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# Stochastic processes on domains, spring 2015

### Lecturer

Petteri Piiroinen

10 sp.

### Course description (in English)

"Stochastic process on domains" is an advanced level course in applied mathematics (applied analysis).

The course will be held either in Finnish or English, depending on those who are taking the course. In either case, there will be a material in both languages that contain at least all the concepts and main results.

The aim of the course is to get an introduction to stochastic processes that "live" inside a bounded domains, either in plane or higher dimensional spaces.

The processes we study at the course will be mainly continuous and continuous time random walks whose behaviour can only depend on the state (location) where they happen to be at any given time. In other wors, we will be studying Markov processes whose state space is a bounded domain (in plane or higher dimensional space).

These processes are closely related with boundary value problems and they can for instance be used to obtain more intuition on the problems of medical imaging, for example.

During the course, we will consider continuos time stochastic processes that can be obtained from Brownian motion via differential equations. In addition, we will study processes that need bit stronger tools before they can be reduced to Brownian motion. The processes would could be obtained from Brownian motion with differential equations but in this time, the equations become singular. In other words, we need to think bit harder to handle these cases.

In particular, we will consider processes that will not escape from the given domain by reflecting back once they hit the boundary of the domain.

One of the central concepts for understanding this reflection is the so called local time on the boundary of the process. This will be defined in various different ways, both "intuitively" as well as with Revuz correspondence. This correspondence we will also consider in detail.

The exact plan is still in flux, but the preliminary list of topics is as follows:

• Short introduction to Brownian motion
• Short introduction to martingales and Markov processes
• Short introduction to Itô stochastic integrals
• Stochastic differential equations
• Reflecting Brownian motion
• Infinitesimal generators
• Dirichlet forms
• Additive functionals and Revuz correspondence
• Fukushima decomposition
• Local time at the boundary
• (Traces and parts of the process)

Throughout the course we will try to approach questions and "things" with as simple means as possible. This means that we will carefully avoid getting lost in the jungle of techincal details. But, at the same time, we won't be hiding behind the bushes either.

We wre not following any single book or reference but the course material is build upon various sources (books and articles). More on the "good to read" material will be announced when the course starts.

The course material will be mainly in English, but at least all the concepts and the main results will be provided also in Finnish...

Kurssi "Stokastiset prosessit alueissa" on soveltavan matematiikan (soveltavan analyysin) syventävien opintojen kurssi. Kurssi pidetään joko suomeksi tai englanniksi, riippuen kurssilaisista. Joka tapauksessa, ainakin kaikki käsitteet ja päätulokset on saatavilla myös materiaalina molemmilla kielillä.

Kurssilla on tarkoitus tutustua stokastisiin prosesseihin, jotka "elävät" rajoitetuissa alueissa, joko tasossa tai useampiulotteisissa avaruuksissa.

Kurssilla tarkasteltavat stokastiset prosessit ovat pääosin jatkuvia ja jatkuva-aikaisia satunnaiskulkuja, joiden käytös voi riippua vain tilasta (paikasta) missä ne kulloinkin ovat. Toisin sanoen, tarkastelemme Markovin prosesseja, joiden tilajoukko on alue (joko tasossa tai korkeampiulotteisissa avaruuksissa).

Tällaiset prosessit liittyvät läheisesti reuna-arvotehtäviin ja siten niiden avulla voi esimerkiksi saada lisäintuitiota esimerkiksi lääketieteellisen kuvantamisen kysymyksiin.

Kurssilla tutustutaan jatkuva-aikaisiin stokastisiin prosesseihin, jotka on palautettavissa Brownin liikkeeseen differentiaaliyhtälöiden välityksellä.

Näiden lisäksi tarkastelemme prosesseja, joiden palauttaminen Brownin liikkeeseen vaatii vahvempia työkaluja. Nämä olisi muuten palautettavissa Brownin likkeeseen differentiaaliyhtälöiden välityksellä, mutta välittävistä yhtälöistä tulee singulaarisia ja tämä aiheuttaa hieman ylimääräistä päänvaivaa.

Erityisesti tarkastelemme prosesseja, jotka pysyvät annetussa alueessa heijastumalla takaisin törmätessään alueen reunaan.

Eräs keskeisistä käsitteistä heijastuksen ymmärtämisessä on prosessin lokaali aika alueen reunalla, jonka määräämme usein eri tavoin, sekä "intuitiivisesti" että ns. Revuzin vastaavuuden välityksellä.

•     Lyhyt johdanto Brownin liikestä
•     Lyhyt johdanto martingaaleihin ja Markovin prosesseihin
•     Lyhyt johdanto Itôn stokastiseen integrointiteoriaan
•     Stokastiset differentiaaliyhtälöt
•     Heijasteleva Brownin liike
•     Infinitesimaaliset generaattorit
•     Dirichlet'n muodot
•     Additiiviset funktionaalit ja Revuzin vastaavuus
•     Fukushiman hajotelma
•     Lokaali aika alueen reunalla
•     (Prosessien jäljet ja osat)

Pyrimme kurssilla lähestymään asioita mahdollisimman yksinkertaisesti teknisiä kohtia vältellen, muttei pakoillen.

Kurssilla emme suoranaisesti seuraa yksittäistä kirjaa vaan luentomateriaali rakentuu useista eri lähteistä (kirjoista ja artikkeleista). Näistä kerrotaan tarkemmin kurssin alettua.

Kurssimateriaali on pääsääntöisesti englanniksi, mutta ainakin kaikki käsitteet ja päätulokset tulevat myös suomeksi.

