.23, p.14, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01147441

M. Bossy, J. Jabir, and D. Talay, On conditional mckean lagrangian stochastic models. Probability Theory and Related Fields, vol.151, pp.319-351, 2011.
URL : https://hal.archives-ouvertes.fr/inria-00345524

M. Bossy and D. Talay, Convergence rate for the approximation of the limit law of weakly interacting particles: application to the burgers equation, Ann. Appl. Probab, vol.6, issue.3, pp.818-861, 1996.
URL : https://hal.archives-ouvertes.fr/inria-00074265

M. Bossy and D. Talay, A stochastic particle method for the McKean-Vlasov and the Burgers equation, Mathematics of Computation, vol.66, pp.157-192, 1997.

D. T. Breeden and R. H. Litzenberger, Prices of state-contingent claims implicit in option prices, The Journal of Business, vol.51, issue.4, pp.621-51, 1978.

H. Brezis, Analyse fonctionnelle, Théorie et applications, 1983.

M. Broadie, Y. P. Du, and C. C. Moallemi, Risk estimation via regression, Operations Research, vol.63, issue.5, pp.1077-1097, 2015.

M. Broadie and P. Glasserman, Estimating security price derivatives using simulation. Management science, vol.42, pp.269-285, 1996.

R. Buckdahn and P. Imkeller, Backward stochastic differential equations with time delayed generator, 2009.

K. Bujok, B. M. Hambly, and C. Reisinger, Multilevel simulation of functionals of bernoulli random variables with application to basket credit derivatives, Methodology and Computing in Applied Probability, vol.17, issue.3, pp.579-604, 2015.

P. Cannarsa and T. D'aprile, Lecture notes on measure theory and functional analysis, 2006.

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications I-II

R. Carmona, J. P. Fouque, and L. H. Sun, Mean field games and systemic risk: a toy model, Communications in Mathematical Sciences, vol.13, pp.911-933, 2014.

L. Cattiaux and P. Pédèches, The 2-D stochastic Keller-Segel particle model : existence and uniqueness. working paper or preprint, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01263156

P. Chaudru-de-raynal, Strong well-posedness of mckean-vlasov stochastic differential equation with hölder drift

P. Cheredito and K. Nam, BSE's, BSDE's and fixed point problems. The Annals of Probability, vol.45, pp.3795-3828, 2017.

S. Crépey, R. Élie, W. Sabbagh, and S. Song, When capital is a funding source: The XVA anticipated BSDEs, 2017.

L. Delong and P. Imkeller, On Malliavin's differentiability of time delayed BSDEs driven by Brownian motions and Poisson random measures, 2009.

L. Delong and P. Imkeller, Backward Stochastic Differential Equations with time delayed generatorsResults and counterexamples, The Annals of Probability, vol.20, issue.4, pp.1512-1536, 2010.

L. Delong and P. Imkeller, Applications of time-delayed backward stochastic differential equations to pricing, hedging and portfolio management in insurance and finance, Applicationes Mathematicae, vol.39, pp.463-488, 2012.

J. Dieudonné, D. Huet, and G. Julia, Eléments d'analyse. tome i, fondements de l'analyse moderne, 1979.

D. E. Dominici, The inverse of the cumulative standard normal probability function, Integral Transforms and Special Functions, vol.14, issue.4, pp.281-292, 2003.

B. Dupire, Pricing with a Smile, Risk, vol.7, pp.18-20, 1994.

S. N. Ethier and T. G. Kurtz, Markov processes: characterization and convergence. Wiley series in probability and mathematical statistics. Probability and mathematical statistics, 1986.

J. Y. Fan, K. Hamza, and F. Klebaner, Mimicking self-similar processes, Bernoulli, vol.21, issue.3, pp.1341-1360, 2015.

A. Figalli, Existence and uniqueness of martingale solutions for sdes with rough or degenerate coefficients, Journal of Functional Analysis, vol.254, issue.1, pp.109-153, 2008.

