Given any sequence of r.v.s $\{X_n\}$ s.t. $X_n\in L^1$ for all $n$ and history $\{\mathcal{F_n}\}$ with $X_n\in \mathcal{F_n}$ for all $n$. How can we prove that this sequence of r.v.s can be written as a sum of a supermartingale and submartingale?
Thank you!
Hints:
Define a sequence of events $A_n: = \big\{E[X_{n+1}|\mathcal F_n] \geq X_n\big\}$, and similarly define $B_n:= A_n^c = \big\{E[X_{n+1}|\mathcal F_n] < X_n \big\}$. Note that $A_n,B_n \in \mathcal F_n$ for all $n$.
Define $Y_n: = \sum_{j=0}^{n-1} (X_{j+1}-X_j)1_{A_j}$ and $Z_n:=\sum_{j=0}^{n-1}(X_{j+1}-X_j)1_{B_j}$, with $X_0:=0$. Note that $X_n=Y_n+Z_n$ for all $n$.
Check that $Y_n$ is a supermartingale and $Z_n$ is a submartingale, by computing $E[Y_{n+1}-Y_n|\mathcal F_n]$ and same for $Z$. Conclude.