### Prerequisites

Todennäköisyysteoria and/or Funktionaalianalyysin peruskurssi. The emphasis will be on the stochastic analysis and all the required probabilistic notions will be introduced during the course. This means that even without prior studies in stochastics, but with solid background on the analysis,  the course can be followed.

On the other hand, the basics of Hilbert spaces is needed in any case.

### Lectures

The first lecture is on Wednesdey 14th at 10-12 in lecture room C124. The lectures are held on weeks 3-9 and 11-18, on Wednesdays at 10-12 and Fridays at 10-12 in room C124. There is a break on week 10 (so no lectures on March 4th and March 6th). Furthermore, the Easter holiday 2.-8.4. means that there are no lectures on April 3rd nor on April 8th. The last lecture is on Wednesday April 29th.

1. Wed 14.1.2015: general introduction and preliminary definition of Brownian motion
2. Fri 16.1.2015: basic notions of probability theory and measure theory
3. Wed 21.1.2015: Brownian motion (1D and d-dimensional) and its properties
4. Fri 23.1.2015: definition of Markov process, transition probability density
5. Wed 28.1.2015: transition functions, construction of Markov processes from transition functions, definition of martingale
6. Fri 30.1.2015: properties of supermartingales, Upcrossing inequality, Doob's maximal inequalities, path properties of supermartingales
7. Wed 4.2.2015: Martingale convergence theorem, Feller processes and stopping times.
8. Fri 6.2.2015: Feller processes, stopping times, sigma-algebras at stopping times and strong Markov property
9. Wed 11.2.2015: Feller processes are strong Markov processes and stopping times with supermartingales
10. Fri 13.2.2015: Uniform integrability, Optional Stopping Theorem, introduced the Debut Theorem
11. Wed 18.2.2015: Quasi-left continuity of Feller process, càdlàg property and Debut Theorem
12. Fri 20.2.2015: Example: first exit time of Brownian motion from a ball and its expectation, introduction to stochastic integrals
13. Wed 25.2.2015: Local martingales, continuous semimartingales and quadratic variations
14. Fri 27.2.2015: Ito formula and definition of the stochastic integral with respect to continuous semimartingales
15. Wed 11.3.2015: On proof of Ito formula and properties of stochastic integral.
16. Fri 13.3.2015: Transience and recurrency of Brownian motion in Rd
17. Wed 18.3.2015: Kakutani's representation theorem part 1
18. Fri 20.3.2015: Kakutani's representation theorem part 2 and regularity
19. Wed 25.3.2015: Flat cone condition and introduction to stochastic differential equations
20. Fri 27.3.2015: Connection between parabolic equations and SDEs
21. Wed 1.4.2015: Stochastic differential equations, Ito's existence and uniqueness result
22. Fri 10.4.2015: Ito's existence and uniqueness result
23. Wed 15.4.2015: RBM in 1D, generalization of Ito formula for convex functions and Tanaka's formula
24. Fri 17.4.2015: Local time and Skorohod's equation
25. Wed 22.4.2015: RBM in higher dimensions, Skorohod's equation in higher dimensions, the Submartingale problem
26. Fri 24.4.2015: More on the Submartingale Problem, Solution of Skorohod's equation solves the the Submartingale Problem, RBM is a strong Markov process
27. Wed 29.4.2015: The parabolic equation construction form RBM via Submartingale problem

### Lecture notes

The lecture notes and other material will be posted here in pdf format. The first lecture and the Friday lectures will be posted here shortly after the lectures.

### Exams and excercises

You can return the excersices during the weekend as email as well.

The course is passed by doing excercises. Every second week you get an excercise sheet and you have two weeks to return it to the lecturer. For other methods for obtaining the study points, please discuss with Petteri Piiroinen. You can return the excercise sheet by hand or you can email a scanned copy to Petteri.

The excercises contained misprints (and might have misprints every now an then...) The errors are pinpointed here so as soon as they are spotted.

Additionally the course can be done by taking an exam in the end.
You get bonus points from the excercise to the exam and the grading will be such that any combination of the grade is taken into account (just excercises and excercises+exam) and the grade will be the maximum of these.

Those of you who want to take the end exam please let send me an email. The exam will take place in the first week of May.

In addition, we can discuss at Ratkomo (or a similar location) once we decide on a suitable time. There I can help you with excercises and issues with lectures.

If there is less than week time and/or you spot a mistake then you have think that problem done. If some problem has multiple mistakes which are not announced within 5 days of providing the excercise sheet I will consider that to have been succesfully done (so Ex 1. Problem 1 and Ex 1. Problem 2 is counted as done for everyone). If you spotted the mistake or just ignore it and did the Problem anyway, you will, hence, get that Problem counted twice.

• Ex 1. Problem 1. The numbers j and n are non-negative integers
• Ex 1. Problem 2. the subindex n was missing from the left-hand side
• Ex 1. Problem 10. since the Lemma 3.13 is not yet covered it is replaced with two simpler ones (the original will be postponed to Excersice sheet 2)
• Ex 1. Problem 1. The sets in algebra Gn can have m (not just n) intervals (so this can also be thought as done)
• Ex 1. Problem 2. On the right hand side the constant 2-n multiplying the integral should be 2n
• Ex 1. Problem 1. The indices were off for the sequences.
• Ex 3. Problem 1. The tau_1 should be less than equal to tau_2.
• Ex 3. Problem 11. The x refers to the starting point of the process X
• Ex 3. Problems 7-9. The filtration is also assumed to be right-continuous.

The suggested solutions are being posted here (during Saturdays).

### Registration

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