I have a small doubt as I am currently self-studying stochastic calculus.
In Brownian motion part, the author talked about change of probability measure over Brownian motions. Now we we know that Brownian motion is normally distributed. As well as each step in the Brownian motion has to be symmetric, otherwise the mean $0$ criteria (of the increments) would be violated.
The following is my understanding.
1) $P(H)=0.5, \overline P(H)=0.5$ where $P$ and $\overline P$ are two probability measures, and $H$ is getting Head. Here I am assuming that Brownian motion is simulated by coin tosses. Then these 2 probability meausures would always agree on the above probability values.
2) In addition, I think due to normality of Brownian motions, the change of measure can only change the mean of the Brownian distribution.
Am I right in point # 1 and # 2. Would be grateful if someone can help in this regard.