Finding $\int_0^t \int_0^s e^{-|p-q|}\,dp\,dq$

Question

I am trying to calculate $$\int_0^t \int_0^s e^{-|p-q|}\,dp\,dq$$

What I have tried so far is to decompose this into $$\int_0^{\max(t,s)} \int_0^{\min(t,s)} e^{-|p-q|}dpdq = \int_0^{\min(t,s)} \int_0^{\min(t,s)} e^{-|p-q|}dp dq + \int_{\min(t,s)}^{\max(t,s)} \int_0^{\min(t,s)} e^{-|p-q|}dpdq$$

But I cannot really progress from here. I would greatly appreciate any help.

Answer 1

Break up the integral region into two, one for $p-q\ge0$ and the other for $p-q < 0$.

For $t\ge s > 0$,

$$I_{t\ge s}= \int_0^s dp \left( \int_0^{p} e^{q-p}dq+\int_{p}^t e^{p-q} dq\right)$$ $$= \int_0^s dp \left[e^{-p}(e^p -1)-e^{p}(e^{-t} -e^{-p} )\right]$$ $$= \int_0^s dp (2-e^{-p}-e^{p-t} )$$ $$= 2s + (e^{-s}-1) - e^{-t}(e^{s}-1) $$ $$= 2s + e^{-s}+e^{-t}- e^{s-t}-1 $$

Similarly,

$$I_{t < s}= 2t + e^{-s}+e^{-t}- e^{t-s}-1 $$