Stochastic Differential equations

16 Abril 2018, 10:00 • João Guerra

Stochastic Differential equations: introdution and solution definition.

Examples: geometric Brownian motion SDE, Langevin SDE, Ornstein- Uhlenbeck process with mean reversion SDE. Solutions.

The Vasicek model and Black-Scholes model with stochastic volatility SDE's.

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The Martingale Representation Theorem

12 Abril 2018, 10:00 • João Guerra

The Itô representation Theorem and the martingale representation theorem. 

Examples and Exercises. 

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Itô Formula. Exercises

9 Abril 2018, 10:00 • João Guerra

Multidimensional Itô processes and multidimensional Itô formula. 

Examples. 

Exercises on the application of the one-dimensional and multidimensional Itô formula. 

 

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One dimensional Itô formula

5 Abril 2018, 10:00 • João Guerra

The Borel-Cantelli Lemma - an elementary proof. 

Brief introduction to the notion of differential of a deterministic differentiable function. An example. 

Introduction to the informal notion of differential of a Brownian motion and stochastic differential of a process by analogy with the first-order taylor development and the first order differential of a deterministic function. 

Definition of the space L_{a,T} of processes

Definition of Itô process. Integral and differential form of an Itô process. 

One-dimensional Itô formula: different versions of the formula in integral form and differential form.

The product table for the product of differentials. An example. 

Sketch of the rigorous proof of the one-dimensional Itô formula. 

 

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Properties of the stochastic integral as a process

22 Março 2018, 10:00 • João Guerra

Properties of the stochastic integral as a process: martingale property and continuity of the trajectories.

Informal introduction to the Itô formula as a stochastic "chain rule" for functions of the Brownian motion. 

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