# STOCHASTIC ANALYSIS

## Learning outcomes of the course unit

[knowledge and understanding]

Know, understand and be able to explain all the essential arguments in the section ""Programma esteso"" below (except for the proofs of those marked (**) and (***), but see ""Modalità di verifica dell'apprendimento"" below), which form a strong theoretical understanding of stochastic processes.

[applying knowledge and understanding]

Be able to solve exercises and problems on the course arguments, in particular all the ""homeworks"" assigned during the lessons.

[making judgements]

Be able to check whether a process is well-defined and when it enjoys the properties introduced in the lectures (i.e. be adapted, a semimartingale, M2-loc class, etc.).

[learning skills]

Be able to read and understand scientific texts which build on the knowledge of continuous-time stochastic processes, stochastic integration and stochastic differential equations in dimension 1.

## Prerequisites

Measure spaces, probability spaces, Borel-Cantelli lemmas, random variables, mathematical expectation, modes of convergence for random variables, L^p spaces

## Course contents summary

In the first part of the course we introduce continuous-time stochastic processes and we deal with the new issues arising from this object. In particular, we develop the tools needed for the study of stochastic processes and we show the existence of the Brownian motion.

Second part is devoted to the construction of the stochastic integral and to the study of its properties, in particular through martingales.

In the third part we give a short introduction to stochastic differential equations.

## Course contents

The list below is based on what was done in the recent years in this course and is subject to small changes and adaptations for the needs of the present year. An unpdated list will be given to the students at the end of the course.

1. The measurability of a RV can be checked against a set of generators

2. Bernoulli process

3. Simple symmetric random walk (SSRW)

4. A stochastic process is measurable iff its components are measurable (**)

5. Law of a RV

6. Marginal laws do not determine finite-dimensional laws (hw: build an example)

7. Finite dimensional laws determine the law as a process (hw: proof)

8. Indistinguishable processes, modifications and versions (hw: example of processes which are versions but not modifications)

9. Independence of sigma-fields, events, RVs (hw: X_i independent RVs and f_i measurable functions, then f_i(X_i) independent)

10. Independence can be checked on a pi-system of generators

11. Brownian motion (BM)

12. Kolmogorov Extension Theorem (**)

13. Kolmogorov Regularity Theorem (***)

14. Existence of BM (*)

15. Modes of convergence for sequences of RVs

16. Variance, covariance, covariance matrix

17. Gaussian law, Gaussian vectors, Gaussian processes (hw: X Gaussian vector, then MX Gaussian vector)

18. Covariance function, compatibiity of finite-dimensional laws (**) (hw: which condition on the covariance function yields the continuity of trajectories?)

19. Brownian bridge B_t-tB_1 (hw: it is a Gaussian process, find covariance function)

20. Wiener space

21. Total variation, BV functions

22. Quadratic variation of a process and of BM (*)

23. BM trajectories are not BV and not Hoelder for alfa>1/2 (**)

24. Filtration, adapted process, progressively measurable process

25. Progr.mis implies adapted, viceversa with right-continuity (**)

26. BM with respect to a given filtration; thm: the natural filtration satisfies (**)

27. Conditional expectation; L1 can be omitted in the definition

28. Properties a-k) (see the book [Williams]) except for proof of k) (*)

29. Existence and uniqueness of conditional expectation (**)

30. Simple processes, their stochastic integral, Ito isometry (hw: good definition, linearity)

31. Distribution of I(X) for X simple: expectation, variance, conditional expectation and conditional second moment (*)

32. Distribution of I(X) for X in M²: expectation, variance, conditional expectation and conditional second moment (**)

33. Stochastic integral for M² processes, good definition, linearity, Ito isometry

34. Density of simple processes in M² (***)

35. Martingales (super, sub), BM is a martingale; the stochastic integral process is a martingale

36. X martingale, f convex, then f(X) is a submartingale; quadratic martingale of stochastic integral process

37. Discrete stochastic integral; martingale property is preserved

38. Stopping times (ST) (hw: equivalence of the definitions); stopped processes; representation of a process as a discrete stochastic integral; hitting time

