I am trying to figure out why a $\mathcal{F}_t^B$ Brownian motion is also a Brownian motion in the regular sense. Here $\mathcal F_t^B$ is just the natural filtration.
I am getting confused with the definition of the $\mathcal F_t^B$-Brownian motion. The only definition I have is that for some bounded measurable $f$ $$E[B_t\vert \mathcal F_s]=P_{t-s}f(B_s).$$
- What exactly is $P_{t-s}f(B_s)$?
- What would be a smart approach? The usual definition with independent increments and normal distribution or rather showing it is a Gaussian process with correct covariance?