We most often encounter stochastic integrals in a form which gives a solution as a stochastic process in a time-like variable which is the upper bound of integration, e.g.:
$$ \int_0^t H_s d X_s $$
I'm curious about integrals taking the form:
$$ \int_0^\infty f(s) H_s d X_s $$
where $f(s)$ is a function shrinking suitably rapidly as $s$ increases so that the integral converges, something like $1/(1+s^2)$ or $e^{-s}$. Because the bounds are fixed, this quantity would take on a univariate distribution. I'd also suppose that $H_s$ is bounded so that this converges.
Are any results available about this type of integral for any particular processes $H$, $X$, or function $f$? I am hopeful to see the actual distribution of values this integral would take. I have searched for results of this type and found these papers:
https://arxiv.org/abs/0707.0538
https://link.springer.com/chapter/10.1007/978-3-540-31449-3_12
These works are firstly filled to the brim with definitions that are difficult to wrap my head around given my inexperience in stochastic processes, and moreover do not compute any actual stochastic integrals but rather seek to only classify integrals of this type.