Stochastic Processes

V3F1/F4F1 Stochastic Processes – SS 2021

Tuesday 14.15-16.00 and Friday 10.15-12.00. Online via Zoom.

M. Gubinelli and L. La Rocca.

eCampus page

Tutorials

  (see eCampus)

Examination

• 26/7-1/8 Oral exam, online.

• Mid september. Oral exam, probably online.

• Example questions for the oral exam (PDF)

Description

The course is an introduction to various classes of stochastic processes, namely families of random variables indexed by a discrete or continuous parameter (time or space). Topic that will be covered include

• General definition of conditional expectation

• Martingales in discrete time and their convergence

• Markov chains, their long time asymptotic and convergence to equilibrium    

• Brownian motion: construction and sample path properties

The course is a follow-up of “Einfhrung in die Wahrscheinlichkeitstheorie” and a prerequisite to “Introduction to Stochastic Analysis" 

Prerequisites 

a good knowledge of measure theoretic probability as covered for example in:

1. M. Gubinelli: Einfhrung in die Wahrscheinlichkeitstheorie, WiSe 2020/21 (link)

2. R. Durrett: Probability: Theory & Examples, Chapters 1 and 2

3. D. Williams: Probability with martingales, Part A and C

Lecture Notes

We will follow mainly the lectures notes of my previous course in SS2019: 

• Note 1: Review of Measure spaces, Integration theory (v1.1, 10/4/19)

• Note 2: Conditional expectations (v1.0, 15/4/19)

• Note 3: Martingales (v1.1, 2/5/19)

• Note 4: Asymptotic behavior of martingales (v1.1, 3/5/19)

• Note 5: Closed martingales (v1.0, 13/5/19)

• Note 6: Martingale CLT and some applications of martingales (Kolmogorov's LLN, Kakutani's theorem, Radon-Nikodym theorem) (v1.1, 4/6/219)

• Note 7: Optimal stopping problems (v1.0, 3/6/19)

• Note 8: Markov chains (v1.1, 18/6/19)

• Note 9: Discrete chains, Doob's h-transform (v1.1, 21/7/16, updated version)

Further References

• D. Williams: Probability with martingales, Part B.

• R. Durrett: Probability: Theory & Examples.

Problem sheets

   (See on the eCampus page for the course)

Course Journal

• Lecture 13/4 : Overview of the course. (script)

• Lecture 16/4 : Review of measures spaces, Random variables, Dynkin's, Monotone class and Carathodory extension theorems. (script)

• Lecture 20/4: Integration. Lp spaces, completeness. Product measures and integrals, Fubini-Tonelli.  Uniform integrability (statements). (script)

• Lecture 23/4: Uniform integrability (proofs). Conditional expectations: motivation.  (script)

• Lecture 27/4: Definition and uniqueness of conditional expectations, examples.  (script)

• Lecture 30/4:  Existence of conditional expectation via orthogonal projection in L^2. (script)

• Lecture 4/5: Review of basic properties of conditional expectations, relation with independence, some examples. Proof of some properties of conditional expecations under independence assumptions.  (script)

• Lecture 7/5: Regular conditional probabilities. Filtrations, adapted and previsible processes, stopping times. (script)

• Lecture 11/5:  Sigma-algebra of a stopping time, Wald's identity for sums of independent random variables. (script)

• Lecture 14/5: Martingales, some properties, Doob's decomposition. Quadratic variation. (script ,v.2)

• Lecture 18/5: Martingale transform, Doob's optional sampling theorem, martingale property at stopping times. (script)

• Lecture 21/5: Asymptotics of martingales, martingale convergence theorem. (script)

• Lecture 1/6: Closed martingales. Square-integrable martingales. Doob's maximal and L^p inequalities. (script)

• Lecture 4/6: Martingales closed in L^p and UI martingales. Optional stopping for closed martingales. (script)

• Lecture 8/6: The tail sigma algebra and Kolmogorov's 0/1 law. Backwards martingales. Kolmogorov's law of large numbers. (script)

• Lecture 11/6: Kakutani's theorem. (script)

• Lecture 15/6: Radon-Nikodym theorem. (script)

• Lecture 18/6: Optimal stopping problems. Value process, Snell's envelope and existence and characterisation of optimal stopping times. (script) 

• Lecture 22/6: Markovian optimal stopping problems. Solution of Moser's problem. Random recurrences and the Markov property.  (script) 

• Lecture 25/6: Markov processes. Transition kernel, some examples, general markov property. (script)

• Lecture 29/6: Law of a Markov chain. Canonical space. Strong Markov property. (script)

• Lecture 2/7: Martingale problems, relation with potential analysis, maximum principe. (script)

• Lecture 6/7: Discrete chains, transience and recurrence.  (script)

• Lecture 9/7: Computations of some probabilistic quantities via linear equations involving the generator. Doob's h-transform. (script)

• Lecture 13/7: An application of the h-transform to conditioning of Markov chains. Invariant measures, existence and uniqueness.  (script)

• Lecture 16/7: Conclusion of the discussion of invariant measures. Question session. (script)