# Lévy and Affine Processes

## Lecture: Alessandro Gnoatto

**Date and Time**

- Lectures: Mon 08:00 to 09:45 (Room B 045), Wed 08:00 to 09:45 (Room B 045).
- Lectures start at 08.00 sharp.
- Lecture starts on October 17th, 2012

**Further Information**

- This is a graduate lecture on stochastic processes. We will study tractable examples of continuous time Markov processes, which may be employed for modeling phenomena like financial quantities (interest rates, stock returns etc). The nice analytical features of the stochastic process we will consider are given by the particularly simple form of their characteristic function. For this reason we will start by recalling some well known results from the theory of characteristic functions.

In the first part of the lecture, we will introduce the class of Lévy processes, which nests known examples like Brownian Motion and the Poisson process. In this first part we will talk about infinite divisibility, martingales, random measures and stopping times. After that we will explore the jumps of Lévy processes by looking at Poisson random measures and Poisson integration. This will open up the way to prove the famous Lévy-Ito decomposition, which allows us to understand the structure of the paths of a Lévy process. The Lévy-Ito decomposition permits us to simplify the proof of another important result, which is the Lévy-Khintchine formula. This last formula tells us that Lévy processes/infinitely divisible distributions may be described by means of a triplet describing the drift, the diffusion and the jumps of the process.

The second topic will be an intuitive introduction to operator semigroups, and their importance in Finance (Kolmogorov equations).

Time permitting, the final part of the lecture will be devoted to Affine processes, in the sense of Duffie Filipovic and Schachermayer (2003). We will see that the concept of Lévy triplet may be further generalized by introducing a class of processes where the triplet is state dependent, so instead of having a constant drift, diffusion and jump coefficient, the process will feature also linear drift diffusion and jump coefficients. A famous example in this class is provided by the square root process, introduced in Finance by Cox Ingersoll and Ross (1985). - The lecture will be held in English.
- The lecture addresses: Master Wirtschaftsmathematik (WP61)
- Literature:

Sato, K. I.: Lévy processes and infinitely divisible distributions. Cambridge University press.

Applebaum, D.: Lévy processes and stochastic calculus Cambridge University press.

Keller-Ressel, M.: Affine processes- Theory and applications in Finance – PhD thesis Vienna University of Technology.

Duffie, D. Fililpovic, D., Schachermayer, W.: Affine processes and applications in finance