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1 = Stochastic methods for physics and biology, spring 2010 =
2
3 === Lecturer ===
4
5 [[Stefan Geritz>>doc:mathstatHenkilokunta.Geritz, Stefan]]
6 [[Paolo Muratore-Ginanneschi>>doc:mathstatHenkilokunta.Muratore-Ginanneschi, Paolo]]
7
8 === Scope ===
9
10 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). In particular we will illustrate methods of qualitative, asymptotic and numerical analysis.
11
12 10 cu.
13
14 === Type ===
15
16 Advanced studies.
17
18 === Prerequisites ===
19
20 === Lectures ===
21
22 Weeks 3-9 and 11-18, Monday 14-16 in room C124 and Friday 14-16 in room C123.
23
24 Easter holiday 1.-7.4.
25
26 First lecture: Friday, 22.01.2010
27
28 === Lecture Notes ===
29
30 The lecture notes cover and sometimes integrate the material expounded in the lections. They also give bibliographic references for the same topics. For exercises please refer to the section below.
31
32 |=(((
33 Lectures 1-10
34 )))|=(((
35 Lectures 10-20
36 )))
37 |(((
38 [[Lecture_01>>attach:lecture_01.pdf]]: Background of Probability
39 )))|(((
40 [[Lecture_11>>attach:lecture_11.pdf]]: Kolmogorov-Chentsov theorem
41 )))
42 |(((
43 [[Lecture_02 (rev 29.01)>>attach:lecture_02.pdf]]: Bernoulli variables, Borel-Cantelli lemma
44 )))|(((
45 [[Lecture_12>>attach:lecture_12.pdf]]: White-noise
46 )))
47 |(((
48 [[Lecture_03>>attach:lecture_03.pdf]]: Limit theorems
49 )))|(((
50 [[Lecture_13>>attach:lecture_13.pdf]]: Ito integral
51 )))
52 |(((
53 [[Lecture_04>>attach:lecture_04.pdf]]: Martingales
54 )))|(((
55 [[Lecture_14>>attach:lecture_14.pdf]]: Ito and Stratonovich calculus
56 )))
57 |(((
58 [[Lecture_05>>attach:lecture_05.pdf]]: Numerical generation of random variables
59 )))|(((
60 [[Lecture_15>>attach:lecture_15.pdf]]: SDE, existence and uniqueness
61 )))
62 |(((
63 [[Lecture_06 (rev 12.02)>>attach:lecture_06.pdf]]: Statistical tests
64 )))|(((
65 [[Lecture_16>>attach:lecture_16.pdf]]: Kolmogorov pair and diffusion
66 )))
67 |(((
68 [[Lecture_07>>attach:lecture_07.pdf]]: Fokker-Planck equation I
69 )))|(((
70 [[Lecture_17>>attach:lecture_17.pdf]]: Exit time statistics
71 )))
72 |(((
73 [[Lecture_08>>attach:lecture_08.pdf]]: Fokker-Planck equation II
74 )))|(((
75 [[Lecture_18>>attach:lecture_18.pdf]]: Girsanov formula
76 )))
77 |(((
78 [[Lecture_09>>attach:lecture_09.pdf]]: Brownian motion
79 )))|(((
80 [[Lecture_19>>attach:lecture_19.pdf]]: Numerical integration of SDE's
81 )))
82 |(((
83 [[Lecture_10>>attach:lecture_10.pdf]]: Karhunen-Loève representation
84 )))|(((
85 [[Lecture_20>>attach:Population_dynamics.pdf]]: Population dynamics
86 )))
87
88 [[Probabilistic interpretation of set theoretic concepts >>attach:probabilistic_interpretation_table.pdf]]
89
90 === Exams ===
91
92 There will be two options for taking the course. Passing an exam to be held after the end of the course, in
93 date to be agreed or working out a short project (recommended).
94
95 === Bibliography ===
96
97 1. L.C. Evans, //"An Introduction to Stochastic Differential Equations"//, lecture notes.
98 1. P. E. Kloeden, E. Platen, H. Schurz, //"Numerical Solution Of Sde Through Computer Experiments"//, Springer (Universitext) (2003).
99 1. N. Berglund, B. Gentz //"Noise-Induced Phenomena in Slow-Fast Dynamical Systems. A Sample-Paths Approach"//, Springer (Probability and its Applications) (2005).
100 1. D.J. Higham, //"An algorithmic introduction to numerical simulation of stochastic differential equations"// SIAM Review, Education Section, 43, 2001, 525-546. ([[Link>>url:http://personal.strath.ac.uk/d.j.higham/pubs.html||shape="rect"]] to Higham's publications page.)
101
102 Book extended preview is available on line from [[google books>>url:http://books.google.com/?utm_source=createcontent&utm_medium=link&utm_content=booksearch&utm_campaign=en||shape="rect"]]
103
104 === [[Registration>>url:https://oodi-www.it.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57072||shape="rect"]] ===
105
106 Did you forget to register? [[What to do>>doc:mathstatOpiskelu.Kysymys4]].
107
108 === Exercise groups ===
109
110 |=(((
111 Group
112 )))|=(((
113 Day
114 )))|=(((
115 Time
116 )))|=(((
117 Place
118 )))|=(((
119 Instructor
120 )))
121 |(((
122 1.
123 )))|(((
124 Wednesday
125 )))|(((
126 16-18
127 )))|(((
128 C124
129 )))|(((
130
131 )))
132
133 [[Exercise set 1>>attach:exercises_01.pdf]]
134 [[Exercise set 2>>attach:exercises_02.pdf]]
135 [[Exercise set 3>>attach:exercises_03.pdf]]
136 [[Exercise set 4>>attach:exercises_04.pdf]]
137 [[Exercise set 5>>attach:exercises_05.pdf]]
138
139 ==== Courseware ====
140
141 The codes below make use of the [[pgplot>>url:http://www.astro.caltech.edu/~~tjp/pgplot/||shape="rect"]] graphic library and of the [[GNU GSL>>url:http://www.gnu.org/software/gsl/||shape="rect"]]
142 scientific library. You need to install these packages in order to use the codes as they are.
143
144 [[A c code to generate histograms>>attach:histo_plot.c]] and its [[initialization code>>attach:histo_plot.ini]]