INGLESE

Elementi di probabilità. Definizione e proprietà dei processi stocastici.
Il moto browniano.
Probabilità condizionata.
Martingale.
Processi di Markov.
Elementi essenziali sull'integrale stocastico e
Equazioni differenziali stocastiche (SDE)
Simulazione.

Paolo Baldi: “Stochastic Calculus, An Introduction Through Theory and Exercises”, Springer

Lo scopo di questo corso è fornire l'approccio matematico alla teoria dei processi stocastici e alcuni algoritmi di base per la loro simulazione.

No

Costituiranno il corso lezioni frontali teoriche, risoluzione di esercizi e algoritmi di simulazione in R. Per ogni lezione verranno forniti anche appunti scritti dal docente.

Allo studente sarà richiesta la presentazione orale di un progetto comprensivo di domande teoriche e di alcuni esercizi.

Facendo riferimento al testo di Paolo Baldi: “Stochastic Calculus, An Introduction Through Theory and Exercises”, Springer,
i contenuti del corso sono di seguito elencati:
CHAPTER 1- Elements of Probability
1.1- Probability spaces, random variables
1.2- Variance, covariance, law of a r.v.
1.3- Independence, product measure
1.4- Probabilities on Rm
1.5- Convergence of probabilities and random variables
1.6- Characteristic functions
1.7- Gaussian laws
1.8- Simulation
1.9- Measure-theoretic arguments
CHAPTER 2- Stochastic Processes
2.1- General facts
2.2- Kolmogorov’s continuity theorem (no proof)
CHAPTER 3- Brownian Motion
3.1- Definition and general facts
3.2- The law of a continuous process, Wiener measure
3.3- Regularity of the paths (no proof)
3.4- Asymptotics (no proofs)
3.5- Stopping times
3.6- The stopping theorem (no proof)
3.7 The simulation of the Brownian motion
CHAPTER 4 – Conditional Probability
4.1- Conditioning
4.2- Conditional expectations
4.3- Conditional laws (only the definition)
CHAPTER 5 –Martingales
5.1- Definitions and general facts
5.2- Discrete time martingales
5.3- Discrete time martingales: a.s. convergence

CHAPTER 6 - Markov Processes
6.1 Definitions and general facts
6.2 The Feller and strong Markov properties
CHAPTER 7-The Stochastic Integral
7.1- Introduction
7.2- Elementary processes
7.3- The stochastic integral (definition and properties)
7.4- The martingale property
CHAPTER 8-Stochastic Calculus
8.1- Ito’s formula
CHAPTER 9- Stochastic Differential Equations (SDE)
9.1 - Definitions
9.2 – Examples
CHAPTER 11- Simulation
11.1 Numerical approximation of an SDE
11.5 Other Schemes: The Euler simulation scheme.
Nel programma del corso sono compresi anche alcuni esercizi dei capitoli su elencati.

English

Elements of Probability. Definition and properties of Stochastic Processes.
The Brownian Motion.
Conditional Probability.
Martingales.
Markov Processes. Essentials on the stochastic integral and
Stochastic Differential Equations (SDE)
Simulation.

Paolo Baldi: “Stochastic Calculus, An Introduction Through Theory and Exercises”, Springer

The purpose of this course is to provide the mathematical approach to the theory of stochastic processes and some basic algorithms for their simulation.

No

Theoretical lectures, resolutions of exercises and simulation algorithms by R will constitute the course. Notes written by the teacher will be also provided for each lecture.

The oral presentation of a project including theoretical questions and some exercises will be required to the student.

Contents of the course: SPATIAL RANDOM PROCESSES (2023/2024)
Referring to the book of Paolo Baldi: “Stochastic Calculus, An Introduction Through Theory and Exercises”, Springer

CHAPTER 1- Elements of Probability
1.1- Probability spaces, random variables
1.2- Variance, covariance, law of a r.v.
1.3- Independence, product measure
1.4- Probabilities on Rm
1.5- Convergence of probabilities and random variables
1.6- Characteristic functions
1.7- Gaussian laws
1.8- Simulation
1.9- Measure-theoretic arguments
CHAPTER 2- Stochastic Processes
2.1- General facts
2.2- Kolmogorov’s continuity theorem (no proof)
CHAPTER 3- Brownian Motion
3.1- Definition and general facts
3.2- The law of a continuous process, Wiener measure
3.3- Regularity of the paths (no proof)
3.4- Asymptotics (no proofs)
3.5- Stopping times
3.6- The stopping theorem (no proof)
3.7 The simulation of the Brownian motion
CHAPTER 4 – Conditional Probability
4.1- Conditioning
4.2- Conditional expectations
4.3- Conditional laws (only the definition)
CHAPTER 5 –Martingales
5.1- Definitions and general facts
5.2- Discrete time martingales
5.3- Discrete time martingales: a.s. convergence

CHAPTER 6 - Markov Processes
6.1 Definitions and general facts
6.2 The Feller and strong Markov properties
CHAPTER 7-The Stochastic Integral
7.1- Introduction
7.2- Elementary processes
7.3- The stochastic integral (definition and properties)
7.4- The martingale property
CHAPTER 8-Stochastic Calculus
8.1- Ito’s formula
CHAPTER 9- Stochastic Differential Equations (SDE)
9.1 - Definitions
9.2 – Examples
CHAPTER 11- Simulation
11.1 Numerical approximation of an SDE
11.5 Other Schemes: The Euler simulation scheme.
Some exercises of all above chapters are also included in the program of the course.