39. Optional Stopping Theorem in discrete time (*) (hw: last case with Fatou lemma)

40. SSRW with two barriers: probability of hitting (*) and expected time (**)

41. Sum, min and max of two ST is a ST

42. Sigma-field of a ST; good definition in the case of a ST deterinistic; measurability of X_tau

43. Optional Sampling Theorem in discrete time (**) (hw: example of submartingale and ST tau s.t. X_tau is not integrable)

44. Corollary of Optional Sampling thm (**)

45. Maximal inequality in discrete time (*)

46. Hitting time in contiuous time, for adapted continuous processes and a closed set; second case without proof

47. Reward: 1 point more at exam registration to whoever finds the error in Caravenna's lecture notes

48. Sigma-field of a ST in continuous time (hw: measurability of X_tau)

49. Uniform Integrability property (UI); the conditional expectations of a fixed RV X with respect to any family of sigma-fields is UI (**)

50. Optional Sampling Theorem in continuous time (***)

51. Maximal inequality in continuous time (*)

52. Stopped martingale is a martingale (in continuous time) (**)

53. Continuityof trajectories of stochastic integral process (*)

54. Localization Theorem for Ito integrals (**)

55. Stochastic integral for M²_loc processes; good definition, it generalizes the stochastic integral on M² (**)

56. Properties of stochastic integral for M²_loc processes; integral up to a ST (without proof)

57. Continuity of the stochatic integration operator with respect to the right topologies (**)

58. Ito Processes; their quadratic variation; stochatic integral with respect to an Ito process

59. Ito Formula (without proof)

60. Stochastic Differential Equations (SDE): strong and weak solutions, uniqueness pathwise and in law

61. Well-posedness thm for SDEs with Lipschitz coefficients (without proof)

62. Some examples of exactly solvable SDEs: geometric BM, Orstein-Uhlenbeck process

Remark. The proofs presented in the lessons must be studied at different levels, according to their complexity:

- the simplest and most immediate ones, that can be thought of as simple verifications, must be always known

- the other ones are marked with (*), (**) or (***) according to their complexity: the ones marked (*) must be always known; the ones marked (**) and (***) are in general not part of the exam, but for a small set of them fixed by the teacher some days before the examination (at most 3 af which at most one (***))

- definitions and statements of theorems must be always knowni; if any proof is not marked with (*), (**) or (***), it must be known

- homework given during the lessons count as (**)

## Recommended readings

Francesco Morandin - Lecture notes 2016 and 2018

Francesco Morandin - Lecture notes 2020 (developed during the course and available online after each lesson)

Francesco Caravenna - Moto browniano e analisi stocastica

Daniel Revuz, Marc Yor - Continuous Martingales and Brownian Motion

Ioannis Karatzas, Steven E. Shreve - Brownian Motion and Stochastic Calculus

David Williams - Probability with Martingales

Paolo Baldi - Equazioni differenziali stocastiche e applicazioni

Bernt Øksendal - Stochastic Differential Equations: An Introduction with Applications

## Teaching methods

Traditional classes (48 hours). Arguments are presented in a formal way, with proofs for most statements. Much stress is given to the motivations and we include some examples of applications. There are no exercise sessions scheduled, but homework is regularly assigned during lessons and students are encouraged to do it at home and possibly ask for solutions during the teacher office hours.

## Assessment methods and criteria

The examination is in the form of an interview, based on a program given by the teacher at least 3 days before. The program is formed by:

A) one exercise or problem

B) two arguments marked (**) or (***) from the updated ""programma esteso""

C) one homework

The arguments in the updated ""programma esteso"" which are neither market (**) nor (***) can be asked for without notice, as well as all the definitions and the statements of the theorems.

Parts B) and C) can be substituted as a whole by choosing one of the advanced arguments from a list that will be made available by the teacher.

To pass the exam the student should master the mathematical language and formalism. She must know the mathematical objects and the theoretical results of the course and she should be able to use them with ease. She should also be able to prove theorems by herself.

## Other informations

There will be an e-learning website, where the student can find video and blackboard trascriptions for each lesson, since the teaching is done through tablet PC