So I am trying to show that $E[e^{W_s + W_t}] = e^{\frac{t+s}{2}}e^{\min(s,t)}$
Where $W_s, W_t$ are Brownian motions.
I had an idea that I could express each Brownian motion as a function of $Z \sim N(0,1)$ and then use $Z$'s mgf to solve it. Such that,
$E[e^{W_s + W_t}] = E[e^{\sqrt{s}Z + \sqrt {t} Z}]= E[e^{(\sqrt{s} + \sqrt {t}) Z}] = e^{\frac{(\sqrt{s} + \sqrt{t})^2}{2}} = e^{\frac{(s+t)}{2}}e^{\sqrt{s}\sqrt{t}}$
I am kinda stuck at this point, and I dont understand why I cant come up with the correct answer with my approach.
Could anybody help me out here?
Thanks in advance!