I need. to find $E\exp(luX_{t})$, where $ u \in \mathbb R$

Question

$$ X_{t} = \int _{0}^{t}h(t)dW_{t} $$ where $h(t)$ is a determined function (not random) and $W_{t}$ is Wiener process.

My thoughts so far:

1)use Ito's formula and find X's probability density.

2)Try to do it by definition of Ito's integral maybe.

3)I know for sure that $X_{t}$ is a gauss process, so maybe there's something there, but still, I'd need to find density first.

Nothing came out so far. So any help is appreciated.

Answer 1

$X_t$ is normal with mean $0$ and variance $\int_0^{t}h(s)^{2}ds$. Hence $Ee^{luX_t}=e^{l^{2}u^{2}\int_0^{t}h(s)^{2}ds}$.