Laurea Magistrale (M.Sc.) in Stochastics and Data Science

Partial and Stochastic differential equations

Partial and Stochastic differential equations

Academic year 2022/2023

Sommario del corso

• Course objectives
• Results of learning outcomes
• Course delivery
• Learning assessment methods
• Program
• Suggested textbooks and readings
• Teaching tools

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

At the end of the course, students will know several important methods to study mathematical models involving partial and stochastic differential equations; in particular they will know basic elements of the theory of Partial Differential Equations (PDEs), Ito stochastic integral and the related stochastic differential equations. Particular attention will be devoted between the interaction of the two theories, e.g., the students will understand relations between stochastic differential equations, the Laplace equations and the Kolmogorov (parabolic) equations. They will be able to apply the theory of stochastic differential equations to solve problems in PDEs arising in applied sciences.

Course delivery

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

Please check this pageopen_in_new for the teaching modalities foreseen for the a.y. 2022/23.

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.

Program

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

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• Teaching materials
• Exams and finals
• Go to Moodle
• Visit the forums