Let $(X_k)_{1\leq k \leq n}$ be a Walsh-Paley $L^p$-martingale (a dyadic martingale) with values in a Banach space $X$.
Why does there exist a dyadic martingale $(Y_k)_{1\leq k \leq n}$ with the same joint distribution than $(X_k)$ with some functions $\phi_k\in L^p([0,1]^k,X)$ and $\varphi_k\in L^p([0,1])$ such that $$ (Y_{k}-Y_{k-1})(\theta_1,\ldots,\theta_{k+1})=\phi_k(\theta_1,\ldots,\theta_{k})\varphi_k(\theta_{k+1}) $$ and $\int_0^1\varphi_k=0$ for any $k$?