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

Clément, Emmanuelle; Gloter, Arnaud; Nguyen, Huong

ESAIM: Probability and Statistics, Tome 22 (2018), pp. 58-95.

This work focuses on the asymptotic behavior of the density in small time of a stochastic differential equation driven by a truncated $\alpha$-stable process with index $\alpha \in (0,2)$. We assume that the process depends on a parameter $\beta ={(\theta ,\sigma )}^T$ and we study the sensitivity of the density with respect to this parameter. This extends the results of [E. Clément and A. Gloter, Local asymptotic mixed normality property for discretely observed stochastic dierential equations driven by stable Lévy processes. Stochastic Process. Appl. 125 (2015) 2316-1352.] which was restricted to the index $\alpha \in (1,2)$ and considered only the sensitivity with respect to the drift coefficient. By using Malliavin calculus, we obtain the representation of the density and its derivative as an expectation and a conditional expectation. This permits to analyze the asymptotic behavior in small time of the density, using the time rescaling property of the stable process. upper boundupper bound

Classification : 60G51, 60G52, 60H07, 60H20, 60H10, 60J75
Mots clés : Lévy process, density in small time, stable process, Malliavin calculus for jump processes

[1] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edn. Vol. 116 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2009). | DOI | MR | Zbl

[2] K. Bichteler, J.-B. Gravereaux and J. Jacod, Malliavin Calculus for Processes with Jumps. Vol. 2 of Stochastics Monographs. Gordon and Breach Science Publishers, New York (1987). | MR | Zbl

[3] N. Bouleau and L. Denis, Dirichlet Forms Methods for Poisson Point Measures and Lévy Processes. Vol. 76 of ty Theory and Stochastic Modelling. Springer, Cham (2015). With emphasis on the creation-annihilation techniques. | DOI | MR

[4] E. Ç"I"Nlar, Probability and Stochastics. Vol. 261 of Texts in Mathematics. Springer, New York (2011). | MR | Zbl

[5] E. Clément and A. Gloter, Local asymptotic mixed normality property for discretely observed stochastic differential equations driven by stable Lévy processes. Stochastic Process. Appl. 125 (2015) 2316–2352. | DOI | MR | Zbl

[6] E. Clément, A. Gloter and H. Nguyen, LAMN property for the drift and volatility parameters of a SDE driven by a stable Lévy process. Preprint (2017). | HAL | Numdam | MR | Zbl

[7] A. Debussche and N. Fournier, Existence of densities for stable-like driven SDE’s with Hölder continuous coefficients. J. Funct. Anal. 264 (2013) 1757–1778. | DOI | MR | Zbl

[8] L. Denis, A criterion of density for solutions of Poisson-driven SDEs. Probab. Theory Related Fields 118 (2000) 406–426. | DOI | MR | Zbl

[9] N. Fournier and J. Printems, Absolute continuity for some one-dimensional processes. Bernoulli 16 (2010) 343–360. | DOI | MR | Zbl

[10] Y. Ishikawa and H. Kunita, Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE with jumps. Stochastic Process. Appl. 116 (2006) 1743–1769. | DOI | MR | Zbl

[11] D. Ivanenko and A. Kulik, Malliavin calculus approach to statistical inference for Lévy driven SDE’s. Methodol. Comput. Appl. Probab. 17 (2015) 107–123. | DOI | MR | Zbl

[12] P. Jeganathan, On the asymptotic theory of estimation when the limit of the log-likelihood ratios is mixed normal. Sankhy 44 (1982) 173–212. | MR | Zbl

[13] A. Kulik, On weak uniqueness and distributional properties of a solution to an SDE with α-stable noise. Preprint (2015). | arXiv | MR

[14] J. Picard, On the existence of smooth densities for jump processes. Probab. Theory Related Fields 105 (1996) 481–511 . | DOI | MR | Zbl