At the moment I am following a uni course on Financial mathematics, the current subject is Brownian Motion. A subject I have now encountered a couple of times which I don't really understand is the Expected value of a BM to a power, for example $E[W^2(t)]$, which equals t. This particular example can be solved by rewriting the formula for variance, but I'm looking for a broader way of solving these. I have seen the use of moment generating functions for normal distributions pop up here and there, but my book does not really explain this use of the MGF and I have not found a good explanation of how to use these online as well. Could anyone maybe explain how the MGF should be used in this instance and why it works?
Thanks in advance!