Let $X_t$ be a stochastic process defined on $\mathbb R$ such that all sample paths are right-continuous, non-decreasing, bounded between 0 and 1, $X_{-\infty}=0$, $X_\infty = 1$.
What can be said about the integrals
$$\int_{-\infty}^\infty s \, {X_s}^m \, dX_s$$
where $m=0,1,...,n$.
Can we prove that those integrals make sense (so that indeed $X_t$ is a semimartingale and that $s \, {X_s}^m$ is in $L^1(X)$) and are finite $a.s$ ? Can we get a nice expression for them ? Any pieces of information, even for some $m$'s, are welcome.
What are the tools to study such integrals besides change of variable formula and generalized Itô's lemma ? I only know those results for processes on $\mathbb R^+$, is it safe to assume they work in a similar way on $\mathbb R^-$ ?