What happens if you define an Ito/Stratonovich-like Integral using random evaluation/quadrature points?

I am reading Bernt Øksendal's "Stochastic Differential Equations" and came across his wonderful discussion of how the Ito and Stratonovich integrals differ: (pg. 24, 6th edition)

In general it is natural to approximate a given function $f(t,\omega)$ by $ \sum_j f(t^*_j,\omega)\cdot \chi_{[t_j,t_{j+1})}(t) $ where the points $t^*_j$ belong to the intervals $[t_j,t_{j+1}]$, and then define $\int_S^T f(t,\omega) dB_t(\omega)$ as the limit (in a sense that we will explain of $\sum_j f(t^*_j,\omega)[B_{t_{j+1}} - B_{t_j}](\omega)$ as $n\to\infty$. However, the example above [omitted here] shows that – unlike the Riemann-Stieltjes integral – it does make a difference here what points $t^*_j$ we choose. The following two choices have turned out to be the most useful ones:

1. $t^*_j = t_j$ (the left end point), which leads to the Itô integral [notation omitted] and
2. $t^*_j = (t_j+t_{j+1})/2$ (the mid point), which leads to the Stratonovich integral [notation omitted]

This led me to wonder about the following question: What would happen if we chose the "quadrature point" (my choice of words) $t^*_j$ randomly within the given interval? Would this produce a third possible interpretation of integration with respect to Brownian motion? Or would it merely reproduce one of the two existing integrals? Or perhaps it would produce something distinct, but trivially expressible in terms of the two integrals referenced above.