Autocorrelation function of integral of cont. white noise

Question

Let $W(t)$ be continuous time white noise, that is, a wide-sense stationary (WSS) zero-mean Gaussian process with autocorrelation function $R_W (\tau) = σ^2\delta(\tau)$. Calculate the auto correlation function of $X(t) = \int_0^t W(r) dr$.

Can anyone help me understand how they get these integration limits in the following step in the calculation?

Answer 1

By the very definition, we have

$$\delta(\tau) = \begin{cases} 0, & \tau \neq 0, \\ 1, & \tau = 0 \end{cases}.$$

Therefore,

$$\delta(r-q) = \begin{cases} 0, & r \neq q, \\ 1, & r=q \end{cases}.$$

Suppose that $s<t$, i.e. $\min\{s,t\}= s$. For any $q \in (s,t]$ and $r \in [0,s]$ we have

$$\delta(r-q)=0$$

since $r-q<0$. Consequently,

$$\int_0^s \int_s^t \mathbb{E}(W_q W_r) \, dq \, dr = 0.$$

This implies the identity you want to prove.

The argumentation for the case $t \leq s$ works analogously; I leave it to you.

Answer 2

The integration domain on the left is a rectangle $s\times t$ starting from the origin.

Because of the Dirac $\delta$, that on the right is a line segment along the main diagonal, starting from the origin and extending until the first of $s,t$ is met.