Jump filtering and efficient drift estimation for Lévy-driven SDE's

2016/03/16 by Gloter, Arnaud, Loukianova, Dasha, Mai, Hilmar
#62F12 #62M05 #FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1603.05290

Abstract

The problem of drift estimation for the solution X of a stochastic differential equation with Lévy-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 Δ_n^3-ε→ 0, where n is the number of observations and Δ_n is the maximal sampling step. This result relaxes the condition nΔ_n² → 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 Lévy-driven models from finance.

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