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Partial and Stochastic differential equations


Partial and Stochastic differential equations


Academic year 2023/2024

Course ID
Bruno Toaldo
Marino Badiale
2nd year
Teaching period
First semester
D.M. 270 TAF B - Distinctive
Course disciplinary sector (SSD)
MAT/05 - mathematical analysis
Type of examination
It is crucial to have attended the courses: Analysis, Probability Theory, Stochastic Processes

Sommario del corso


Course objectives

The course aims to put the student in a position to understand the mathematical formulation of various models of applied sciences involving partial and stochastic differential equations. The course uses analytical and probabilistic concepts and tools that are developed in the courses "Analysis", "Probability Theory" and "Stochastic Processes"; these concepts will be briefly mentioned before using them. The proofs of the main results of the course are carried out completely. They show important links between Analysis and Probability. To improve the skills of reading and study the teachers propose the reading of some scientific articles. The course suggests an approach to the research in Mathematical Analysis with probabilistic methods, and vice versa.


Results of learning outcomes

Knowledge and understanding

At the end of the course, students will have acquired knowledge and a solid understanding of various essential methods for studying mathematical models that involve partial and stochastic differential equations. This includes grasping the fundamental concepts behind the theory of Partial Differential Equations (PDEs), the Ito stochastic integral, and the associated stochastic differential equations.

Applying knowledge and understanding

Furthermore, students will be able to apply their acquired knowledge and understanding effectively. They will have the capability to use these mathematical tools to address real-world problems in the field of applied sciences.

Making judgements

A crucial aspect of their learning journey will involve making informed judgments and decisions. Students will develop the capacity to assess and make choices based on their understanding of how stochastic differential equations relate to problems in PDEs. They will gain the ability to discern when and how to apply specific mathematical techniques in different contexts.

Communication skills

Throughout the course, students will enhance their communication skills. They will learn how to articulate and explain complex mathematical concepts from Partial and Stochastic Differential Equations and their applications in a clear and concise manner, facilitating better comprehension and collaboration with others.

Learning skills

Lastly, the course will foster valuable learning skills. Students will develop the capacity to continue their mathematical education independently. They will be equipped with the tools necessary to explore and expand their knowledge in the realm of mathematical modeling, PDEs, and stochastic differential equations, enabling lifelong learning and growth in this field.



1.The diffusion equation:

Derivation of the equation; the fundamental solution; uniqueness and maximum principles; mean-value formulas; some existence results; reaction-diffusion equation; energy methods; some nonlinear problems.

2. The Laplace equation:

Fundamental solution; mean-value formulas; properties of harmonic functions; Green’s function; sub- and superharmonic functions: the Perron method; energy methods.

3. Stochastic calculus:

Brownian trajectories and p-variation, Ito integral and its properties, Ito Lemma, Ito SDEs (existence and uniqueness), examples.

4. PDE problems and diffusion:

Dirichlet problem and diffusion, parabolic equation and diffusion, Feynman-Kac formula, Black and Scholes equation


5. Dirichlet problem:

BM as a martingale; harmonic functions, Brownian martingale and the spherical mean value property; probabilistic representation of the solution for the Dirichlet problem


Course delivery

The teaching course is composed of 48 hours of lectures. Supporting material will be made available on Moodle in due time.

The lectures will be in presence with exceptions in accordance with university regulations.


Learning assessment methods

The final exam will be an oral test. The test will consist of a discussion on the topics of the course, in particular on theorems and proofs.

Suggested readings and bibliography



  • Closed
    Enrollment opening date
    01/09/2022 at 00:00
    Enrollment closing date
    30/06/2023 at 23:55
    Last update: 08/09/2023 10:39
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