Brownian Motion Calculus
Ubbo FWiersema
John Wiley & Sons Ltd 2008

First the Brownian motion process needs to be specified. This is the
subject of Chapter 1. Then an integral of the form  T
t=0 σ S(t)dB(t)
needs to be defined; that requires a new concept of integration which is
introduced in Chapter 3; the other term,  T
t=0 μS(t) dt, can be handled
by ordinary integration. Then the value of S(T ) needs to be obtained
from the above equation. That requires stochastic calculus rules which
are set out in Chapter 4, and methods for solving stochastic differential
equations which are described in Chapter 5. Once all that is in place, a
method for the valuation of an option needs to be devised. Two methods
are presented. One is based on the concept of a martingale which is
introduced in Chapter 2. Chapter 7 elaborates on the methodology for
the change of probability that is used in one of the option valuation
methods. The final chapter discusses how computations can be made
more convenient by the suitable choice of the so-called numeraire.