# Continuous and Discrete Stochastic Analysis

## Registro delle lezioni

Docente Alessandra Caraceni, Franco Flandoli

## Lecture

• 07 Nov 2023 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Prima lezione su fondamenti di Calcolo delle Probabilità. Probability density functions, examples of exponential and Gaussian pdf, proof of the property of area one. Heuristic language of random variables. Mean (expected value) and variance computed by means of a density.

• 17 Nov 2023 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Definitions of algebra, σ-algebra, generated σ-algebra; probability measure (or distribution), Probability space (Ω,F,P); first properties. ∙ Examples of Ω, of F, of P: in particular finite and discrete cases, role of probability of singletons; examples, Bernoulli scheme. ∙ Borel σ-algebra on metric spaces, Borel sets in R, measurable functions on (Ω,F) with values in (R,B(R)).

• 20 Nov 2023 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Rigorous definition of random variable and of probability law of a r.v. Discussion of the link between the heuristic vision of a r.v. as an unpredictable quantity with some kind (objective or subjective) of statistical regularity, and the rigorous denition of r.v. and its law. Example of the Bernoulli scheme: r.v.s X1;...;XN; SN and their laws. In particular, the law of SN is related to the theorem which links Bernoulli and Binomial r.v.s (sum of independent B (1; p) is B (N; p)). Definition of conditional probability and independence.

• 24 Nov 2023 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Discrete r.v.'s, examples (Bernoulli, Binomial, Poisson, Geometric). Definition of mean value in the discrete case and computation of the mean value for each one of the examples. Link between Bernoulli r.v.'s and geoemetric time. Lack of memory for exponential r.v. Simple r.v.'s and their integral; monotone and uniform (if bounded) approximation of a r.v. by simple r.v.'s and definition of integral.

• 27 Nov 2023 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Expected value in general and its particular forms for discrete and continuous r.v.'s. Basic rules (linearity, positivity, Hölder inequality, properties for independent r.v.'s). ∙ Definition of covariance and correlation coefficient (invariance by scaling and bound in [-1,1]), problem of being equal to zero, approximatively in examples. ∙ Definition of convergence a.s., in probability and L^{p}, stratement of links. Dominated convergence theorem of Lebesgue.

• 01 Dic 2023 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Weak Law of Large Numbers (with proof), both with convergence in Probability and L², with preliminaries on Markov and Chebyshev inequalities. Definition of uncorrelated r.v.'s and additional properties of the Variance (scale, additivity on uncorrelated variables). Variants of weak LLN. ∙ Random vectors, joint density and marginals, their links (also in the case of independent r.v.'s). ∙ Standard normal (Gaussian) random vector, graphical interpretations.

• 04 Dic 2023 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

General Gaussian random vectors, covariance matrix, different definitions (as affine transformation of the standard one; by density, by projections). Principal component analysis; equivalence of uncorrelated and independent components of a Gaussian vector. ∙ Ergodic theorem in weak form.

• 11 Dic 2023 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Definition of weak convergence of measures (and the corresponding convergence in law of r.v.'s). Illustration by the example of N(0,(1/(n²))) converging to the delta Dirac δ₀. Characterization by cpdf (Cumulative disribution function) and link to the example. ∙ Characteristic function, computation for the standard Gaussian and for generic Gaussian, link to the example.

• 15 Dic 2023 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Statement of the Central Limit Theorem. Centered r.v.'s. Development of a sum as mean (first order) plus Gaussian fluctuations (second order). Formulation of the CLT by means of probability of intervals. ∙ Concentration of Binomial distribution and link to CLT. ∙ Tails of sums: computation of Large Deviation theory type to see the dependence on the distribution, opposite to the Gaussian universality. ∙ Proof of CLT by characteristic functions, developing some properties of characteristic functions for the purpose.

• 18 Dic 2023 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Speranza e probabilità condizionali, rispetto ad una sigma algebra: introduzione intuitiva al concetto di informazione contenuta in una sigma-algebra (es. sigma algebre associate ad un processo stocastico), caso di sigma algebra finitamente generata, corrispondente definizione di speranza condizionale e probabilità condizionale. Cenno alla definizione generale basata su spazi di Hilbert e proiezione sul sottospazio delle v.a. misurabili rispetto alla sigma algebra di condizionamento. Cenni alla passeggiata casuale come primo esempio di processo stocastico, legame con proprietà note di binomiali e TLC, esercizio sul massimo raggiunto entro un tempo n.

• 15 Gen 2024 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Abstract definition of WN (white noise) and PPP (Poisson point process). Motivation from random points, without and with intensities, with different densities, in the plane. Rare event convergence theorem. Other links between sum of Bernoulli and limit distributions. Poisson process and Brownian motion from PPP and WN.

• 22 Gen 2024 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Introduction to stochastic processes depending on a real parameter: r.v. and trajectories, continuity, filtration and adaptedness, modification and indistinguishable, statement of theorem on modification of continuous processes. Statement of Kolmogorov theorem on continuity of a modification, and Holder property. Definition of Brownian motion, comments on its properties and trajectories and outlook on their irregularity. Existence of a rough BM from white noise: construction of white noise, its action on test functions, its integration. Existence of a continuous version by Kolmogorov theorem.

• 26 Gen 2024 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

General concept of stationarity, stationarity of a process with time parameter, weak and strong stationarity, example of exponential decay of correlation and Ornstein Uhlenbeck process. Also white noise is stationary. Different views on white noise and link with Brownian motion (its derivative). Numerical simulation of Ornstein Uhlenbeck process; explanation of the sqart(dt) in the noise term of the simulation. Stationary increment processes and their link with stationary processes; pictures and BM example. A few words about Fractional noise.

• 07 Feb 2024 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Wiener integral: costruction by simple functions and density. Esercises. Two sided Brownian motion. Stationary Ornstein Uhlenbeck process. Esercises. Rescaling and approximation of white noise, or physical model with inertia which approximates Brownian motion. Esercises. Ito integral, first for simple processes, then for general adapted, by the isometry formula (to be proved next time).

• 14 Feb 2024 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

More details on the definition of Ito integrals. First properties, including the statement of a Doob's inequality for the p-average. Exercise on the regularity w.r.t. to time and parameters. Discussion of extensions (e.g. Young integral). Stochastic differential equations: definition of strong solution, pathwise uniqueness. Cauchy-Lipschitz theorem.

• 16 Feb 2024 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Qualitative study of a deterministic and stochastic equation (double well). Fokker-Planck equation, first elements. Exercises.

• 21 Feb 2024 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Stratonovich integral, quadratic variation, Ito processes, first rules. Ito formula, with quadratic variation. Idea of proof. Examples. Additional rules on quadratic variation. Exercises. Proof of Fokker-Planck equation.

• 28 Feb 2024 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Derivation, by Ito formula, of the measure-valued formulation of Fokker-Planck equation, from a solution to the SDE. Discussion of the existence of a regular density. Strong formulation in the case of regular density. Stationary solutions, statement of the link with stationary processes. Explicit formula in the case of gradient systems. Example of two-well potential.

• 06 Mar 2024 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Kolmogorov equation, link with expected values (starting from a Kolmogorov solution), ideas about inverting the process. Duality with Fokker-Planck and use for uniqueness of FP.

• 08 Mar 2024 (2h 00m)

Franco Flandoli - Corso (attività didattica) - In presenza

Statement of Veretennikov Theorem, ideas about path by path uniqueness, proof by Ito-Tanaka trick in the case of Holder drift.