Stochastic Question: $d \int B_s ds = ?$

Question

Stochastic Question: $d \int_0^t B_s ds = ?$

$B_s$ is the standard Brownian motion at time $s$. This is an Ito integral. Operator $d$ is defined in the standard Ito sense.

For those who understands this question, this background is plenty. I know at least one other people who agrees.

Answer 1

Notation such as $dX_t=B_t\,dt$ is just shorthand for $$ X_t = X_0 + \int_0^t B_s\,ds $$ So if $X_t = \int_0^t B_s\,ds$, then $dX_t=B_t\,dt$. Or, in other words, $$ d\left({\int_0^t B_s\,ds}\right) = B_t\,dt, $$ simply because that's what the notation means.