1. Introducing Stochastic Calculus and its rules.

2. Introducing Stochastic Differential Equations.

3. Showing their utility in modelling evolutions with noise in different contexts such as Finance, Engineering, Chemistry, Physics.

Knowledge and understanding

Lectures and exercise sessions will allow students to

- know the fundamental elements of stochastic processes (Brownian motion, martingales and Markov processes), stochastic calculus and stochastic differential equations;

- understand the applications: there are very many situations (in finance, telecommunications, control, ...) where stochastic processes, and in particular diffusions, are a natural model;

 - understand the connections with other branches of pure mathematics, in particular with partial differential equations.

Ability in applying knowledge and understanding

Lectures and exercise sessions will allow students to

- manipulate the basic notions of stochastic calculus acquiring elements of the theory and ability to work with them;

- have an adequate knowledge of the techniques to formalize and to model evolutions with noise in different contexts of science and technology;

- learn the constructive mathematical proofs of some key result of the theory;

- acquire the ability to present in a clear and precise way the models and the results of the theory.

1. Elements of Probability. Probability spaces, random variables. Variance, covariance, probability distribution, density. Independence. Random vectors. Convergence of random variables. Characteristic functions. Gaussian laws. Measurability theorems.

2. Stochastic processes. Filtrations. Trajectories. Equivalent processes, modifications, indistinguishable processes. Finite-dimensional laws and Existence Kolmogorov’s theorem. Kolmogorov’s continuity theorem. Stopping times.

3. Brownian motion. Definition and basic properties. Finite-dimensional distributions. The White Noise.

4. Conditional probability. Conditional expectations. The augmented Brownian filtration.

5. Martingales. Definitions and basic properties of continuous time Martingales.

6. The stochastic integral. Elementary processes. The stochastic integral. Ito Isometry. The stochastic integral as a process. Stopping theorems. Local martingales.

7. Stochastic calculus. Stochastic differential of an Ito process. Ito's Lemma. Girsanov’s Theorem. The martingales of the Brownian filtration.

8. Stochastic Differential Equations. A class of SDE. Definition of solutions. Existence and Uniqueness theorems for the solution. SDE and Markov processes. Connections between SDE and PDE. Feynman-Kac formula.

The only prerequisite is a first course in Probability or equivalent and general notions in Measure and Integration theory are required.

The exam consists of a written test (compulsory) and an oral examination.

The written test consists of two parts denoted, respectively, A and B. Part A contains theoretical questions in order to ascertain the understanding of the basic notions (definitions, theorems, proofs, counterexamples) of stochastic processes, stochastic calculus and stochastic differential equations.

Part B contains 2 exercises, each one containing 7 – 10 questions, on

- stochastic processes, and in particular Brownian Motion, and their properties;

- conditioning, martingales and their applications in the investigation of stochastic processes;

- stochastic calculus: stochastic integral and Ito’s formula;

- stochastic differential equations: existence and uniqueness of the solution and its properties.

During the written test it is not allowed neither to consult documents or archives nor to communicate with other people.  In particular it is not allowed to keep on the mobile phone or some other device allowing to communicate, or to use internet, to consult documents.

The candidate will be asked to solve problems with computations and to justify rigorously all passages. He/she will be required to possess the elements of the theory and to be able to manipulate the theoretical notions.

The evaluation of the written test will take into account the clarity in the exposition of the used procedures, the skill to motivate them, the accuracy in the computations, the familiarity with mathematical language.

The maximum grade of A+B is 30/30. In order to pass the exam, the student must get at least 18/30. The oral exam is decided case by case. For instance, if a student gets 16/30 or 17/30 in A+B then she/he is allowed to take the oral exam. In any case, a student who gets a sufficient grade in A+B (i.e. at least 18/30) can ask to take an oral exam. To get the laude the oral exam is mandatory.

No mid-term test is planned.

The written and the oral test have to be passed in the same exam session.

The oral exam consists of questioning on topics of the course.

The candidate will be asked to introduce the fundamental notions of the stochastic calculus, to summarize the main properties of mathematical tools of stochastic calculus, to state and to prove the main theoretical results. Moreover, the candidate is supposed to be able to present rigorously with an adequate and specialized dictionary and mathematical formalism the results and the applications and to possess skill at making connections between the teaching topics.

The exam has the goal of checking whether the student has acquired the following skills:

• knowledge of some basic concepts of stochastic analysis which are helpful in several applications
• knowledge of the proofs of some fundamental theorems in stochastic integration theory, stochastic calsulus and stochastic differential equations
• understanding of the range of applicability of various results through counterexamples
• ability to state rigorous definitions of the presented notions
• ability to prove some of the presented theorems
• ability to apply the presented theory to solve given problems 
• finding simple counterexamples
• communicating mathematical concepts in a clear and rigorous way

More information can be found in the Sillabo of the teaching published on BeeP, https://beep.metid.polimi.it/.

Draft of a forthcoming book, courtesy of the Author.