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.
Lesson period: Second semester
(In case of multiple editions, please check the period, as it may vary)
1. Theory of Stochastic Processes 1.1. Stochastic Processes 1.1.1. Introduction to stochastic processes and principal examples of Stochastic processes (Gaussian, Levy, Markov) 1.1.2. Study of Markov Processes - Characterization of a Markov process. The Markov transition function. - Transition semigroups and infinitesimal operator associated to a Markov Process - The homogeneous case 1.1.3. Markov processes and martingales: the Dynkin formula. 1.1.4. Brownian Motion and the Wiener Process - Existence of the Wiener process. - Characterization as a 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. - The quadratic variation - Behavior of the trajetories of a Wiener Process: the reflection principle, the law of iterated logarithm, a.s. no differentiability, a.s. continuity - The multidimensional case
1.2. Ito's Integral 1.2.1. Definition and Properties : integral as L2 limit and limit in probability 1.2.2.Ito's process: properties and martingales 1.2.3. The Stochastic differential 1.2.4. Ito 's Formula 1.2.5. Brownian exponentials 1.2.6. Martingale Representation Theorem
1.3. Stochastic Differential Equations 1.3.1. Existence and uniqueness theorems 1.3.2. The Markov Property of Solutions 1.3.3. Kolmogorov Equations 1.3.4. Feyman-Kac Theorem 1.3.5. Girsanov Theorem 1.3.6. SDE and PDE: Cauchy problem 1.3.7. Multidimensional SDE
1.4. Stability (not all the years) 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. The Stochastic Integral - Ito vs Stratonovich - Properties of the Ito's integral 2.5. Differential Equations - Simulation of ordinary differential equations (ODE): Euler and Heun methods - Simulare equazioni differenziali stocastiche (SDE): Eulero-Maruyama and Milstein Methods - Strong and Weak Convergence for the stochastic case. Consistency. Stability. 2.6. Examples and Applications. - Population dynamics - Interacting particle systems - SDE in Finance
Prerequisites for admission
Knowledge of the theory of probability with a particular reference to the construction of probability spaces, the real random vectors , the conditional expected value, and the various types of convergence.
Lectures are taken with the aid of a blackboard.
Lab lectures are taken in a computer room.
- V.Capasso, D. Bakstein An Introduction to Continuous-Time Stochastic Processes. Theory, models, and applications to Finance, Biology and Medicine. Birkhauser, Boston, 2015. - P. Baldi, Stochastic Calculus: An Introduction Through Theory and Exercises, Springer, 2017 - Lecture notes
Assessment methods and Criteria
The exam consists in a oral examination and an evaluation of the lab.
The lab part consists in the evaluation of the last two assigned homework. The aim is to evaluate the capability of the student of modelling, simulating and perform a qualitative analysis of some random dynamics.
During the oral examination the student should describe and discuss some of the results of the program with the aim to evaluate the knowledge and the comprehension of the argument, as well as the applicative ability.
Both evaluations, weighted with the CFU, give a contribution to the final mark. Grades are expressed in thirties.