For any stopping time T, the pre-T σ-algebra FT is defined via Definition 2.12 makes precise the intuitive notion that a stopping time is one for which we know that it has occurred at the moment it...

For any

stopping time T, the pre-T σ-algebra F_T
is defined via

Definition 2.12 makes precise the intuitive notion that a stopping time is one for which we know that it has occurred at the moment it occurs. formalizes the idea that the σ-algebra FT contains all the information that is available up to and including the time T. A strong Markov process, defined precisely in Chapter 6, is a process which possesses the Markov property, as described at the beginning of this section, but with the constant time t generalized to be an arbitrary stopping time T. Theorem 2.15 below shows that Brownian motion is also a strong Markov process. First we establish a weaker preliminary result.