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 benefited from insights gained by attending lectures given by T. Kurtz.
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. The notes contain proofs which spell out every minute detail, since on a first look at the subject the purpose of some of the steps in a proof often seems elusive. The notes intent to show 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.