The syllabus for the final exam is basically chapters 1, 2, 3, and 4 in Durretts book plus the note on Brownian motion. Even if there have been questions about definitions and proofs from the syllabus on earlier exams, such problems will not be given this year. It is, however, always an advantage to be familiar with standard ways to reason.
Some parts of chapters 1-4 will not be tried on the exam. They include:
Subsection 1.5.2 on the Metropolis-Hastings Algorithm.
Subsection 3.2.2 on Cost Equations.
Section 3.3 on Age and Residual Life.
Section 4.2.1: It suffices to know about the Yule process.
Section 4.6: Forget everything from Example 4.30 onwards.
The section on It?'s formula in the note on Brownian motion will not be tried on the exam.
This ...