I am struggling to compute the following partial derivative of an integral
$\dfrac{\partial }{\partial t} \int_{a}^{\infty} x(t) dF(x(t)) $,
where x is a random variable that depends on the deterministic variable $t$ and $F$ is the CDF of $x$.
Suppose, we could take the partial derivative inside the integral, and we could rewrite $dF(x(t)) = f(x(t))dx$, then we would have
$ \int_{a}^{\infty} \dfrac{\partial }{\partial t} x(t) f(x(t))dx$
and apply the product rule. However, $dx(t)$ still depends on $t$. Should I further substitute the differential $dx(t)$ until I reach an expression in $dt$ ? Is this approach valid? Is there a less complicated way to compute the partial derivative?
Any ideas would be greatly appreciated.
If I am correct in thinking that the CDF of a function if defined as: $$F[x](t)=\int_0^t x(\tau)d\tau$$ then we can say that: $$dF[x](t)=d\left(\int_0^t x(\tau)d\tau\right)$$ so if we got back to your integral we have: $$I=\int_a^\infty x(t)d\left(\int_0^t x(\tau)d\tau\right)$$ now if we do IBP we get: $$I=\left[x(t)F[x](t)\right]_{t=a}^\infty-\int_a^\infty x'(t)F[x](t)$$ However I am unsure as to what this represents, I am also confused as to how you are differentiating wrt $t$ when in a way it is the integration variable. Maybe try and use LIR first and see where that gets you?
EDIT: This is not a complete answer just a suggestion too big for a comment