Jump filtering and efficient drift estimation for Lévy-driven SDEs
| Autor: | Hilmar Mai, Dasha Loukianova, Arnaud Gloter |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Statistics and Probability
010102 general mathematics Lévy-driven SDE Sampling (statistics) Order (ring theory) Estimator maximum likelihood estimation ergodic properties 01 natural sciences Measure (mathematics) 010104 statistics & probability Stochastic differential equation 62M05 Bounded function efficient drift estimation Jump Applied mathematics 60J75 0101 mathematics Statistics Probability and Uncertainty 62F12 Likelihood function high frequency data Mathematics |
| Zdroj: | Ann. Statist. 46, no. 4 (2018), 1445-1480 |
| ISSN: | 0090-5364 |
| DOI: | 10.1214/17-aos1591 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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