## Seminar on Probability and Statistics

Seminar information archive ～06/23｜Next seminar｜Future seminars 06/24～

Organizer(s) | Nakahiro Yoshida, Hiroki Masuda, Teppei Ogihara, Yuta Koike |
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### 2008/02/06

14:50-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Estimating the Degree of Activity of jumps in High Frequency Data

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/18.html

**Jean JACOD**(Universite Paris 6)Estimating the Degree of Activity of jumps in High Frequency Data

[ Abstract ]

Suppose that a continuous-time process X = (X_t )_{t >= 0} is observed at finitely many times, regularly spaced, on the fixed time interval [0, T ]. We suppose that this process is an It\\^o semimartingale, with a non-vanishing diffusion coefficient, and with jumps. The aim is to estimate the so-called ”Blumenthal-Getoor” index of the (partially observed) path on [0, T ], which is the (random) infimum of all reals r such that the sum \\sum_{s\\le T} |\\Delta X_s|^r is finite (\\Delta X_s denotes the jump size at time s). When X is a L'evy process, this infimum is non-random, and also independent of T , and has been introduced by Blumenthal and Getoor. Under appropriate assumptions, unfortunately rather restrictive, we provide an estimator, which is consistent when the step size between observations goes to 0, and satisfies in addition a Central Limit Theorem. We also show the (surprising) values that this estimator takes, when applied to real financial data.

[ Reference URL ]Suppose that a continuous-time process X = (X_t )_{t >= 0} is observed at finitely many times, regularly spaced, on the fixed time interval [0, T ]. We suppose that this process is an It\\^o semimartingale, with a non-vanishing diffusion coefficient, and with jumps. The aim is to estimate the so-called ”Blumenthal-Getoor” index of the (partially observed) path on [0, T ], which is the (random) infimum of all reals r such that the sum \\sum_{s\\le T} |\\Delta X_s|^r is finite (\\Delta X_s denotes the jump size at time s). When X is a L'evy process, this infimum is non-random, and also independent of T , and has been introduced by Blumenthal and Getoor. Under appropriate assumptions, unfortunately rather restrictive, we provide an estimator, which is consistent when the step size between observations goes to 0, and satisfies in addition a Central Limit Theorem. We also show the (surprising) values that this estimator takes, when applied to real financial data.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/18.html