Forward integral is introduced by Francesco RussoPierre Vallois as a generalization of Ito integral. For simplicity, let $B$ be a standard Brownian motion and let $\phi$ be a measurable process. The forward integral of $\phi$ with respect to $B$ is defined to be

$\int_0^t \phi_s d^-B_s \overset{\Delta}{=} \lim_{\epsilon\rightarrow \infty} \int_0^t\phi(s)\frac{B_{(s+\epsilon)\wedge t}-B_s}{\epsilon}\,ds$

whenever the limit exists in probability, in which case $\phi$ is said to be forward integrable w.r.t. $B$. It was shown that if $\phi$ is adapted, then this coincides with the Ito integral.

I was wondering how do one check whether a (non-adapted)process is forward-integrable(as it's not obvious to see whether the limit in probability exists or not). Namely, I would like to know

(1) Are there any general criterion or sufficient condition(except adaptivity) for a process to be forward integrable?

(2) Can someone provide me some reference, or even show me some examples of processes that are forward integrable(or some examples that are not)?

(3) Can one construct a process that is forward integrable, and its natural filtration contains the brownian filtration?

Any comment and help are greatly appreciated!


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