It just occurred to me that the natural log cannot take on negative values, so if X(t) is a Geometric Brownian Motion and it is less than zero at some point, then ln(X(t)) can't happen.
So a statement like this:
"If X(t) is a Geometric Brownian Motion then ln(X(t+s)/X(t)) = ln(X(t+s)) - ln(X(t)) is the increment of the corresponding Arithmetic Brownian Motion" is not true in general, am I right?
So a statement like this:
"If X(t) is a Geometric Brownian Motion then ln(X(t+s)/X(t)) = ln(X(t+s)) - ln(X(t)) is the increment of the corresponding Arithmetic Brownian Motion" is not true in general, am I right?
Does Geometric Brown Motion imply Arithmetic Brownian Motion?
