0
$\begingroup$

$$ X_{t} = \int _{0}^{t}h(t)dW_{t} $$ 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.

$\endgroup$
2
$\begingroup$

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

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.