, def Simulation ( n , alpha , sigma , X0 , Euler_sub_pas ) : for j in range ( 1 , 1 1 ) : nEuler=n * Euler_sub_pas Xt =

, V=np . random . uniform (?np . p i / 2 , np . p i / 2 ) W= np . random . exponential

. Xt-[-i-]=-xt, ) / nEuler ) +sigma *

, def SDE_Levy ( n , alpha , sigma , X0 , Euler_sub_pas ) : nEuler=n * Euler_sub_pas Xt =

. Xtemp=, * ( Euler_sub_pas +1) for j in range, vol.1

, V=np . random . uniform (?np . p i / 2 , np . p i / 2 ) W= np . random . exponential

, ? alpha ) / alpha ) )

, PYTHON CODE for j in range

=. Xt and E. X0,

Y. Aït, -. Sahalia, and L. Hansen-hansen, Handbook of Financial Econometrics: Applications, volume 2 of Handbooks in Finance

Y. Aït, -. Sahalia, and J. Jacod, Volatility estimators for discretely sampled Lévy processes

, Ann. Statist, vol.35, issue.1, pp.355-392, 2007.

Y. Aït, -. Sahalia, and J. Jacod,

, Fisher's information for discretely sampled Lévy processes

, Econometrica, vol.76, issue.4, pp.727-761, 2008.

Y. Aït, -. Sahalia, and J. Jacod, Estimating the degree of activity of jumps in high frequency data

, Ann. Statist, vol.37, issue.5A, pp.2202-2244, 2009.

C. Larry and . Andrews, Special functions of mathematics for engineers

, SPIE Optical Engineering Press, 1998.

D. Applebaum,

, Lévy processes and stochastic calculus, Cambridge Studies in Advanced Mathematics, vol.116

O. E. Barndorff-nielsen, T. Mikosch, and S. I. Resnick, Lévy processes: theory and applications

K. Bichteler, J. Gravereaux, and J. Jacod,

, Malliavin calculus for processes with jumps, vol.2, 1987.

J. Birge and V. Linetsky,

, Modeling financial security returns using lévy processes

, Operations Research and Management Science: Financial Engineering, vol.15, 2008.

D. Blackwell, The comparison of experiments

, Proceedings, Second Berkeley Symposium on Mathematical Statistics and Probability, p.93102, 1951.

R. M. Blumenthal and R. K. Getoor, Sample functions of stochastic processes with stationary independent increments

, J. Math. Mech, vol.10, pp.493-516, 1961.

N. Bouleau and L. Denis,

, Dirichlet forms methods for Poisson point measures and Lévy processes, of Probability Theory and Stochastic Modelling, vol.76

. Springer, , 2015.

A. Brouste and H. Masuda, Efficient estimation of stable lévy process with symmetric jumps. Statistical Inference for Stochastic Processes, 2018.

A. D. Bull, Near-optimal estimation of jump activity in semimartingales

, Ann. Statist, vol.44, issue.1, pp.58-86, 2016.

P. Carr and H. Geman, The fine structure of asset returns: An empirical investigation, The Journal of Business, vol.75, issue.2, pp.305-332, 2002.

E. Ç?nlar, Probability and stochastics, Graduate Texts in Mathematics, vol.261

E. Clément and A. Gloter,

, Local asymptotic mixed normality property for discretely observed stochastic differential equations driven by stable Lévy processes

, Stochastic Process. Appl, vol.125, issue.6, pp.2316-2352, 2015.

E. Clément and A. Gloter, Estimating functions for SDE driven by stable Lévy processes, 2018.

E. Clément, A. Gloter, and H. Nguyen,

, Asymptotics in small time for the density of a stochastic differential equation driven driven by a stable Lévy process

. Esaim-probab and . Stat, , 2018.

E. Clément, A. Gloter, and H. Nguyen,

, LAMN property for the drift and volatility parameters of a SDE driven by a stable Lévy process

. Esaim-probab and . Stat, , 2018.

A. Debussche and N. Fournier, Existence of densities for stable-like driven SDE's with Hölder continuous coefficients

, J. Funct. Anal, vol.264, issue.8, pp.1757-1778, 2013.

L. Denis, A criterion of density for solutions of Poisson-driven SDEs

, Probab. Theory Related Fields, vol.118, issue.3, pp.406-426, 2000.

P. D. Ditlevsen, Anomalous jumping in a double-well potential

, Phys. Rev. E, vol.60, pp.172-179, 1999.

N. Fournier and J. Printems, Absolute continuity for some one-dimensional processes
DOI : 10.3150/09-bej215

URL : https://hal.archives-ouvertes.fr/hal-00693025

, Bernoulli, vol.16, issue.2, pp.343-360, 2010.

V. Genon-catalot and J. Jacod, Estimation of the diffusion coefficient for diffusion processes: random sampling

. Scand, J. Statist, vol.21, issue.3, pp.193-221, 1994.

-. Valentine-genon, J. Catalot, and . Jacod,

, On the estimation of the diffusion coefficient for multi-dimensional diffusion processes

, Ann. Inst. H. Poincaré Probab. Statist, vol.29, issue.1, pp.119-151, 1993.

E. Gobet,

, Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach

, Bernoulli, vol.7, issue.6, pp.899-912, 2001.

E. Gobet, LAN property for ergodic diffusions with discrete observations

J. Hájek, A characterization of limiting distributions for regular estimators, Ann. Inst. H. Poincaré Probab. Statist, vol.38, issue.5, pp.711-737, 2002.

, Z. Wahrscheinlichkeitsth. Verw. Geb, vol.14, pp.323-330, 1970.

