Chapter 5
Stochastic processes and
PDEs
There is a close connection between solving an sde and solving boundary value
problems for a certain type of partial dierential equations (pdes) which are
known as parabolic. Parabolic pdes are, roughly speaking, th
Chapter 12
Conditional expectations
and martingales
12.1
Conditional expectations
To start with, we take another look at conditional probabilities. You might want
to consult Section 2.3 to refresh your memory: there we introduced conditional
probability d
Chapter 8
Measure-theoretic
probability
In this chapter we will try to familiarize you with the language of measuretheoretic probability theory. A probability will be seen as a map giving dierent
weights, between 0 and 1, to possible future events or outc
MATHEMATICAL METHODS
AUTUMN 2009
LECTURE 9: THE ITO STOCHASTIC INTEGRAL
(LECTURE NOTES, CHAPTERS 11 (AND 10)
1. Introduction
We will assume from now that our Brownian motion (Wt )t0 is dened
on some probability space (, F, P) (for which one may take BM if
Mathematical Methods for
Financial Engineering, I
Autumn 2009
Raymond Brummelhuis
Department of Economics, Mathematics and Statistics,
Birkbeck College, University of London,
Malet Street, London WC1E 7HX
October 1, 2009
2
Chapter 1
Introduction
Market pr
MATHEMATICAL METHODS
AUTUMN 2009
LECTURE 8: MEASURE THEORETIC PROBABILITY
(SUMMARY OF LECTURE NOTES, CHAPTERS 8 AND 9)
RAYMOND BRUMMELHUIS
BIRKBECK
Probability Spaces (lecture Notes, sections 8.1 - 8.3)
Measure theoretic probability: axiomatics based on t
Chapter 6
Multivariable Ito Calculus
As is the case for ordinary Calculus, there exists a multi-variable version of
Ito Calculus involving more than one Brownian. It is relevant for modelling
situations in which there are several independent sources of un
Chapter 3
Brownian motion
3.1
Introducing Brownian motion
Brownian motion has become one of the fundamental building blocks of modern
quantitative nance. Indeed, the basic continuous-time model for nancial asset
prices assumes that the log-returns of a gi
Chapter 4
A crash course in It
o
calculus
In this chapter we introduce the basic rules of stochastic calculus or It calculus.
o
We will do this using an intuitive approach which is based on calculus-style
dierentials, postponing the mathematically rigorou
Chapter 7
Characteristic functions of
random variables
We briey leave the subject of stochastic processes to get acquainted with a
powerful tool for analysing sums of independent random variables, the characteristic function of a random variable. To some
Chapter 10
Hilbert spaces
This chapter contains some non-examinable, though useful, foundational material on Hilbert spaces. Hilbert spaces play an important rle in the rigorous
o
construction of the Ito integral in the next chapter. Some basic knowledge