# Stochastic Analysis

One current line of research at the Institute of Probability and Statistics is based in the area of Stochastic Analysis; this includes:

**Stochastic partial differential equations**

Semilinear stochastic partial differential equations have a broad spectrum of applications, including natural sciences and economics. The analysis of such equations, for example existence and uniqueness theorems, stability results and numerics, requires techniques from Probability Theory and Functional Analysis.

**Stochastic invariance problems**

The solutions to semilinear stochastic partial differential equations typically evolve in an infinite dimensional Hilbert space. In view of a better analytical tractability, one is often interested in the question when the solution process stays on a finite dimensional submanifold of the Hilbert space. The investigation of such stochastic invariance problems requires knowledge from Probability Theory and Differential Geometry.

(Figure: Trajectory of a solution process on an invariant manifold)

#### Stochastic processes

For applications, for example from financial mathematics, one is often interested in investigating particular classes of analytically tractable stochastic processes. Examples of such classes are affine processes, or Lévy processes, which constitute a subclass of the affine processes. Families of Lévy processes, which have been used for modeling financial data, are tempered stable processes and bilateral gamma processes.

**Selected Publications:**

- Filipović, D., Tappe, S., Teichmann, J. (2014): Invariant manifolds with boundary for jump-diffusions.
*Electronic Journal of Probability***19**(51), 1-28 - Küchler, U., Tappe, S. (2013): Tempered stable distributions and processes.
*Stochastic Processes and Their Applications***123**(12), 4256-4293 - Tappe, S. (2013): The Yamada-Watanabe Theorem for mild solutions to stochastic partial differential equations.
*Electronic Communications in Probability***18**(24), 1-13 - Tappe, S. (2012): Existence of affine realizations for Lévy term structure models.
*Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences***468**(2147), 3685-3704 - Filipović, D., Tappe, S., Teichmann, J. (2010): Jump-diffusions in Hilbert spaces: Existence, stability and numerics.
*Stochastics***82**(5), 475-520

Further research articles in the field of Stochastic Analysis, which have been written at the Institute of Probability and Statistics - partially together with researchers at home and abroad - can be found on the homepage of Stefan Tappe. There, the pdf-files of the mentioned articles are available as well.