thanks steve, that makes perfect sense!
some further questions on the same topic:
1. I was looking at the ASET which offers an alternative answer to the same question - they instead use the conditions:
E[(W(t)-W(s)]=0 and Cov[W(t),W(s)]=min(s,t)
Whereas in the examiners report they used:
Var[W(t)]=0 and Var(W(t))=E[(W(t))^2] - E[W(t)]^2
- If we get this type of question where we asked to prove something is Standard Brownian Motion, can we use any combination of these conditions or we have to use certain conditions with each other?
- Also, What other conditions can we use?
2. Looking at the notation in the notes - I am confused as to the difference between W(t) and B(t)?
I was under the impression W(t) is used to denote General BM, whilst B(t) denotes standard BM. However, in this question W(t) denotes Standard BM.
Are these terms simply used interchangeably or ?
3. Final Question I promise: Geometric BM :
S(t)=exp[w(0)+sigmaB(t)+mu(t)]
I don't understand how from this we can derive that S(t) has a Lognormal[w(0)+mu(t),(sigma^2)t] distribution?
Last edited by a moderator: Nov 11, 2015