Stochastic
Calculus
Alan Bain

1. Introduction
The following notes aim to provide a very informal introduction to Stochastic Calculus,
and especially to the Ito^ integral and some of its applications. They owe a great deal to Dan
Crisan's Stochastic Calculus and Applications lectures of 1998; and also much to various
books especially those of L. C. G. Rogers and D. Williams, and Dellacherie and Meyer's
multi volume series `Probabilities et Potentiel'. They have also beneted from insights
gained by attending lectures given by T. Kurtz.
The present notes grew out of a set of typed notes which I produced when revising
for the Cambridge, Part III course; combining the printed notes and my own handwritten
notes into a consistent text. I've subsequently expanded them inserting some extra proofs
from a great variety of sources. The notes principally concentrate on the parts of the course
which I found hard; thus there is often little or no comment on more standard matters; as
a secondary goal they aim to present the results in a form which can be readily extended
Due to their evolution, they have taken a very informal style; in some ways I hope this
may make them easier to read.
The addition of coverage of discontinuous processes was motivated by my interest in
the subject, and much insight gained from reading the excellent book of J. Jacod and
A. N. Shiryaev.
The goal of the notes in their current form is to present a fairly clear approach to
the Ito^ integral with respect to continuous semimartingales but without any attempt at
maximal detail. The various alternative approaches to this subject which can be found
in books tend to divide into those presenting the integral directed entirely at Brownian
Motion, and those who wish to prove results in complete generality for a semimartingale.
Here at all points clarity has hopefully been the main goal here, rather than completeness;
although secretly the approach aims to be readily extended to the discontinuous theory.
I make no apology for proofs which spell out every minute detail, since on a rst look at
the subject the purpose of some of the steps in a proof often seems elusive. I'd especially
like to convince the reader that the Ito^ integral isn't that much harder in concept than
the Lebesgue Integral with which we are all familiar. The motivating principle is to try
and explain every detail, no matter how trivial it may seem once the subject has been
understood!
Passages enclosed in boxes are intended to be viewed as digressions from the main
text; usually describing an alternative approach, or giving an informal description of what
is going on { feel free to skip these sections if you nd them unhelpful.
In revising these notes I have resisted the temptation to alter the original structure
of the development of the Ito^ integral (although I have corrected unintentional mistakes),
since I suspect the more concise proofs which I would favour today would not be helpful
on a rst approach to the subject.
These notes contain errors with probability one. I always welcome people telling me
about the errors because then I can x them! I can be readily contacted by email as
alanb@chiark.greenend.org.uk. Also suggestions for improvements or other additions
are welcome.
Alan Bain
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