No, this approach to the mean-variance model should not be interpreted as an optimal hedging strategy in terms of asset $B$. Given $w_A$, solving for $Y$ simply provides the solution for the total dollar amount of the $B$ side of the portfolio that completes and satisfies the $w_A + w_B = 1$ constraint, if that constraint is in fact kept intact somewhere in ...
Note that for the replicating portfolio to be self-financing it suffices that (1):
where I have changed the notation by designating by $B_t$ the money market account:
Hence, because the portfolio is self-financing, its dynamics are: