## The Annals of Statistics

- Ann. Statist.
- Volume 37, Number 1 (2009), 184-222.

### Testing for jumps in a discretely observed process

Yacine Aït-Sahalia and Jean Jacod

#### Abstract

We propose a new test to determine whether jumps are present in asset returns or other discretely sampled processes. As the sampling interval tends to 0, our test statistic converges to 1 if there are jumps, and to another deterministic and known value (such as 2) if there are no jumps. The test is valid for all Itô semimartingales, depends neither on the law of the process nor on the coefficients of the equation which it solves, does not require a preliminary estimation of these coefficients, and when there are jumps the test is applicable whether jumps have finite or infinite-activity and for an arbitrary Blumenthal–Getoor index. We finally implement the test on simulations and asset returns data.

#### Article information

**Source**

Ann. Statist., Volume 37, Number 1 (2009), 184-222.

**Dates**

First available in Project Euclid: 16 January 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1232115932

**Digital Object Identifier**

doi:10.1214/07-AOS568

**Mathematical Reviews number (MathSciNet)**

MR2488349

**Zentralblatt MATH identifier**

1155.62057

**Subjects**

Primary: 62F12: Asymptotic properties of estimators 62M05: Markov processes: estimation

Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

**Keywords**

Jumps test discrete sampling high-frequency

#### Citation

Aït-Sahalia, Yacine; Jacod, Jean. Testing for jumps in a discretely observed process. Ann. Statist. 37 (2009), no. 1, 184--222. doi:10.1214/07-AOS568. https://projecteuclid.org/euclid.aos/1232115932