I am trying to solve following multiple integral where the inner integral is stochastic.
$$ \int_0^t \int_0^s e^{- a (s-u)} \,dW(u) \,ds$$
where a is a constant and $W$ is a Brownian motion. I can find the distrubtion of the inner integral by using Ito Isometry
$$ \int_0^s e^{-a(s-u)} \,dW(u) \sim \mathcal{N}\left(0, \frac{1}{2a} [ 1 - e^{-2 a s} ]\right) $$
So my question is how do I solve an integral of a distribution, which in this example is $$ \int_0^t \mathcal{N}\left(0, \frac{1}{2a} [ 1 - e^{-2 a s} ] \right) \, ds$$
Thank you for all the responses!