# Stochastic methods in physics and biology, spring 2016

**Teacher:** Paolo Muratore-Ginanneschi

**Teaching Assistant**: Joonas Turunen

**Scope:** 10 cr

**Type:** Advanced studies

**Teaching:**

**Topics:**

**Prerequisites:**

## News

- No lecture on Fri 04.03 and on Fri 01.01. We will agree on the dates for making up for these lectures.

## Teaching schedule

Weeks 3-9 and 11-18, Monday and Friday 14-16 in room B120.

First lecture Friday 22.01

Easter holiday 24.-30.3.

## Exams

Written test of 2h and 30 min maximal duration to be held in a date to be agreed after the end of the course

First exam call: FRIDAY 13.05 from 11:00 to 15:00 in either A111 or B123.

## Course material

The main reference for the course, besides lecture notes, is

- Pavliotis, Grigorios A.
*Stochastic processes and applications : diffusion processes, the Fokker-Planck and Langevin equations,*Springer 2014

(e-reading available from the Helsinki University Library)

Lecture notes

Lectures 1-10 | Lectures 10-20 |
---|---|

Lecture 01: Kolmogorov axioms (set - event correspondence table ) | Lecture 11 : Some exactly solvable stochastic differential equations |

Lecture 02: On independence, conditional probability and Bayes' formula | Lecture 12 : PDE's and martingales |

Lecture 03: Stochastic processes, stationarity and a.s. convergence | Lecture 13 : Dynkin's and Feynman-Kac formulas |

Lecture 04 : Stationarity and ergodicity | Lecture 14: Diffusions in domains with boundaries |

Lecture 05: Wiener process and Markov processes | Lecture 15 : Langevin's equation |

Lecture 06: Kolmogorov's equations | Lecture 16: A succint overview of Smoluchowski equation |

Lecture 07: Ito calculus without probability | Lecture 17: Statistics of exit times from a domain |

Lecture 08: Stochastic integrals | Lecture 18: One dimensional diffusions |

Lecture 09: Stochastic differential equations | Lecture 19: Feller's analysis of boundary conditions for 1d diffusions |

Lecture 10: A martingale intermezzo | Lecture 20: Numerical methods |

## Presemo

The course as a dedicated presemo room (web space). To access this web space just follow the foregoing link. Students are encouraged to use the presemo room for any civilized activity pertaining the course. We will also try to use the presemo room during the lectures for checking the student response to multi-answer questions. For this reason, we encourage students to have a mobile device with them when attending classes.

## Registration

Did you forget to register? What to do?

## Exercises

Exercise sessions on Fridays 12-14 in B321, starting from 29.1.2016. Extra points for the exam from solved exercises: 25%-> 1p, 50%-> 2p, 75%-> 3p. One additional point after participating in at least 80% of the exercise sessions.

### Assignments

- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12

### Exercise classes

Group | Day | Time | Room | Instructor |
---|---|---|---|---|

1. | Fri | 12-14 | B321 | Joonas Turunen |

## Bibliography

The following list comprises some extra references which I found useful while preparing course material.

Gardiner C. W., "Handbook of stochastic methods for physics, chemistry and the natural sciences" (Springer), 1994, 13, 442.

- Nelson E. "Dynamical Theories of Brownian Motion" Princeton University Press 1967 (freely available)
Evans L. C., "An Introduction to Stochastic Differential Equations", Berkeley lecture notes.

van Handel R., "Stochastic Calculus, Filtering, and Stochastic Control", CalTech lecture notes (2007).

Klebaner F. C., "Introduction to stochastic calculus with applications" (Imperial College Press), 2005, 416.

Higham D. J., "An algorithmic introduction to numerical simulation of stochastic differential equations", SIAM Review, Education Section, 43, 2001, 525-546.

- Boffetta G., Vulpiani A. "La Probabilità in Fisica" (in italian) Springer, 2012, 236
Durrett R., "Essentials of Stochastic Processes" Springer, 2012, 265

Durrett R., "Probability models for DNA sequence evolution", Springer 2008, 432 (e-reading from UH)

Steele, J. M., "Stochastic calculus and financial applications", Springer, 2001, 300.

## Course feedback

Course feedback can be given at any point during the course. Click here.