Levy processes, stochastic analysis and financial modelling with jump processes

• This course will be offered online starting November 2nd.
• Further information is available on the moodle system.

Lévy processes form a very fundamental class of continuous time processes including Brownian motion and the Poisson process as special cases. They are heavily used in stochastic modelling, especially in mathematical finance. In this course the theory of Lévy processes and the intrinsically related infinitely divisible distributions will first be studied. Thereafter, stochastic integration and stochastic analysis for Lévy processes and general semimartingales is discussed in detail. Finally,  financial market models based on Lévy processes are introduced and analysed.

In particular the following topics are part of the course:

• Stylized facts of financial data and why use Lévy processes in finance?

• Distributional and path-wise characterisations of Lévy processes

• Examples of Lévy processes and infinitely divisible distributions

• Stochastic integration and stochastic differential equations (SDEs) for general semi-martingales
• Stability of SDEs
• Continuous time Markov processes, Lévy-driven SDEs and their associated infinitesimal generators, partial integro-differential equations and their use in derivative pricing
• Incomplete financial market models in continuous time
• Exponential Lévy models in finance
• Stochastic volatility models
• Option pricing by integral transform techniques

Students will

• be familiar with the complex techniques of Levy processes, stochastic analysis for general semimartingales, Markov processes and financial mathematics in continuous time using models with jumps;
• have knowledge of the required probabilistic and analytic techniques;
• be able to solve complex questions in the context of Levy processes, stochastic analysis and financial mathematics;
• recognize links to further mathematical areas such as functional analysis, Fourier analysis, statistics, optimization and numerical analysis