How to prove that the multi-period market satisfies Non-arbitrage given that the single-period market admits Non-arbitrage Here is the question: Let $(Ω,\mathscr F,\mathbb P,\mathbb F= (\mathscr F_k)_{k=0,...,T})$ be a filtered probability space and $S=(S_k)_{k=0,...,T}$ a discounted price process. Show that the following are equivalent: a) $S$ satisfies Non-arbitrage. b) For each $k = 0, . . . , T − 1$, the one-period market $(S_k, S_{k+1})$ on $(Ω, \mathscr F_{k+1}, \mathbb P, (\mathscr F_k, \mathscr F_{k+1}))$ satisfies Non-arbitrage. I know how to prove from a) to b). But proving from b) to a) seems quite difficult. So if anyone can help, please share your idea here. Thanks!

No-arbitrage is equivalent to existence of state prices or stochastic discount factor. The proof usually is based on separating hyperplane theorem (see Duffie's Dynamic Asset Pricing Theory for example) but the conclusion is that prices do not admit arbitrage from date $t_1$ to date $t_2$ if and only if there exist state prices $\pi_{t_1,t_2}$ such that $S_{t_1}=E^{t_1}[\pi_{t_1,t_2}S_{t_2}]$.

Part b) implies existence of $\pi_{k,k+1}$ such that $S_{k}=E^{k}[\pi_{k,k+1}S_{k+1}]$ for $k=0,...,T-1$. For any $t_1