Electronic Journal of Statistics

Parametric estimation for discretely observed stochastic processes with jumps

Hoang-Long Ngo

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Abstract

We consider a two dimensional stochastic process (X,Y), which may have jump components and is not necessarily ergodic. There is an unknown parameter θ within the coefficients of (X,Y). The aim of this paper is to estimate θ from a regularly spaced sample of the process (X,Y). When the dynamic of X is known, an estimator is constructed by using a moment-based method. We show that our estimators will work if the Blumenthal-Getoor index of the jump part of Y is less than 2. What is perhaps the most interesting is the rate at which the estimators converge: it is $1/{\sqrt{n}}$ (as when the underlying processes are not contaminated by jumps) when that index is not greater than 1. When the dynamic of X is unknown, we introduce a spot volatility estimator-based approach to estimate θ. This approach can work even if the sample is contaminated by microstructure noise.

Article information

Source
Electron. J. Statist., Volume 4 (2010), 1443-1469.

Dates
First available in Project Euclid: 9 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1291903545

Digital Object Identifier
doi:10.1214/10-EJS590

Mathematical Reviews number (MathSciNet)
MR2741208

Zentralblatt MATH identifier
1329.62367

Subjects
Primary: 60M09 60J75: Jump processes
Secondary: 60M05 62G05: Estimation

Keywords
Blumenthal-Getoor index discrete observation jump process microstructure noise non-ergodic process nonparameric estimation parametric estimation

Citation

Ngo, Hoang-Long. Parametric estimation for discretely observed stochastic processes with jumps. Electron. J. Statist. 4 (2010), 1443--1469. doi:10.1214/10-EJS590. https://projecteuclid.org/euclid.ejs/1291903545


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