J. Hájek, Local asymptotic minimax and admissibility in estimation

, Theory of Statistics, vol.1, pp.175-194, 1972.

P. Hall and C. Charles-heyde, Martingale limit theory and its application

I. A. Ibragimov, R. Z. Has'minskii, and S. Kotz, Statistical Estimation: Asymptotic Theory, Stochastic Modelling and Applied Probability, vol.16

Y. Ishikawa and H. Kunita,

, Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE with jumps

, Stochastic Process. Appl, vol.116, issue.12, pp.1743-1769, 2006.

D. Ivanenko and A. Kulik,

, Malliavin calculus approach to statistical inference for Lévy driven SDE's

, Methodol. Comput. Appl. Probab, vol.17, issue.1, pp.107-123, 2015.

D. Ivanenko, A. M. Kulik, and H. Masuda,

, Uniform LAN property of locally stable Lévy process observed at high frequency

. Alea-lat, Am. J. Probab. Math. Stat, vol.12, issue.2, pp.835-862, 2015.

J. Jacod,

, The Euler scheme for Lévy driven stochastic differential equations: limit theorems

A. Probab, , vol.32, pp.1830-1872, 2004.

J. Jacod and P. Protter,

, Discretization of processes, Stochastic Modelling and Applied Probability, vol.67

. Springer, , 2012.

A. Janicki and A. Weron,

, Simulation and chaotic behavior of ?-stable stochastic processes, Monographs and Textbooks in Pure and Applied Mathematics, vol.178

M. Dekker and . Inc, , 1994.

P. Jeganathan,

, On the asymptotic theory of estimation when the limit of the log-likelihood ratios is mixed normal

, Sankhy ¯ a Ser. A, vol.44, issue.2, pp.173-212, 1982.

P. Jeganathan,

, Some asymptotic properties of risk functions when the limit of the experiment is mixed normal

, Sankhy ¯ a Ser. A, vol.45, issue.1, pp.66-87, 1983.

B. Jing, X. Kong, and Z. Liu, Modeling high-frequency financial data by pure jump processes

, Ann. Statist, vol.40, issue.2, pp.759-784, 2012.

B. Jing, X. Kong, Z. Liu, and P. Mykland, On the jump activity index for semimartingales

, J. Econometrics, vol.166, issue.2, pp.213-223, 2012.

R. Kawai and H. Masuda,

, On the local asymptotic behavior of the likelihood function for Meixner Lévy processes under high-frequency sampling

, Statist. Probab. Lett, vol.81, issue.4, pp.460-469, 2011.

R. Kawai and H. Masuda,

, Local asymptotic normality for normal inverse Gaussian Lévy processes with high-frequency sampling

. Esaim-probab and . Stat, , vol.17, pp.13-32, 2013.

V. N. Kolokoltsov,

, Markov processes, semigroups and generators, de Gruyter Studies in Mathematics, vol.38

X. Kong, Z. Liu, and B. Jing, Testing for pure-jump processes for high-frequency data

, Ann. Statist, vol.43, issue.2, pp.847-877, 2015.

A. Kulik,

, On weak uniqueness and distributional properties of a solution to an SDE with ?-stable noise, 2015.

Y. A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes

. Springer, , 1998.

L. and L. Cam, Limits of experiments

, Proc. 6th Berkeley Symp, vol.1, pp.245-261, 1972.

L. L. , C. , G. Lo, and Y. , Asymptotics in statistics: Some basic concepts

, Springer Series in Statistics, 1990.

L. L. , C. , G. Lo, and Y. , Asymptotics in statistics: some basic concepts

, Business Media, 2000.

H. Luschgy and G. Pagès, Moment estimates for Lévy processes

, Electron. Commun. Probab, vol.13, pp.422-434, 2008.

H. Masuda,

, Joint estimation of discretely observed stable Lévy processes with symmetric Lévy density

, J. Japan Statist. Soc, vol.39, issue.1, pp.49-75, 2009.

H. Masuda,

, Non-gaussian quasi-likelihood estimation of sde driven by locally stable Lévy process, 2017.

T. Mikosch, S. Resnick, H. Rootzén, and A. Stegeman, Is network traffic approximated by stable Lévy motion or fractional Brownian motion?, Ann. Appl. Probab, vol.12, issue.1, pp.23-68, 2002.

J. Picard, On the existence of smooth densities for jump processes

, Probab. Theory Related Fields, vol.105, issue.4, pp.481-511, 1996.

J. Picard, Density in small time at accessible points for jump processes

, Stochastic Process. Appl, vol.67, issue.2, pp.251-279, 1997.

G. G. Roussas, Contiguity of probability measures: some applications in statistics, vol.63

K. Sato, Lévy Processes and Infinitely Divisible Distributions

, Cambridge Studies in Advanced Mathematics, 1999.

W. Schoutens, Lévy processes in finance: Pricing financial derivatives

, Wiley Series in Probability and Statistics, 2003.

P. Tankov and R. Cont, Financial Modelling with Jump Processes
URL : https://hal.archives-ouvertes.fr/hal-00002693

H. Chapman, CRC Financial Mathematics Series, 2015.

V. Todorov,

, Jump activity estimation for pure-jump semimartingales via self-normalized statistics

, Ann. Statist, vol.43, issue.4, pp.1831-1864, 2015.

V. Todorov and G. Tauchen,

, Limit theorems for power variations of pure-jump processes with application to activity estimation

, Ann. Appl. Probab, vol.21, issue.2, pp.546-588, 2011.

A. W. Van-der and . Vaart,

, Asymptotic statistics, vol.3

W. Feller, An Introduction to Probability Theory and its Applications, vol.2, 1971.