Stochastic Calculus and Applications

A.Y. 2017/2018
Overall hours
Learning objectives
The main scope of the course is to give an introduction tothe methods ofnstochastic calculus, with particular attention to some application in Biology, Medicine and Finance.

The students are guided from definitions and fundamental results of the theory of the stochastic processes to the formulation of the systems of stochastic differential equations of the Ito's type. To the analysis of such equations and to the modelling via stochastic differential equations in some application fields is devoted the second part of the course. As a complement of the theory, a simulation laboratory is based on the concept of the learning by doing: via simulation some of the most important properties of the stochastic processes and in particular of the Wiener process are guessed; the possible counterpart of some deterministic models, and the numerical solution of some PDE via the simulation of SDE system are discussed.
Expected learning outcomes
Student learn how to treat and discuss the mail properties of the Markov stochastic processes and of the Winer process, in particular. He is able to understan the main probabilistic consequences of the constructiin of the Ito stochastic integral, above all the ones related to the martigales. He gain a knowledge of the stochastic differential equation and their relation to PDEs.
Besides the theoretical knowledge, he learn how it is possible to introduce the randomness modelling some situation already known from the deterministic point of view. He knows how simulate a system of SDEs and quantify the properties of the solutions via some statistical procedure.
Course syllabus and organization

Single session

Lesson period
Second semester
Course syllabus
0. Introduction
0.1. Conditional expectation with respect to a sigma algebra: definition and basic properties
0.2. Martingale: definition and basic properties

1. Theory of Stochastic Processes
1.1. Stochastic Processes
1.1.1. Infinite joint distributions on the space of trajectories. Compatible measure systems, and probability space projective systems: the Theorem of Kolmogorov-Bochner.
1.1.2. Example of compatible sistems
- Gaussian Processes
- Processes with independent components and processes with independent increments
- Markov Processes
1.1.3. Study of Markov Processes
- Characterization of a Markov process. The Markov transiction function.The Chapman-Kolmogorov equation as a compatibility condition.
- Semigroups and infinitesimal operator associated to a Markov Process
1.1.4. Markov processes and martingales: the Dynkin formula.
1.1.3. Brownian Motion and the Wiener Process
- Existence of the Wiener process: Donsker's theorem
- Characterization as gaussian process
- The Wiener process as a martingale. The squared wiener process a submartingale.
- The Levy's theorem as a characterization of the Wiener process.
- Behavior of the trajetories of a Wiener Process
- The multidimensional case

1.2. Ito's Integral
1.2.1. Definition and Properties
1.2.2. Stochastic Integrals as Martingales
1.2.3. The Stochasti Differenziale
1.2.4. Ito 's Formula
1.2.5. Martingale Representation Theorem

1.3. Stochastic Differential Equations
1.3.1. The Markov Property of Solutions
1.3.2. Girsanov Theorem
1.3.3. Kolmogorov Equations
1.3.4. Feyman-Kac Theorem
1.3.5. SDE and PDE: Cauchy problem
1.3.6. Multidimensional SDE

1.4. Stability
1.4.1 Asymptotic behavior and stability
1.4.2. Invariant distribution


2. Generation and Simulation of Stochastic Processes

2.1. Simulation of random variables
2.2. Stochastic Processes and Random walks
- Simulation and study via parameters and distribution estimators
- Rescaling of Random Walks
2.3.Brownian motion
- Properties
- Quadratic Variations
- Study of the differentiability of Brownian motion
2.4. Simulation of Stochastic Integral
- Ito vs Stratonovich
2.5. Simulation of Stochastic Differential Equation (SDE)
- Eulero-Maruyama and Milstein Methods
- Strong and Weak Convergence. Consistency. Stability.
2.6. Examples and Applications.
- Population dynamics
- Interacting particle systems
- SDE in Finance
MAT/06 - PROBABILITY AND STATISTICS - University credits: 9
Practicals: 10 hours
Laboratories: 24 hours
Lessons: 42 hours
Educational website(s)
Monday, 10:30 am - 1:30 pm (upon appointment, possibly suppressed for academic duties)
Department of Mathematics, via Saldini 50, office 1017.
upon appointment via email
Via Saldini 50, second floor