Show that a standard Brownian motion is a martingale with respect to the filtration , where . Moreover, show that is a martingale with respect to the filtration .
First part I have done, but for the second part I arrive at a point where I have this...
which is apparently equal to . Why is this?
(More info if needed)
and continuing I arrive at the above result I'm stuck at.
No! It says that is the n-th moment of a Gaussian random variable of mean 0 and variance , that's all. For , is the variance itself (mean = 0), so it is . We can also write , so that is a standard Gaussian random variable (if then )); then and there are formulas for . By the way, if is odd (by symmetry).