JUMP FILTERING AND EFFICIENT DRIFT ESTIMATION FOR LÉVY-DRIVEN SDE'S

Arnaud Gloter (1) , Dasha Loukianova (1) , Hilmar Mai (2)

1 LaMME - Laboratoire de Mathématiques et Modélisation d'Evry
2 CREST - Centre de Recherche en Économie et Statistique

The problem of drift estimation for the solution $X$ of a stochastic differential equation with L\'evy-type jumps is considered under discrete high-frequency observations with a growing observation window. An efficient and asymptotically normal estimator for the drift parameter is constructed under minimal conditions on the jump behavior and the sampling scheme. In the case of a bounded jump measure density these conditions reduce to $n\Delta_n^{3-\eps}\to 0,$ where $n$ is the number of observations and $\Delta_n$ is the maximal sampling step. This result relaxes the condition $n\Delta_n^2 \to 0$ usually required for joint estimation of drift and diffusion coefficient for SDE's with jumps. The main challenge in this estimation problem stems from the appearance of the unobserved continuous part $X^c$ in the likelihood function. In order to construct the drift estimator we recover this continuous part from discrete observations. More precisely, we estimate, in a nonparametric way, stochastic integrals with respect to $X^c$. Convergence results of independent interest are proved for these nonparametric estimators. Finally, we illustrate the behavior of our drift estimator for a number of popular L\'evy--driven models from finance.

Mots clés: Lévy-driven SDE; efficient drift estimation; maximum-likelihood estimation; high frequency data; ergodic properties

Domaines: Mathématiques [math]; Statistiques [math.ST]

Arnaud Gloter, Dasha Loukianova, Hilmar Mai. JUMP FILTERING AND EFFICIENT DRIFT ESTIMATION FOR LÉVY-DRIVEN SDE'S. 2016. ⟨hal-01287823v2⟩