Convex Stochastic Optimization

The course is an introduction to dynamic programming and duality in convex stochastic optimization. We apply the theory to stochastic control and financial mathematics. Dynamic programming gives optimality conditions in terms of Bellmann equations that form a basis for various numerical methods to solve practical problems. Duality builds on the general conjugate duality framework of convex optimization that is then applied to convex stochastic optimization. As special cases, we will learn that familiar results in optimization and financial mathematics, like duality in linear programming and the fundamental theorem of asset pricing, are special cases of the theory.

Lecture notes (PDF, 765 KB)

Exercises

Exercise session 16.10 : a lecture

Exercise session 23.10: Exercises 2.2.1 - 2.2.5 in the lecture notes

Exercise Session 30.10: Holiday, no exercises.

Exercise Session 8.11.: Thm. 2.5, Ex 2.4.1., Thm 2.8, Ex 2.4.2, Lem 2.9

Exercise Session 15.11.: Ex 2.5.1 , Thm 2.15, Thm 2.17, Ex 2.5.2, Lem 2.20

Exercise Session 22.11.: Thm 2.21, Ex 2.7.1, Thm 2.24, Ex. 2.7.3, Ex. 2.7.4

Exercise Session 29.11: Everything in Section 2.8.

Exercise Session 6.12: Postponed.

Exercise Session 13.12: Ex 2.9.1, Thm 3.1, Thm 3.9.

Exercise Session 20.12: A lecture.

Exercise Session 10.1. A lecture.

Exercise Session 17.1. A lecture .

Convex Analysis- Rockafellar

Variational Analysis -Rockafellar and Wets

Functional analysis -Rudin

Stochastic Finance -Föllmer and Schied

Target Participants: Master students in Mathematics and Financial and Insurance Mathematics. Master students in mathematics and bachelor students are also welcome.

Pre-requisites: Probability

Applicable credits: 9ECTS