**Prerequisites**The students can apply the definition of a continuous time martingale (Definition 1.3.1, Karatzas and Shreve). The students can integrate with respect to continuous, local martingales (Chapter 3.3.2, Karatzas and Shreve).

The students can apply the Ito formula (Theorem 3.3.3, Karatzas and Shreve).

The students can apply Girsanov's change of measure theorem (Theorem 7.19, Liptser and Shiryaev).

The students can apply the Burkholder-Davies-Gundy inequalities (Theorem 3.3.28, Karatzas and Shreve).

**The students can apply the definition of a strong solution of a stochastic differential equation (Definition 5.2.1, Karatzas and Shreve).**

It will be sufficient to learn about these topics by following a simultaneous course on Stochastic Differential Equations.

Ioannis Karatzas and Steven E. Shreve. Brownian Motion and Stochastic Calculus. Springer-Verlag, New York, second edition, 1991.

Robert S. Liptser and Albert N. Shiryaev. Statistics of Random Processes: I General Theory. Springer-Verlag, Berlin, 2001.

**Aim of the course**This course is a theoretical course on statistics for time-continuous stochastic processes.

It covers in particular statistics for diffusion processes and Lévy processes.

We will start with parametric estimation for stochastic differential equations and then move quickly to nonparametric estimation of the drift and the invariant density under continuous-time observations. The course covers nonparametric estimation of the volatility for high-frequency observations. We outline nonparametric estimation for diffusions with low-frequency data. In the second part of the lecture we study the estimation of Lévy processes. We analyse the spectral estimation of Lévy processes from low-frequency observations in the finite intensity case. We also study their behaviour in the misspecified case of infinite intensity. We derive properties of spectral estimators for general Lévy measures. We investigate high-frequency estimators for compound Poisson processes and for Lévy processes.

After successfully finishing this course, the student is able to:

- derive properties of continuous-time nonparametric estimators of the drift
- derive propoerties of continuous-time nonparametric estimators of the invariant density
- derive properies of nonparametric volatility estimators based on high-frequency data
- explain nonparametric estimation for diffusion processes with low-frequency data
- derive properties of spectral estimators for finite intensity Lévy processes
- analyse finite intensity estimators in the infinite intensity case
- derive properties of estimators for general Lévy measures
- derive properties of high-frequency estimators for compound Poisson processes
- derive properties of high-frequency estimators of the Lévy density

**Lecturers**Jakob Söhl (TUD)

- Docent: Jakob Söhl