Conditioning a martingale increment by earlier increments

Question

I have a $L^1$ - martingale ($E[|X|]<\infty$) defined on $(\Omega,\mathcal F , \mathbb P)$, with constant expectation $EX_t$, and I have to prove that $$E\{(X_v-X_u)|(X_t-X_s)\}=0$$ for $0\le s<t\le u<v$. $$$$ Can I consider $X_t-X_s$ as a filtration, then apply the linearity property?

Answer 1

Notice that $X_t-X_s$ is $\mathcal F_u$-measurable. Therefore by the tower property we a have

$$\begin{array}{rl}E(X_v|X_t-X_s) &= E(E(X_v|\mathcal F_u)|(X_t-X_s)) \\ &=E(X_u|(X_t-X_s)).\end{array}$$ Hence $$\begin{array}{rl}E(X_v-X_u|X_t-X_s) &=E(X_v|X_t-X_s)-E(X_u|(X_t-X_s)) \\&=0.\end{array}$$