Consider the following scalar linear differential equation: $$ \dot x(t) = c x(t) + w(t),\ \ \ x(0)\in\mathbb{R}, $$ where $c\in\mathbb{R}$ and $w(t)$ is a white noise process with $\mathbb{E}[w(t)]=0$ and $\mathbb{E}[w(t)w(s)]=\sigma^2\delta(t-s)$.
My question. What is the value of the cross-covariance term $\mathbb{E}[x(t)w(t)]$?
My (heuristic) attempt. The state evolution can be written as $$ x(t) = e^{ct} x(0) +\int_{0}^t e^{c(t-\tau)} w(\tau) \mathrm{d}\tau. $$ Hence, $$ \mathbb{E}[x(t)w(t)] = e^{ct} \mathbb{E}[x(0)w(t)] + \int_{0}^t e^{c(t-\tau)} \mathbb{E}[w(\tau)w(t)] \mathrm{d}\tau = \sigma^2 \int_{0}^t e^{c(t-\tau)} \delta(t-\tau) \mathrm{d}\tau. $$ The last integral contains a Dirac's delta "function" centered at $t$. But $t$ is also the upper integral limit, so, in some sense, we should consider "half" of the effect of the Dirac's delta, yielding $\mathbb{E}[x(t)w(t)]=\sigma^2/2$.
My argument is totally heuristic and I would like to know if there is a formal way to derive the value of $\mathbb{E}[x(t)w(t)]$. I'm also aware that the definition of white noise, although widely used in physics and engineering, is mathematically tricky and one should always work with the integral of the white noise (i.e., a Wiener process). Thanks for your help.
