Theory and Numerics of Ordinary and Partial Differential Equations

The first lesson of the course will provide a primer on complex variables. Using this mathematical formalism, the focus of the remaining first part of the course will be to introduce linear ordinary and linear partial differential equations, and the "cheap" methods to solve them using Fourier and Laplace transforms. The ordinary and partial differential equations will be placed into a context of applied mathematics (e.g. classic deterministic and stochastic systems) saving the theoretical approach for advanced lectures.

Stochastic Processes and Stochastic Calculus

This course aims at introducing some important stochastic processes (Markov chains, martingales,
Poisson process, Wiener process) and Ito calculus.
Some proofs are sketched or omitted in order to have more time for examples, applications and
In particular, the course deals with the following topics:

Statistics Lab.

- Brief introduction to R (
- Creating random variables.
- Applications to the central limit theorem and the law of large numbers
- Descriptive statistics: (i) Representing probability and cumulative distribution functions in discrete and continuous cases; (ii) calculating mean, variance, concentration indexes, covariance and correlation coeff.
- Statistical inference: (i) Point estimation and properties; (ii) interval estimation and properties; (iii) hypothesis testing and properties.

States and Markets

The main topic is the political economy of state formation in historical and comparative perspective. We will read some classic works by economic historians, sociologists and political scientists and try to relate this work to the modern literature (in economics) on the political economy of state capacity. We will also briefly talk of some theories (old and new) on the origins of federalism, and try to relate this work to the literature on state formation. rules and redistribution.

Optimal Control

Discrete-time optimal control: dynamic programming for finite/infinite horizon and deterministic/stochastic optimization problems. LQ and LQG problems, Riccati equations, Kalman filter. Deterministic continuous-time optimal control: the Hamilton-Jacobi-Bellman equation and the Pontryagin?s principle. Examples of optimal control problems in economics.