Books
Bernt Øksendal: Stochastic Differential Equations, 2007. Springer Verlag. 6th Edition.
Fred Espen Benth: Option Theory with Stochastic Analysis, 2004. Springer Verlag.
Syllabus
From Øksendal's book:
Ch. 2: All
Ch. 3: Sect. 3.1 and 3.2. From Thm 3.2.4 page 31 is not included
Ch. 4: All
Ch. 8: Sect. 8.6, pages 159-164, Thm 8.6.6 is included
From Benth's book:
Ch. 1: All
Ch. 2: All, except section 2.5 and 2.6
Ch. 3: All (this is in reality simply a soft version of chapters 2, 3 and 4 in Øksendal)
Ch. 4: All, except page 79 (Malliavin derivative) and section 4.9
Ch. 5: Sect. 5.1 (Monte Carlo simulation), until subsection 5.1.3, page 105.
All exercises and the compulsory task are part the required knowledge