Expectation Value of Wiener Process

Question

I want to perform this integral, $$\int^t_0\int^{t'}_0 \delta(s-s')dsds'$$ I know that the result should be min(t,t'), as it is the expectation value of the wiener process. I just want to know how to deal with this delta function in 2 dimensions, are we need to integrate on the line s-s'=0? Please help, Thanks in advance.

Answer 1

When viewed as a function of $s$ the delta function is a unit point mass in $s'$. Clearly, $\int_0^{t'}\delta(s-s')\,ds$ is zero if $t'<s'$ and one otherwise. Therefore, the double integral is $$ \int_0^t1_{\{s'\le t'\}}\,ds'=\min(t,t')\,. $$