Lecture plan

This plan is tentative and may be changed.

BØ = Bernt Øksendal’s book

Week Topics Textbook and notes Comments
35 Introduction to measure and integration
Sigma algebras, measures, mesurable functions,
the Lebesgue integral, main theorems
Lp-spaces
Note on Measure and Integration
See also Øksendal Appendix B
36 Review of probability theory
Probability space, stochastic variables, distribution,
moments, characteristic functions, multivariate Gaussian variables
Independence. Borel-Cantelli lemmas
Conditional Expectation
Note on Measure and Integration
Øksendal Appendix A and B
See also Øksendal sec. 2.1
37 Conditional Expectation
Stochastic processes
Definition
Brownian motion
motivation, definition, properties
finite dimensional distributions
Note on Measure and Integration
Note on Brownian motion
Øksendal chp 2.1
Evans Lecture note chp 3.A, 3.B
38 Brownian motion - further remarks,
sample path properties,
Levy-Ciesielski construction
Martingales
definition (discrete and continuous),
filtration, adapted processes,
examples, inequality, closedness in L2,
martingale transform
Note on Brownian motion
Øksendal chp. 2.1 + page 31
Evans chp 2.I and 3
39 Stochastic integrals
White noise and SDEs
Stieltjes integrals
Elementary functions and Ito isometry
The Itô integral
Definition and properties,
martingale, continuity a.s.
Øksendal chp. 3
Note on Martingales and the Itô Integral
Read yourselves:
A comparision of Ito and Stratonovich integrals, Øksendal p. 35-37
40 Itô processes
Itô's formula with proof
Multi-dimensional versions
Stochastic Differential Equations
Strong solutions
Example
Øksendal chp. 4.1-4.2, 5.1
Evans Lecture note chp 4.D, 4.E
(for proofs of Ito's formula)
Read yourselves:
Øksendal chp. 4.3
(proofs not relevant for the exam)
41 Existence and uniqueness of strong solutions
Weak solutions, definition and uniqueness
Øksendal chp. 5
Evans Lecture Note chp. 5.A (the definition)
Read yourselves:
Øksendal chp. 5.3
(discussion on weak solutions and uniqueness)
42 Solutions to some linear SDEs
Pysical Browninan motion
Diffusion processes
Introduction, Markov processes, Itô Diffusions
Øksendal chp. 5
Note on Linear SDEs and Physical Brownian Motion
Note on Informal Comments on Itô Diffusions
Espen is in Spain
Harald Krogstad will give the lectures
43 Project work No lectures,
office hours during lecture times
44 Project work No lectures,
office hours during lecture times
45 Markov processes, Itô Diffusions
Stopping times, Dynkin’s Formula,
the generator,
The Dirichlet problems for elliptic PDEs
Brownian motion in Rn
Øksendal chp. 7
Note on Informal Comments on Ito Diffusions
46 Brownian motion in Rn
hitting and exit times and probabilities
Kolmogorov's backward equation
Øksendal chp. 8.1
Note on Fokker-Planck eqns. and Kolmogorov forward and backward eqns.
Read yourselves:
Proof of Theorem 8.1.1 b) in Øksendal
47 The Fokker-Planck or Kolmogorov's Forward Equations
Øksendal problem 8:8.2
The Resolvent
Feynman-Kac Formula
Itô diffusions and Martingales
Øksendal chp. 8.2-8.3
Note on Fokker-Planck eqns. and Kolmogorov forward and backward eqns.
2013-11-11, Espen Robstad Jakobsen