The main scope of the course is to give an introduction to the methods of stochastic calculus, with particular attention to the Ito's calculus. From definitions and fundamental results of the theory of the stochastic processes, with particular attention to the class of Markov processes, and the of the Wiener process, the students are guided to the formulation of the systems of stochastic differential equations of the Ito's type. A construction of the Ito's integral both as L2 limit and limit in probability is given. The martingality properties of the Itos's process, following the one of the Wiener process are analyzed. Particular interest is devoted to the analysis of the stochastic differential equations and the links to the PDEs. 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. Particular attention to some application in Biology, Medicine and Finance is given.
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.
1. Stochastic calculus: Theory 1.1. The Wiener process or Brownian motion - Definition and first properties of Brownian motion (BM) - Path properties of BM - The multidimensional case
1.2. Ito's Integral - Definitions and properties - Stochastic integrals as local martingales - Ito's formula and applications - Lévy's criterion for BM - Multidimensional stochastic integral
1.3. Stochastic Differential Equations (SDEs) - Strong and weak solutions of SDEs - Existence and uniqueness of strong solutions - The Markov property of solutions - Flow property of solutions - Solving linear SDEs
1.4. SDE and PDE - Dirichlet and Cauchy problems - Feynman-Kac Theorem - Backward and forward Kolmogorov equations
1.4. Change of probability and Brownian martingales - Exponential martingales - Girsanov Theorem - Cameron-Martin formula - Representation theorem of Brownian martingales
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 integrals - Ito vs Stratonovich. The lambda integral. 2.5. Simulation of Stochastic Differential Equation (SDE) 2.5.1 Eulero-Maruyama and Milstein methods 2.5.2 Strong and weak Convergence; consistency; stability. 2.6. Examples and applications.
Prerequisites for admission
Knowledge of the basis of probability theory (in particular, construction of probability spaces, random vectors, conditional expectation, various types of convergence) and of stochastic processes (martingales and Markov processes). Taking the courses Probability and Advanced Probability is strongly recommended.
Lectures in classrooms via blackboard. Lab lectures in a computer room.
- P. Baldi, Stochastic Calculus: An Introduction Through Theory and Exercises, 2017
Other references: - Karatzas, Shevre, Brownian Motion and Stochastic Calculus, Springer, 1988 - F. Caravenna, Moto Browniano e Analisi Stocastica, 2011. Lectures notes available online.
Assessment methods and Criteria
The exam consists in: - an assessment of the lab activity; - an oral examination on the theory;
For the students who have attended the course, the lab assessment is based on an evaluation of the students' participation to the lab activities and a quantitative assessment of the last two homework (mark out of 30).
For the other students: they will be evaluated on the basis of a little report assigned by the teacher and they will be asked one or two questions on the methods explained during the lab. The report is handed out one week before the oral exam, while the questions will be given on the same day as the oral exam and the students will have to answer them in a written form before that exam (mark out of 30).
The oral examination will focus on all the material explained during the course (theory part). Marks are expressed out of 30.