C. Lemma, Assume d ? 3, and take K = M d+2 points for the optimal quantization of ? n , then it holds under (Hµ) and (HF), as M ? +?

F. Abergel, A. Anane, A. Chakraborti, A. Jedidi, and I. Muni-toke, Limit Order Books, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01686122

M. Avellaneda and S. Stoikov, High-frequency trading in a limit order book, Quantitative Finance, vol.8, pp.217-224, 2007.

A. Balata, C. Huré, M. Laurière, H. Pham, and I. Pimentel, A class of finite-dimensional numerically solvable McKean-Vlasov control problems, ESAIM Proceedings and surveys, p.65, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01718751

N. Baradel, B. Bouchard, D. Evangelista, and O. Mounjid, Optimal inventory management and order book modeling, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01710301

D. Belomestny, A. Kolodko, and J. Schoenmakers, Regression methods for stochastic control problems and their convergence analysis, SIAM Journal on Control and Optimization, vol.48, pp.3562-3588, 2010.

A. Balata and J. Palczewski, Regress-Later Monte Carlo for optimal control of Markov processes, 2017.

N. Bäuerle and U. Rieder, Markov Decision Processes with Applications to Finance, 2011.

P. Brémaud, Point Processes and Queues : Martingale Dynamics, 1981.

A. Cartea and S. , Modeling Asset Prices for Algorithmic and High Frequency Trading, Applied Mathematical Finance, vol.20, issue.6, pp.512-547, 2010.

A. Cartea and S. , Risk metrics and fine tuning of high-frequency trading strategies, In: Mathematical Finance, vol.25, issue.3, pp.576-611, 2013.
DOI : 10.1111/mafi.12023

A. Cartea, S. Jaimungal, and J. Ricci, Buy Low Sell High: a High Frequency Trading Perspective, SIAM Journal on Financial Mathematics, pp.415-444, 2014.
DOI : 10.2139/ssrn.1964781

A. Cartea, J. Penalva, and S. , Algorithmic and High-frequency trading, 2015.

R. Cont, S. Stoikov, and R. Talreja, A stochastic model for order book dynamics, Operations Research, vol.58, pp.549-563, 2007.
DOI : 10.2139/ssrn.1273160
URL : https://hal.archives-ouvertes.fr/hal-00497666

S. E. Aoud and F. , A stochastic control approach for options market making, In: World scientific publishing company, vol.1, issue.1, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01061852

P. Fodra and H. Pham, High Frequency trading and asymptotics for small risk aversion in a Markov renewal model, SIAM Journal of Financial Mathematics, vol.6, issue.1, pp.656-684, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01261362

P. Fodra and H. Pham, Semi Markov model for market microstructure, Applied Mathematical Finance, vol.22, issue.3, pp.261-295, 2015.
DOI : 10.1080/1350486x.2015.1037963
URL : https://hal.archives-ouvertes.fr/hal-00819269

L. Fiorin, G. Pagès, and A. Sagna, Product Markovian quantization of a diffusion process with applications to finance, Methodology and Computing in Applied Probability, pp.1-32, 2018.

L. Gyorfi, M. Kohler, A. Krzyzak, and H. Walk, A Distribution-Free Theory of Nonparametric Regression, 2002.

S. Graf and H. Luschgy, Foundations of quantization for probability distributions, vol.1730, 2000.
DOI : 10.1007/bfb0103945

O. Guéant, C. Lahalle, and J. Fernandez-tapia, Dealing with the Inventory Risk, In: Mathematics and Financial Economics, vol.7, pp.477-507, 2012.

F. Guilbaud and H. Pham, Optimal High Frequency Trading with limit and market orders, Quantitative Finance, vol.13, pp.79-94, 2013.
DOI : 10.2139/ssrn.1871969
URL : https://hal.archives-ouvertes.fr/hal-00835633

O. Guéant, The Financial Mathematics of Market Liquidity: From optimal execution to market making, 2016.

W. Huang, C. Lehalle, and M. Rosenbaum, Simulating and analyzing order book data: The queue-reactive model, Journal of the American Statistical Association, vol.110, issue.509, pp.107-122, 2015.
DOI : 10.1080/01621459.2014.982278
URL : https://hal.archives-ouvertes.fr/hal-01172326

C. Huré, H. Pham, A. Bachouch, and N. Langrené, Deep neural networks algorithms for stochastic control problems on finite horizon: convergence analysis, 2018.

T. Ho and H. Stoll, Optimal dealer pricing under transactions and return uncertainty, Journal of Financial Economics, vol.9, pp.47-73, 1979.
DOI : 10.1016/0304-405x(81)90020-9

A. Jacquier and H. Liu, Optimal Liquidation in a level-I Limit order book for large tick stocks, SIFIN 9, pp.875-906, 2018.
DOI : 10.1137/17m1117860
URL : http://arxiv.org/pdf/1701.01327

I. Kharroubi, N. Langrené, and H. Pham, A numerical algorithm for fully nonlinear HJB equations: an approach by control randomization, Monte Carlo Methods and Applications, vol.20, pp.145-165, 2014.
DOI : 10.1515/mcma-2013-0024
URL : https://hal.archives-ouvertes.fr/hal-00905899

C. Lehalle, M. Othmane, and M. Rosenbaum, Optimal liquidity-based trading tactics, 2018.

L. Massoulié, Stability results for a general class of interacting point processes dynamics, and applications, Stochastic Processes and their Applications, vol.75, pp.1-30, 1998.

M. Muja and D. Lowe, Fast Approximate Nearest Neighbors with Automatic Algorithm Configuration, International Conference on Computer Vision Theory and Applications, 2009.

G. Pagès, H. Pham, and J. Printems, Optimal quantization methods and applications to numerical problems in finance, Handbook on Numerical Methods in Finance, pp.253-298, 2004.

I. Rosu, A dynamic model of the limit order book, Review of Financial Studies, vol.22, pp.4601-4641, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00515873