The given stochastic process $X(t)$ with dynamics is,
$$ dX(t) = -aX(t)dt +\sigma dW(t) $$ $$ X(0)=0 $$
I want to integrate this. The answer to this is given in my text as,
$$ X(t) = \sigma \int_0^t e^{-a(t-u)} dW(u) $$
How does this happen? When I applied Ito's Lemma on the $dX(t)$ equation, I ended up with
$$ X(t) = X(0)+\sigma \int_0^t dW(u) + (1/2) \int_0^t dt $$
which is absolutely wrong.
What mistake am I doing while applying the lemma? Where does the exponent come from? I cannot figure out how to integrate a given stochastic process, and I'd genuinely appreciate any help!
Important Edit After posting this, I tried using integrating factors to solve for $X(t)$ such that the equation becomes,
$$ e^{at}dX(t)= e^{at}(-aXdt + \sigma dW_t)$$ and on integrating, I get $$ X(t)= e^{-at}\int_0^t-e^{au}aXdu + e^{-at}\int_0^te^{au}\sigma dW_u $$
which is still not the required answer. Can you please help me figure out why the deterministic part of the RHS becomes zero?
We will apply Ito's formula to the process $e^{at}X_t$. Let $f(t,x) := e^{at}x$, so by Ito's formula \begin{align*} d(e^{at}X(t)) = df(t,X(t)) &= \partial_t f(t,X_t)dt + \partial_x f(t,X_t)dX(t) + \frac 12 \partial_{xx}f(t,X_t)dX(t)dX(t) \\ &= ae^{at}X(t)dt + e^{at}(-aX(t)dt + \sigma dW(t)) \\ &= \sigma e^{at}dW(t). \end{align*} Re-writing in integral form, this says $$e^{at}X(t) = \int_0^t \sigma e^{as}dW(s) $$ so solving for $X(t)$ gives $$X(t) = e^{-at}\int_0^t\sigma e^{as}dW(s) = \sigma \int_0^t e^{-a(t-s)}dW(s).$$