I have this seemingly simple problem which I haven't been able to solve. I have two standard Brownian motions, $B$ and $W$, on the same probability space and under the same filtration (I am not so sure whether this last part is sensible). What is $\int_0^tB_s\,dW_s$?
The product rule gives $\int_0^tB_s\,dW_s + \int_0^tW_s\,dB_s = B_tW_t$. Obviously the expressions for $\int_0^tB_s\,dW_s$ and $\int_0^tW_s\,dB_s$ should be symmetric but that doesn't really narrow down the possible solutions that much. I played with a few two-variable functions using Ito's lemma but no result so far.
I also gave Riemann sum a shot but I get the same result as above (because of the way I manipulate the terms in the sum). Maybe there is another way to express the terms in the sum that leads to what I want but I doubt that for some reason.
I feel like this has a very simple solution which I am not seeing but I wanted to post it here in any case.