1) 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 integration. Knowledge about statistics is useful but not necessary.

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.

2) 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 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

- derive properties of high-frequency estimators for compound Poisson processes

- derive properties of high-frequency estimators of the Lévy density

3) Rules about Homework/Exam

Oral exam determines 100% of the total grade. The retake is an oral exam as well.

4) Lecture notes/Literature

Lecture Notes 2022 (File attached)

5) Lecturer

Jakob Söhl (TUD)

- Docent: Jakob Söhl