Suppose have a constant stochastic process $X(s)=k$. Suppose we have some other function $B_{s}$ and we are interested in computing
$$\int_{0}^{t} X(s) dB_{s} =\int_{0}^{t} k dB_{s}=k\int_{0}^{t} dB_{s} = kB_{s}|_{0}^{t}=k(B_{t}-B_{0})$$
Now, suppose we have a step function $\phi(s)$ and wish to integrate it. Suppose we have $N$ rectangles underneath the interval $[a,b]$. Then, according to this video, one should write
$$\int_{a}^{b} \phi(s) dB_{s} = \sum_{i=1}^N\int_{a}^{b} k_{i} dB_{s} = \sum_{i=1}^Nk_{i}\int_{a}^{b} dB_{s} = \sum_{i=1}^Nk_{i}(B_b-B_a)$$
However, to me this doesn't make sense because, given my prior result, if I wish to look at each rectangle under the curve, then it would only be on a small interval since the prior result only applies to a single interval. So to me, it should look like
$$\int_{a}^{b} \phi(s) dB_{s} = \sum_{i=1}^N\int_{a_i}^{b_i } k_{i} dB_{s} = \sum_{i=1}^Nk_{i}(B_{b_{i}}-B_{a_{i}})$$
where $\bigcup\limits_{i=1}^{n} [a_{i},b_{i}]=[a,b]$. Do either of these formulations make sense? If so, which one is correct (if either)?
