- Lecturer
- Martin Larsson, martin.larsson@math.ethz.ch
- Time and location
- Tuesdays, 10-12, HG F3
Description
A basic goal in mathematical finance is to develop market models that combine statistical flexibility with analytical tractability. A common class of such models are affine, and more generally polynomial, jump-diffusions. This course will develop the theory of polynomial jump-diffusions, the mathematical tools needed to study them, and discuss a selection of applications. The aim of this course is to develop the theory of polynomial jump-diffusions, the mathematical tools needed to study them, and discuss a selection of applications. Specifically, the goal is to cover the following topics:
- - Introduction to affine and polynomial processes
- - Semimartingales and their characteristics; jump-diffusions
- - Affine and polynomial jump-diffusions; the moment formula; the exponential-affine transform formula
- - Existence and uniqueness theory: Martingale problems; the positive maximum principle; SDE methods
- - Applications: Stochastic volatility; term structure of interest rates
Lecture notes and literature
Lecture notes are available here, and will be continually updated and expanded throughout the course. For further reading on the general theory of stochastic processes and semimartingale theory, see the book Limit Theorem for Stochastic Processes by Jacod and Shiryaev, especially Chapters I and II. Another nice source are the stochastic calculus notes on George Lowther's blog almostsure.
Errata for lecture notes:
- - In Theorem 4.9(ii), the quantities |f_n(y)|+|Gf_n(y)| should be bounded uniformly in both n and y, not just in y.
- - In Example 3.29, the drift coefficient is missing a minus sign. It should be b(x) = - \lambda x.
- - In Example 4.15, \R_+-valued should be replaced by \R-valued in the last sentence.
- - In Proposition 5.22, there is a sign error inside the integrals: The expressions should be e^{-B(\tau)^\top \xi} - 1 + B(\tau)^\top \xi, rather than e^{-B(\tau)^\top \xi} - 1 - B(\tau)^\top \xi.