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Stochastic methods for physics and biology, spring 2012


Paolo Muratore-Ginanneschi


10 cu.


Advanced studies



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 Review of Probability: random variables and their expectations (rev 04.02)

Lecture_11 Karhunen–Loève representation of the Brownian motion

Lecture_02 Review of Probability: classical theorems (part I) (rev 30.01)

Lecture_12 Ito and Stratonovich stochastic integrals

Lecture_03 Review of Probability: classical theorems (part II) (rev 04.02)

Lecture_13 Stochastic differential equations

Lecture_04 Probability and Classical Statistical Mechanics (rev 07.02)

Lecture_14 Backward Master equation and backward Kolmogorov equation

Lecture_05 Random Walk (rev 13.03)

Lecture_15 Forward Master equation and forward Kolmogorov equation

Lecture_06 Conditional expectation and Martingales (rev 13.03)

Lecture_16 Exit time statistics (rev 20.04)

Lecture_07 Markov jump processes (rev 01.03)

Lecture_17 Diffusions in one dimension: analysis of boundary conditions

Lecture_08 Continuity of paths, Kolmogorov-Chentsov theorem

Lecture_18 Hamilton-Bellman-Jacobi equation

Lecture_09 Ito lemma for paths of finite quadratic variation


Lecture_10 Brownian Motion




The aim of the course is to introduce the basic concepts of the theory
of stochastic differential equations (SDE) needed in applications (applied
mathematics, physics and biology). In particular we will illustrate
methods of qualitative, asymptotic and numerical analysis of SDE.
Among the scopes of the course is also to provide an elementary
introduction to stochastic control and filtering theory.


  1. Gardiner, C. W. Handbook of stochastic methods for physics, chemistry and the natural sciences Springer, 1994, 13, 442
  2. L.C. Evans, "An Introduction to Stochastic Differential Equations", Berkeley lecture notes.
  3. R. van Handel, "Stochastic Calculus and Stochastic Control", CalTech lecture notes (2007)
  4. D.J. Higham, "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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