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Seminar on Probability and Statistics
Seminar information archive ～04/19｜Next seminar｜Future seminars 04/20～
| Organizer(s) | Nakahiro Yoshida, Teppei Ogihara, Yuta Koike |
|---|
2009/12/21
15:00-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)
Thomas Simon (Universite de Lille 1)
Absolute continuity of Ornstein-Uhlenbeck processes
https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/13.html
Thomas Simon (Universite de Lille 1)
Absolute continuity of Ornstein-Uhlenbeck processes
[ Abstract ]
Let X be a multidimensional Ornstein-Uhlenbeck process, solution to the S.D.E.
dX = AX + dB
where A is a real nxn matrix and B a Lévy process. We show that when A is non-singular, the law of X_1 is absolutely continuous if and only if the jumping measure of B fulfils a certain geometric condition with respect to A and the Gaussian part of B, which we call the exhaustion property. This optimal criterion is much weaker than for B, which might be very singular and genuinely one-dimensional. The proof uses a certain time derivation procedure and basic arguments from controllability theory.
[ Reference URL ]Let X be a multidimensional Ornstein-Uhlenbeck process, solution to the S.D.E.
dX = AX + dB
where A is a real nxn matrix and B a Lévy process. We show that when A is non-singular, the law of X_1 is absolutely continuous if and only if the jumping measure of B fulfils a certain geometric condition with respect to A and the Gaussian part of B, which we call the exhaustion property. This optimal criterion is much weaker than for B, which might be very singular and genuinely one-dimensional. The proof uses a certain time derivation procedure and basic arguments from controllability theory.
https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/13.html
