Contents |
Course Title
Special Topics: Geometry of Stochastic Processes
Course Code
IAM 752
Credit
(3-0)3
Prerequisites
Consent of the instructor.
Content
Motivation: We will investigate several aspects of fractal and Riemannian geometry related to stochastic processes. In particular stochastic processes of fractional dimension and matrix-valued stochastic processes are traditionally applied in physics, and more recently also in finance. E.g. Wishart models (Gaussian random matrices) are applied to model stochastic volatility for credit risk, and hyperbolic geometry turns out to be essential for the calibration of SABR-LIBOR market models. Non-semimartingale processes such as fractional Brownian motion are used to model memory effects of the markets resulting in self similarities in price dynamics.
Aims
The primary aim is to familiarize with structures and methods from fractal geometry and Riemannian geometry, which are relevant for advanced modeling in stochastic models of finance and physics.
Learning Outcomes
Upon successful completion of the course, it is expected that the student will be familiar with some of the advanced techniques of fractal and Riemannian geometry in stochastic finance.
The inclined research student will be prepared to start mathematical research on the topics related to geometry in stochastic finance.
Suggested Textbooks
Lecture notes will be prepared and further references to research papers will be given during the course.
Some of the content of this course is covered in the following:
About fractal stochastic processes in:
- P. Embrechts, C. Klüppelberg, T. Mikosch: Modelling Extremal Events. Springer 1997
- A.N. Shiryaev: Essentials of Stochastic Finance. World Scientific, 1999.
About matrix valued stochastic processes in:
- M.L. Mehta, Random matrices, 2nd ed. (Academic Press, San Diego, 1991).
About Riemannian geometry in the paper:
- P. Henry-Labordere: Unifying the BGM and SABR models: A short ride in hyperbolic geometry. SSRN Preprint, 2007
Outline
The tentative program of this course is as follows:
- 1.week: self-similar stochastic processes
- 2.week: fractional Brownian motion
- 3.week: fractionality in processes related to commodities
- 4.week: Kolmogorov equation and heat kernel equation
- 5. week: Riemannian geometry related to stochastic processes
- 6.week: connections, curvature, Kolmogorov equation, heat kernel
- 7.week: Stratonovich integral, gauge transformations, change of measure
- 8.week: heat kernel expansion
- 8.week: local volatility of multi asset processes and flat geometry
- 9.week: stochastic volatility and Riemann surfaces
- 10.week: SABR models and hyperbolic geometry
- 11.week: SABR-LIBOR market models and hyperbolic geometry
- 12.week: matrix valued stochastic models, random matrices
- 13.week: orthogonal, unitary, and symplectic/chiral Gaussian ensembles
- 14.week: Wishart models applied to stochastic volatility in credit risk
Resources
- LaTEX