Stochastic Differential Equations


Available soon

General characterization





Responsible teacher

Maria Fernanda de Almeida Cipriano Salvador Marques


Weekly - 4

Total - Available soon

Teaching language



Available soon


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.

 Mid-term evaluation consists of an individual component and / or a group component according to the FCT NOVA evaluation regulation.

Students who do not succeed in the continuous assessment may try to pass a final Exam.

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


2-Stochastic integrals


3-Itô formula


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


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

Kolmogorov equations


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


Programs where the course is taught: