SDE's

8 Outubro 2018, 11:30 • João Guerra

SDE's driven by Brownian motion. 

The Orsnstein-Uhlenbeck process with mean reversion. 

The mean reversion effect. The stationary distribution for the Orsnstein-Uhlenbeck process with mean reversion. 

Financial applications: the Vasicek model and the Black-Scholes model with stochastic volatility. 

 

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Multidimensional Itô formula and SDE's

4 Outubro 2018, 11:30 • João Guerra

Multidimensional Itô formula: examples of application. 

Stochastic differential Equations: determinist equations and stochastic equations. How to solve them. 

Examples: the stochastic differential equation of the geometric Brownian motion; the Langevin equation (Ornstein-Uhlenbeck process). 

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Ito Formula

1 Outubro 2018, 11:30 • João Guerra

Calculate a stochastic integral by definition: example.

Ito formula and Ito processes: motivation and examples.

One-dimensional Ito formula: example related to finance.

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Stochastic integrals

28 Setembro 2018, 12:00 • João Guerra

Stochastic integrals: motivation, definition of stochastic integrals for simple processes and definition of stochastic integrals for processes in L^2_{a} (adapted processes in L^2). 

Examples and exercises.

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Martingales in continuous time and Brownian motion

26 Setembro 2018, 11:30 • João Guerra

Martingales in discrete time: 

Examples: Double betting system, the discounted price of an asset as a martingale in the risk-neutral probability measure. 

Martingales in continuous time: definition, examples and exercises. 

Processes defined as functions of the Brownian motion.

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