Tanaka-Formel

Die Tanaka-Formel ist eine Resultat aus der stochastischen Analysis über die Darstellung der Lokalzeit eines stetigen Semimartingals bei $0$ . Die Formel stammt von dem japanischen Mathematiker Tanaka Hiroshi.

Es existiert eine Verallgemeinerung von Paul-André Meyer.

Tanaka-Formel

Sei $X=(X_{t})_{t\geq 0}$ ein stetiges Semimartingal in $\mathbb {R}$ und $L^{0}$ seine Lokalzeit bei $0$ . Weiter definieren wir folgende Vorzeichenfunktion

\operatorname {sgn} (x-):=1_{(0,\infty )}(x)-1_{(-\infty ,0]}(x).

Dann lautet die Tanaka-Formel

L_{t}^{0}=|X_{t}|-|X_{0}|-\int _{0}^{t}\operatorname {sgn} (X_{s}-)\mathrm {d} X_{s},\quad t\geq 0.

Allgemeiner definiert man die Lokalzeit $L_{t}^{x}$ an einem Punkt $x\in \mathbb {R}$ als die Lokalzeit bei $0$ für den Prozess $X-x$ , dann erhält man folgende Tanaka-Formel

L_{t}^{x}=|X_{t}-x|-|X_{0}-x|-\int _{0}^{t}\operatorname {sgn} \left((X_{s}-x)-\right)\mathrm {d} X_{s},\quad t\geq 0.

^[1]

Literatur

Philip E. Protter: Stochastic Integration and Differential Equations. Springer, 2004, ISBN 3-540-00313-4 (englisch).
K. L. Chung, R. J. Williams: Introduction to Stochastic Integration. Springer, 2010, ISBN 978-1-4612-8837-4 (englisch).

Einzelnachweise

↑ Olav Kallenberg: Foundations of Modern Probability. Hrsg.: Springer. 2021, S. 662, doi:10.1007/978-3-030-61871-1.

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