Let $(X_n)_n$ be a submartingale such that $M=\sup_n E [X^+ _n ] < ∞$ where $X^+ _n = max(0, X_n)$.
Prove: there exist a random variable $X_∞ ∈ L^1 (Ω, P)$ such that $X_n → X_∞$ almost surly as $n → ∞$.
Hint: Use the Doob-Meyer decomposition and martingale convergence theorem.
I didn't use the hint and want to ask if you think my answer is correct. Also I'm wondering what is the solution using the hint.
$$ X_n \leq E[X_{n+1}|\mathcal{F}_n]$$ $$ E[|X_n|]=E[X_n ^+]+E[X_n ^- ]$$ If $N=\inf_n \{-E[X_n ^-]\} =-\infty$ then we get $$ E[X_0] \leq E[X_{n}]\rightarrow -\infty$$ contradiction because $E[X_0]$ is finite
Now $E[|X_n|]<M+|N|$ and so from $L^1$ martingale we get the convergence to $X_\infty \in L^1$
Doob decomposition: $X_n=M_n+V_n$ with $(M_n)$ a martingale and $(V_n)$ increasing, predictable (i.e. $V_n$ is $\mathcal F_{n-1}$ measurable for $n\ge 1$), and $V_0=0$.
Show that $E|M_n| = 2E\left[M_n^+\right] -E[M_0]$.
Notice that $M_n^+\le X_n^+$ for $n\ge 0$.
Use 1., 2., and the basic hypothesis to enable use of martingale convergence theorem.
What does 3. tell you about $\lim_n X_n$?