15. ÖMG-Kongress
Jahrestagung der Deutschen Mathematikervereinigung

16. bis 22. September 2001 in Wien

Sektion 12 - Wahrscheinlichkeitstheorie, Statistik
Montag, 17. September 2001, 16.30, Hörsaal 7


Stochastic Equations Driven by Symmetric Stable Processes

Hans-Jürgen Engelbert, Friedrich-Schiller-Universität Jena



We study stochastic equations

$\displaystyle X_t=x_0+\int_0^t b(u,X_{u-})\, dZ_u,\quad t\geq0,$

driven by one-dimensional symmetric stable processes $ Z$ of index $ \alpha$ with $ 0<\alpha\leq 2$. Here $ b: [0,+\infty)\times R\rightarrow R$ denotes a measurable diffusion coefficient and $ x_0\in R$ is the initial value. As special cases for the driving process $ Z$, Brownian motion ($ \alpha=2$) and the Cauchy process ($ \alpha=1$) are included. We are interested in general conditions for existence and uniqueness of weak solutions. The basic tool is time change of symmetric stable processes. Using the property that appropriate time changes of stochastic integrals with respect to symmetric stable processes are again symmetric stable processes with the same index, we present a new approach which completely unifies the treatment of two quite different cases: the continuous case ($ \alpha=2$) and the purely discontinuous case ( $ 0<\alpha<2$).

E-Mail: engelbert@minet.uni-jena.de

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