We will use (parts of) the book:
I.Karatzas and S.Shreve: Methods of Mathematical Finance, 1998, 2nd Edition. Springer. ISBN: 0-387-94839-2.
In addition the following book of related contents might be useful:
I.Karatzas: Lectures on the Mathematics of Finance, 1997. Amer.Math.Soc..
As for some background material in stochastic calculus the following two books can be considered:
B. ?ksendal: Stochastic Differential Equations, 2007, 6th Edition. Springer. ISBN: 978-3-540-04758-2.
I.Karatzas and S.Shreve: Brownian Motion and Stochastic Calculus, 1998, 2nd Edition. Springer. ISBN: 0-387-97655-8.