Probabilità e Processi Socastici

Syllabus

Michele Gianfelice 

a.y. 2016/2017

gianfelice@mat.unical.it 
http://www.mat.unical.it/~gianfelice/didattica/didattica.html
                                              
Recap of the fundamental notions of measure and probability theory 
- Probability spaces, distribution function of a random variable and of a random vector (r.v.) .
-- Generalized Markov-Chebichev inequality.
- Conditional expectation and conditional probability w.r.t. a sigma-algebra
  - correlation between two events
  - conditional expectation w.r.t. a r.v.
  - regular version of a conditional probability
  - conditional expectation as projection operator on the Hilbert space of square integrable functions
- Sequences of random variables
  - almost sure (a.s.) convergence
  - convergence in probability
  - convergence in mean of order p, p>0
  - convergence in distribution (a.k.a. in law, or weak convergence of a sequence of probability measures)
  - relations among the various notions of convergence of r.v.'s
    - uniformly integrable sequences of r.v.'s
    - Borel-Cantelli Lemma
- Characteristic function of a r.v.
  - definition and properties of the characteristic function nof a r.v.
  - uniqueness theorem
  - inversion formula
  - relation between convergence in distribution of a sequence of r.v.'s and pointwise convergence of the associated sequence of characteristic functions
     - continuity theorem
       - weak law of large numbers for i.i.d. r.v.'s
       - central limit theorem for i.i.d. r.v.'s 

Gaussian laws
- Gaussian r.v.: definition and properties
- Gaussian n-dimensional r. vector: definition and properties

Discrete time martingales
- stopping times
- stopped martingale
- martingale transform
- Doob-Meyer decomposition
- L^2-martingales
- L^1-martingales and regular martingales

The Wiener process
- definition and properties of the Wiener measure
- self-similar processes
- continuity and non-differentiability of the trajectories of the Brownian motion
- Brownian bridge in [0,1]
- Brownian motion with conditions on the starting and the ending point
- Brownian motion with drift
- geometric Brownian motion 

Introduction to stochastic analysis
- Wiener stochastic integral
- Ito stochastic integral
- (Fisk-)Stratonovich stochastic integral
- Ito linear stochastic differential equations
  - additive noise case
  - multiplicative noise case
  - the general case
  - existence and uniqueness theorem
- Black-Scholes option pricing formula for european options 

Bibliography:

•	A. N. Shiryaev Probability, II edition	Springer Verlag 1996
            chapter II, chapter III paragraphs 1,2,3, chapter IV paragraph 1, Chapter VII  
            paragraphs 1,2,3,4.
•	P. Malliavin Integration and Probability  Springer Verlag 1995
            chapter I, chapter II paragraphs 4,5,6, chapter IV paragraphs 1,2,3,4,5.
•	S. Karlin, H. P. Taylor A first course in stochastic processes, Vol.I, II edition Academic press 1975
            chapter 6, chapter 7.
•	Da Prato Giuseppe Introduction to Stochastic Analysis and Malliavin Calculus Scuola Normale Superiore di Pisa - Collana Appunti 2007
•	Mikosch Thomas Elementary Stochastic Calculus with Finance in view World Scientific 1998
            everything but paragraph 3.4 .