N. Fournier and M. Hauray, Propagation of chaos for the Landau equation with moderately soft potentials, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01257022

N. Fournier and B. Jourdain, Stochastic particle approximation of the keller-segel equation and twodimensional generalization of bessel processes, Ann. Appl. Probab, vol.27, issue.5, p.2017
URL : https://hal.archives-ouvertes.fr/hal-01171481

A. Friedman, Partial differential equations of parabolic type, 1964.

A. Friedman, Stochastic Differential Equations and Applications, 1975.

J. Gatheral, T. Jaisson, and M. Rosenbaum, Volatility is rough. Quantitative Finance, vol.18, pp.933-949, 2018.

M. Giles and A. Haji-ali, Multilevel nested simulation for efficient risk estimation

M. B. Giles, Multilevel monte carlo methods, Acta Numerica, vol.24, pp.259-328, 2015.

D. Giorgi, V. Lemaire, and G. Pagès, Weak error for nested Multilevel Monte Carlo. working paper or preprint, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01817386

P. Glasserman, P. Heidelberger, and P. Shahabuddin, Variance reduction techniques for estimating Valueat-Risk. Management Science, vol.46, pp.1349-1364, 2000.

E. Gobet, J. G. López-salas, P. Turkedjiev, and C. Vázquez, Stratified Regression Monte-Carlo Scheme for Semilinear PDEs and BSDEs with Large Scale Parallelization on GPUs, SIAM Journal on Scientific Computing, vol.38, issue.6, pp.652-677, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01186000

E. Gobet and S. Pagliarani, Analytical approximations of BSDEs with nonsmooth driver, SIAM Journal on Financial Mathematics, vol.6, issue.1, pp.919-958, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01003913

E. Gobet and P. Turkedjiev, Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions, Math. Comp, vol.85, issue.299, pp.1359-1391, 2016.
URL : https://hal.archives-ouvertes.fr/hal-00642685

R. D. Gordon, Values of mills' ratio of area to bounding ordinate and of the normal probability integral for large values of the argument, The Annals of Mathematical Statistics, vol.12, issue.3, pp.364-366, 1941.

M. B. Gordy and S. Juneja, Nested simulation in portfolio risk measurement, Management Science, vol.56, issue.10, pp.1833-1848, 2010.

H. Guennoun and P. Henry-labordère, Local volatility models enhanced with jumps, 2016.

J. Guyon, Path-dependent volatility, Risk, 2014.

J. Guyon, Cross-dependent volatility, Risk, 2016.

J. Guyon and P. Henry-labordère, Being particular about calibration. Risk magazine, pp.88-93, 2012.

I. Gyöngy, Mimicking the one-dimensional marginal distribution of processes having an Ito differential, vol.71, pp.501-516, 1986.

J. K. Hale, Ordinary Differential Equations. Dover Books on Mathematics Series, 2009.

P. Hall and C. C. Heyde, Probability and Mathematical Statistics: A Series of Monographs and Textbooks, pages ix -xi, 1980.

K. Hamza and F. C. Klebaner, A family of non-gaussian martingales with gaussian marginals, J. Appl. Math. and Stochastic Analysis, 2007.

J. Ha?kovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2d keller-segel system, Journal of Statistical Physics, vol.135, issue.1, pp.133-151, 2009.

P. Henry-labordere, X. Tan, and N. Touzi, An explicit martingale version of the one-dimensional Brenier's theorem with full marginals constraint, Stochastic Processes and their Applications, vol.126, pp.2800-2834, 2016.

F. Hirsch, C. Profeta, B. Roynette, and M. Yor, Constructing Self-Similar Martingales via Two Skorokhod Embeddings, pp.451-503, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00610188

D. G. Hobson, Fake exponential brownian motion, Statistics & Probability Letters, vol.83, issue.10, pp.2386-2390, 2013.

J. Jabir, D. Talay, and M. Tomasevic, Mean-field limit of a particle approximation of the onedimensional parabolic-parabolic Keller-Segel model without smoothing, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01668926

S. L-johnson-norman, N. Kotz, and . Balakrishnan, Lognormal distributions, continuous univariate distributions, vol.1, 1994.

B. Jourdain, T. Lelièvre, and R. Roux, Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force process, ESAIM: M2AN, vol.44, issue.5, pp.831-865, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00370821

B. Jourdain and S. Méléard, Propagation of chaos and fluctuations for a moderate model with smooth initial data, vol.34, pp.727-766, 1998.

B. Jourdain, S. Méléard, and W. A. Woyczynski, Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws, Bernoulli, vol.11, issue.4, pp.689-714, 2005.

