 Solution

At time $n-1$, the self-financing portfolio can be valued using the holdings about to be carried over the next period:Subtracting this from $V_n={\varphi }_n{S}_n+{\psi }_n{B}_n$ gives the gains identityConversely, suppose the gains identity and the valuation $V_n={\varphi }_n{S}_n+{\psi }_n{B}_n$ hold. Rearranging the identity gives $V_{n-1}={\varphi }_n{S}_{n-1}+{\psi }_n{B}_{n-1}$. Comparing with the valuation using $({\varphi }_{n-1},{\psi }_{n-1})$ shows that rebalancing at $n-1$ costs nothing. Applying this at each rebalancing time recovers the self-financing condition.