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I hope you can help me with this problem. In my lecture notes, my professor stated that for a state price deflator $\phi\in L_{n+1}^2(P, F)$ (F being a filtration) and a strictly positive price process $(I_t)_{t\in \{0,...,n\}}$, $(\phi_t \cdot I_t)_{t\in \{0,...,n\}}$ is a (P,F) martingale. How can I prove this?

I know that you can write $E(\phi_{t+1} I_{t+1}|\mathcal{F}_t)=E(E(\phi_{t+1} I_{t+1}|\mathcal{F}_{t+1})|\mathcal{F}_t)$. But I do not know how to proceed in order to show that this is equal to $\phi_t I_t$. Do you have any hints for me?

I appreciate any kind of help. Thanks a lot.

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