Dynamic Optimization

Professor: : Naila Hayek, Professeur des Universités en Mathématiques Appliquées à l’Université Paris II.

Objective: The aim of this course is to provide some useful mathematical models and tools in economics and finance. The first part of the course is devoted to reminders of static optimization: optimization without constraints and under equality and inequality constraints, to which are added depths and applications. The second part of the course focuses on the dynamic optimization: Calculus of Variations and Optimal Control.

Outline:

I – Static optimization

1/ Optimization without constraint and under constraints

- The Lagrange multipliers; the Kuhn and Tucker theorem

2/ The generalized Lagrangian

3/ Applications: portfolio management , hedging on a futures market

II - Dynamic Optimization

1/ Calculus of Variations

- Necessary conditions and sufficient conditions of optimality; Euler-Lagrange Equation

- Applications: consumption-saving problem, ...

2/ Deterministic Optimal Control

- Necessary conditions and sufficient conditions of optimality; the Pontryagin Maximum Principle

- Applications: portfolio management in the presence of transaction costs, ...

3/ Introduction of Dynamic Programming

Grading: Final exam at the end of the semester.

Bibliography:

• Demange G & Rochet J.C. Méthodes mathématiques de la finance. 3e edition Economica. 2005 ;
• Hayek N. & Leca J.P. Mathématiques pour l’Economie. Analyse-Algèbre. 5e edition Dunod. 2015.