Sumários

Stochastic Differential Equations

29 Setembro 2026, 11:30 • João Guerra

Examples of solutions of stochastic differential equations: the geometric Brownian motion SDE, the Langevin SDE, the Ornstein-Uhlenbeck process SDE, the Ornstein-Uhlenbeck process with mean reversion SDE.

Financial applications of the Ornstein-Uhlenbeck process with mean reversion SDE.

Discussion of exercise on the application of Itô formula.

Discussion of exercises and introduction to stochastic differential equations

28 Setembro 2026, 12:00 • João Guerra

Discussion of exercises about martingales in continuous time and applications of Itô formula.

Introduction to stochastic differential equations. Deterministic ordinary differential equations and stochastic differential equations. Existence and uniqueness of solutions to SDE's: sufficient conditions (Linear growth property and Lipschitz property) and example of explosion of solution in finite time.

Example of solving the geometric Brownian motion SDE by simple application of the Itô formula.

The Itô formula

22 Setembro 2026, 11:30 • João Guerra

The Itô formula: motivation and idea using the Taylor formula and the chain rule.

Itô process definition.

Examples of application of the one-dimensional Itô formula.

The multidimensional Itô formula. Examples of application.

Stochastic integrals

21 Setembro 2026, 12:00 • João Guerra

Stochastic integrals: motivation and the Riemann-Stieltjes integral.

Stochastic integrals of simple processes: definition, examples and properties.

Stochastic integrals of adapted processes: definition and properties. Example.

Martingales. Exercises.

15 Setembro 2026, 11:30 • João Guerra

Martingales in discrete and continuous time. Definitions, properties and examples.

The symmetric random walk as an example of martingale in discrete time.

Martingales related with Brownian motion.

Exercises about the inverting property of the Brownian motion and properties of conditional expectation.