Sumários
18 Março 2025, 11:30
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João Guerra
Infinitesimal generator of a diffusion. Definition and examples.
PArabolic PDE's and a stochastic representation formula for the solution: the Feynman-Kac formula.
Example.
Discussion of exercises on Stochastic Differential Equations.
13 Março 2025, 11:00
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João Guerra
Systems of linear stochastic differential equations. Solution formula.
The exponential of a matrix: definition and basic properties.
How to calculate the exponential of a matrix A if the matrix is diagonalizable, by using eigenvalues and eigenvectors.
Examples.
Introduction to numerical approximations of solutions of SDE's: the Euler scheme and the Millstein scheme.
Discussion of exercises about stochastic differential equations.
11 Março 2025, 11:30
•
João Guerra
SDE's linear in x for the diffusion coefficient and the integrating factor method to solve them. Example.
General one-dimensional linear SDE's. Method of solution.
The existence and uniqueness theorem for one-dimensional SDE's with weaker assumptions on the diffusion coefficent. Example.
Discussion of exercises on the Ito formula and stochastic differential equations.
6 Março 2025, 11:00
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João Guerra
Theorem of existence and uniqueness for solutions of stochastic differential equations. Examples.
Linear growth condition and Lipschitz condition: examples and discussion.
The Ornstein-Uhlenbeck process with mean reversion SDE - solution and properties of the solution. Applications to the Vasicek model of interest rates and the stochastic volatility Black-Scholes model.
27 Fevereiro 2025, 11:00
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João Guerra
Deterministic ordinary differential equations and stochastic differential equations.
Solutions of stochastic differential equations.
Example: The SDE for geometric Brownian motion: how to solve it.
The Langevin Equation or the Ornsteind-Uhlenbeck process SDE - how to solve it.
Discussion of exercises about the application of Itô formula.