First are related definitions from my lecture note:
A $\textbf{filtration}$ on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ is a sequence $(\mathcal{F}_{n})_{n \in \mathbb{Z}_{+}}$ of sub-sigma fields of $\mathcal{F}$ such that $\mathcal{F}_{n} \subseteq \mathcal{F}_{n+1}$ for all $n$.
A $\textbf{stochastic process}$ is a collection of random variables defined on the same probability space.
A stochastic process $(X_{n})_{n \in \mathbb{Z}_{+}}$ is $\textbf{adapted}$ to the filtration $(\mathcal{F}_{n})_{n \in \mathbb{Z}_{+}}$ if $X_{n}$ is $\mathcal{F}_{n}$-measurable for all $n$.
A process $(X_{n}, \mathcal{F}_{n})_{n \in \mathbb{Z}_{+}}$ is a martingale if
$(\mathcal{F}_{n})_{n \in \mathbb{Z}_{+}}$ is a filtration and $(X_{n})_{n \in \mathbb{Z}_{+}}$ is adapted to $(\mathcal{F}_{n})_{n \in \mathbb{Z}_{+}}$.
$X_{n}$ is integrable for all $n$.
$\mathbb{E} [X_{n+1} | \mathcal{F}_{n}]=X_{n}$ for all $n$.
Then I have an exercise:
I'm stuck at simplifying $\operatorname{Var} [X_{n}]-2 \operatorname{Cov}(X_{n+1},X_n)$. I feel that the exercise gives insufficient hypothesis to simplify this expression.
Could you please elaborate on this point? Many thanks!
My attempt:
We have $$\begin{aligned} &\mathbb E [M_{n+1} | \mathcal F] \\ ={} &\mathbb E \Big [ \big (X_{n+1} - \mathbb E[X_{n+1}] \big )^2- \operatorname{Var} [X_{n+1}] \Big | \mathcal F \Big ] \\ \overset{(1)}{=}{} &\mathbb E \Big [ \big (X_{n+1} - \mathbb E[X_{n}] \big )^2- \operatorname{Var} [X_{n+1}] \Big | \mathcal F \Big ]\\ ={} &\mathbb E \Big [ \big ( (X_{n+1}-X_n)+(X_n - \mathbb E[X_{n}]) \big )^2- \operatorname{Var} [X_{n+1}] \Big | \mathcal F \Big ] \\ ={} &\mathbb E \Big [ (X_{n+1}-X_n)^2 + 2(X_{n+1}-X_n)(X_n - \mathbb E[X_{n}])+ (X_n - \mathbb E[X_{n}])^2- \operatorname{Var} [X_{n+1}] \Big | \mathcal F \Big ] \\ \overset{(2)}{=}{} & \mathbb E \big [ (X_{n+1}-X_n)^2 \big ] + 2 \mathbb E \big[(X_{n+1}-X_n)(X_n - \mathbb E[X_{n}]) \big | \mathcal F \big ]+ (X_n - \mathbb E[X_{n}])^2- \operatorname{Var} [X_{n+1}] \\ \overset{(3)}{=}{} & \mathbb E \big [ (X_{n+1}-X_n)^2 \big ] + 2 (X_n - \mathbb E[X_{n}]) \mathbb E [ X_{n+1}-X_n | \mathcal F ]+ \big (X_n - \mathbb E[X_{n}]) \big )^2- \operatorname{Var} [X_{n+1}] \\ \overset{(4)}{=}{} & \mathbb E \big [ (X_{n+1}-X_n)^2 \big ] + \big (X_n - \mathbb E[X_{n}]) \big )^2- \operatorname{Var} [X_{n+1}] \\ \overset{(5)}{=}{} & \mathbb E \Big [ \big ( (X_{n+1}- \mathbb E[X_{n+1}])-(X_n - \mathbb E[X_{n}]) \big )^2 \Big] + \big (X_n - \mathbb E[X_{n}]) \big )^2- \operatorname{Var} [X_{n+1}] \\ ={} & \mathbb E \big [ (X_{n+1}- \mathbb E[X_{n+1}])^2 \big ] -2 \mathbb E\big [(X_{n+1}- \mathbb E[X_{n+1}])(X_n - \mathbb E[X_{n}]) \big ] + \mathbb E \big [(X_n - \mathbb E[X_{n}])^2 \big] \\& \quad + \big (X_n - \mathbb E[X_{n}]) \big )^2- \operatorname{Var} [X_{n+1}] \\ ={} & \operatorname{Var} [X_{n+1}] -2 \operatorname{Cov}(X_{n+1},X_n) + \operatorname{Var} [X_{n}]+ \big (X_n - \mathbb E[X_{n}]) \big )^2- \operatorname{Var} [X_{n+1}] \\ ={} & \big (X_n - \mathbb E[X_{n}]) \big )^2 + \operatorname{Var} [X_{n}]-2 \operatorname{Cov}(X_{n+1},X_n) \end{aligned}$$
(1), (4), and (5) follow from $\mathbb E[X_{n+1}] = \mathbb E[X_{n}]$.
(2) follows from $(X_{n+1}-X_n)$ is independent of $\mathcal F$ and $(X_n - \mathbb E[X_{n}])^2$ is $\mathcal F$-measurable.
(3) follows from $(X_n - \mathbb E[X_{n}])$ is $\mathcal F$-measurable.