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Stochastic methods in physics and biology, spring 2014


Paolo Muratore-Ginanneschi 

Matteo Marcozzi (assistant)


10 sp.

The aim of the course is to introduce the basic concepts of the theory of stochastic differential equations needed in applications (applied mathematics, physics and biology).



Advanced studies


The course is intended for undergraduate students of mathematics, physics. Prior courses in advanced calculus and linear algebra are required (Diff.Int. 1-2 and Lineaarialgebra 1, or Mapu 1-2).


Weeks 4-9 and 11-18, Monday 14-16 in room D123, and Friday 14-16 in room C123.

Easter holiday 17.-23.4.

Lecture Notes

The lecture notes cover and sometimes integrate the material expounded in the lections. They also give bibliographic references for the same topics.

Lectures 1-10

Lectures 10-20

Lecture 01: Probability spaces and random variables (v 11.02)Lecture 11: Karhunen–Loève representation of the Brownian motion
Lecture 02:: Independence, conditional expectations and inequalities (v 11.02)Lecture 12: Calculus for paths of finite quadratic variation (v 04.04)
Lecture 03: Classical limit theorems (v 11.02)Lecture 13: Ito integrals (v. 08.04)
Lecture 04: Sequences of random variables and martingales (v 07.02)Lecture 14: Stratonovich integral, stochastic differential equations (v 08.04)
Lecture 05: From Bernoulli variables to Random Walk (v 11.02) 
Lecture 06: Continuous Time Random Walks and Montroll-Weiss equation 
Lecture 07: Asymptotic analysis of the Montroll–Weiss process 
Lecture_08: Girsanov formula for continuous Markov Processes 
Lecture 09: Kolmogorov axioms and Kolmogorov–Chentsov theorem (v. 21.03) 
Lecture 10: Brownian motion (v. 21.03) 



  1. Schuss Z., "Theory and Applications of Stochastic Processes: An Analytical Approach" (Springer), 2010, 170, 468.
  2. Gardiner C. W., "Handbook of stochastic methods for physics, chemistry and the natural sciences" (Springer), 1994, 13, 442.

  3. Evans L. C., "An Introduction to Stochastic Differential Equations", Berkeley lecture notes.

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

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

  6. Higham D. J., "An algorithmic introduction to numerical simulation of stochastic differential equations", SIAM Review, Education Section, 43, 2001, 525-546. (Link to Higham's publications page.)


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Exercise session

1.Friday16-18B321Matteo Marcozzi

Exercise Notes

Notes 01 
Notes 02 
Exercise Session07 
Exercise Session 08 
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