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