= Solution
Let $Y$ and $\widetilde Y$ be normalized martingale deflators. For any date $T$ and bounded $\mathcal F_T$-measurable $X_T$, condition g supplies a strategy whose only prescribed cash flow is $X_T$ at $T$. Applying the martingale identity from part b gives
$$
\mathbb E[Y_TX_T]=\pi_0^H
=\mathbb E[\widetilde Y_TX_T].
$$
Thus $\mathbb E[(Y_T-\widetilde Y_T)X_T]=0$ for every bounded $\mathcal F_T$-measurable $X_T$. Taking indicators, or the sign of the difference, shows $Y_T=\widetilde Y_T$ almost surely. Since $T$ was arbitrary, the normalized martingale deflator is unique.
Back to article page
