I am reading the proof of the Strong Markov property for Ito diffusions In Oksendal 6th edition p117 Theorem 7.2.4, and I do not understand where the fact that the stopping time has to be almost surely finite comes in.
The proof in Oksendal imitates that of the Markov property for Ito diffusion, which itself, at heart, relies on the fact that an Ito diffusion is time-homogeneous.
Please could you let me know where the a.s. finiteness of the stopping time is required in the proof?