What is the moment generating function of a function of a wiener process?

Question

If I have that $B(t)$ is a wiener process (brownian motion), what is the moment generating function of

$X(t) = e^{B(t)-t/2}$?

Answer 1

The moment generating function of $X_t$ will be defined by

$$M_X(\lambda)=\mathbb E( e^{X_t\lambda})$$ In this case your random variable $X_t:= e^{B_t-t/2}$ is a functional of a Gaussian r.v., the the above expectation can be calculated as follows

$$M_X(\lambda)=\int_{\mathbb R} e^{\lambda e^{x-t/2}}\frac{1}{\sqrt{2\pi t}}e^{-x^2/2t}dx$$

The integrand is clearly not in $L^1$ so you can claim that no m.g.f exists for this particular random variable.