I. Karatzas and S. E. Shreve, Methods of mathematical finance, 1998.

N. E. Karoui, S. Peng, and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, vol.7, pp.1-71, 1997.

T. G. Kurtz, Equivalence of Stochastic Equations and Martingale Problems, pp.113-130, 2011.

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

T. Lelievre, F. Otto, M. Rousset, and G. Stoltz, Long-time convergence of an Adaptive Biasing Force method. working paper or preprint, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00153946

A. Lipton, The vol smile problem, vol.15, pp.61-65, 2002.

D. B. Madan and M. Yor, Making markov martingales meet marginals: with explicit constructions, Bernoulli, vol.8, issue.4, pp.509-536, 2002.

C. Marchioro and M. Pulvirenti, Hydrodynamics in two dimensions and vortex theory, Comm. Math. Phys, vol.84, issue.4, pp.483-503, 1982.

H. P. Mckean, A class of Markov processes associated with nonlinear parabolic equations, Proc. Natl. Acad. Sci. USA, vol.56, pp.1907-1911, 1966.

S. Méléard, Stochastic particle approximations for two-dimensional navier-stokes equations, Dynamics and Randomness II, pp.147-197, 2004.

J. L. Menaldi, Stochastic differential equations with jumps, 2014.

S. Menozzi and V. Lemaire, On some non asymptotic bounds for the euler scheme, Electron. J. Probab, vol.15, pp.1645-1681, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00445494

M. Musiela and M. Rutkowski, Martingale methods in financial modelling, 2005.

K. Oleszkiewicz, On fake Brownian motions, Statistics and Probability Letters, vol.78, issue.11, p.1251, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00622142

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

E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equation, Lecture Notes in CIS, vol.176, pp.200-217, 1992.

E. Pardoux and A. Rascanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, vol.69, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01108223

S. Peng and Z. Yang, Anticipated backward stochastic differential equations. The Annals of Probability, vol.37, pp.877-902, 2009.

V. Piterbarg, Markovian projection method for volatility calibration, Risk Magazine, vol.20, pp.84-89, 2007.

Y. Ren, D. Madan, and M. Qian, Calibrated and pricing with embedded local volatility models, Risk, pp.138-143, 2007.

R. T. Rockafellar and S. Uryasev, Optimization of conditional Value-at-Risk, Journal of risk, vol.2, pp.21-42, 2000.

D. W. Stroock and S. R. Varadhan, Multidimensional diffusion processes, vol.233

. Springer-verlag, , 1979.

A. Sznitman, Topics in propagation of chaos, Ecole d'Eté de Probabilités de Saint-Flour XIX -1989, pp.165-251, 1991.

D. Talay and M. Tomasevic, A new McKean-Vlasov stochastic interpretation of the parabolic-parabolic Keller-Segel model: The one-dimensional case, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01673332

D. Talay and L. Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Analysis and Applications, vol.8, issue.4, pp.483-509, 1990.
URL : https://hal.archives-ouvertes.fr/inria-00075490

D. Talay and O. Vaillant, A stochastic particle method with random weights for the computation of statistical solutions of mckean-vlasov equations, Ann. Appl. Probab, vol.13, issue.1, pp.140-180, 2003.
URL : https://hal.archives-ouvertes.fr/inria-00072260

R. Temam, Navier-Stokes Equations, 1979.

V. C. Tran, A wavelet particle approximation for mckean-vlasov and 2d-navier-stokes statistical solutions, Stochastic Processes and their Applications, vol.118, pp.284-318, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00260978

Z. Yang and R. J. Elliott, Anticipated backward stochastic differential equations with continuous coefficients, Communications on Stochastic Analysis, vol.7, issue.2, pp.303-319, 2013.

G. Yin and C. Zhu, Properties of solutions of stochastic differential equations with continuous-statedependent switching, Journal of Differential Equations, vol.249, issue.10, pp.2409-2439, 2010.

D. M. Young and R. T. Gregory, A survey on Numerical Mathematics, vol.1, 1973.

J. Zhang, A numerical scheme for BSDEs, Ann. Appl. Probab, vol.14, issue.1, pp.459-488, 2004.

X. Zhang, Degenerate irregular sdes with jumps and application to integro-differential equations of fokkerplanck type, Electron. J. Probab, vol.18, p.25, 2013.