This question is regarding an answer to the question below:
Expectation regarding Brownian Motion
This is a formula regarding getting expectation under the topic of Brownian Motion. $$ \begin{align*} E[W(s)W(t)] &=E[W(s)(W(t)−W(s))+W(s)^2]\\ &=E[W(s)]E[W(t)−W(s)]+E[W(s)^2]\\ &=0+s =\min(s,t). \end{align*} $$
One of Michael Hardy's comment is: "The step that says $$E[W(s)(W(t)−W(s))]=E[W(s)]E[W(t)−W(s)]$$ depends on an assumption that $t>s$."
So, finally, my question is how does the assumption $t>s$ play out in the $$E[W(s)(W(t)−W(s))]=E[W(s)]E[W(t)−W(s)]?$$
What if $t\leq s$?
Thanks a lot! Love the smart math stack exchange crowd!
For $t>s$ you know from the definition of a Brownian Motion that $W_t-W_s$ is independent of $W_s$ and this implies
$$\mathbb{E}(W_s \cdot (W_t-W_s)) = \mathbb{E}(W_s) \cdot \mathbb{E}(W_t-W_s)$$
If $t \leq s$ that's not true, but you can use
$$\mathbb{E}(W_s \cdot W_t) = \mathbb{E}((W_s-W_t) \cdot W_t +W_t^2)$$
instead (where $W_s-W_t$ is now independent of $W_t$) and do the same calculations as above.