# Stochastic Differential Equations

## Objectives

-Main objectives:

Being an optional course of the Master degree in Mathematics and Applications, this course intends to give appropriate knowledge for the study of the evolution of random phenomena described by diffusion processes.

-Main objectives related to knowledge:

Knowledge of Itô calculus. Knowledge of the classical theorem of existence and uniqueness of solution for a stochastic differential equation. Knowledge of the Markov property for the solution of a stochastic differential equation. Knowledge of the relationship between stochastic differential equations and certain deterministic partial differential equations.

-Main competencies:

know the Itô calculus. know how to solve linear stochastic differential equations. know how to study problems of existence and uniqueness for stochastic differential equations. know how to use the Markov property to manipulate relationships between stochastic differential equations and equations with partial derivatives.

## General characterization

##### Code

8540

##### Credits

6.0

##### Responsible teacher

Maria Fernanda de Almeida Cipriano Salvador Marques

##### Hours

Weekly - 4

Total - 56

##### Teaching language

Português

### Prerequisites

Knowledge of measure theory and Lebesgue integration. Knowledge of ordinary differential equations.

### Bibliography

*1 -- Hui.Hsiung Kuo: Introduction to Stochastic Integration. Springer. 2006*

*2 -- Bernt Oksendal: Stochastic Differential Equations. Sringer. 1998*

*3 -- Paul Malliavin: Integration and Probability. Springer-Verlag. 1995*

### Teaching method

*The professor gives the course by lectures, where he explains all topics referred to in the syllabus.*

### Evaluation method

The evaluation is performed by Continuous Evaluation or Exam.

1- Continuous Evaluation

The Continuous Evaluation is performed through Exercise Resolutions, one Test and a Work. The Resolutions of Exercises have the classification of 20 values, the Test has the classification of 20 values and the Work has the classification of 20 values.

The classification of Continuous Evaluation is obtained by doing the arithmetic mean of the 3 classifications obtained.

If the Continuous Evaluation classification is higher than or equal to 9.5, the student is approved with this rounded classification to the units.

2 - Exam

All students enrolled in the course (and not yet approved) can realize the Exam.

The final classification will be equal to 1/3 (classification obtained in the Resolution of the Exercises) + 2/3 (classification of the Exam).

If the final classification is higher than or equal to 9.5 the student is approved with this rounded classification to the units.

If the final classification is less than or equal to 9.4 the student is not approved.

## Subject matter

*1-Brownian Motion- Wiener integral- Conditional expectation- Martingales*

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*2-Stochastic integrals*

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*3-Itô formula*

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*4- Applications of the Itô Formula- Exponential process- Transformation of probability measures-Girsanov theorem*

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*5- Stochastic Differential equations-Existence and uniqueness-Markov property-Diffusion processes- Semigroups and*

*Kolmogorov equations*

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*6-Linear stochastic differential equations- Applications to Finance- Feynman-Kac Formula*