Martingale Convergence Theorem

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space . Let $(X_{n}){n \geq 1}$ be a discrete-time Submartingale wrt Discrete-Time Filtration $\mathcal{F}{*} := (\mathcal{F}{n} \subset \mathcal{B}){n \in \mathbb{N}}$ so that $\sup\limits_{n \geq 1} \mathbb{E}(X_{n})^{+} < \infty$. Then there exists $X \in L^{1}(\mathcal{B})$ so that $X_{n} \to X$ almost surely.

Letting $a < b \in \mathbb{R}$, we uniformly bound the number of Complete Upcrossings with the Upcrossing Inequality. First note that if $a \geq 0$ $$(X_{n} - a)^{+} = 1_{X_{n} \geq a}X_{n} -1_{X_{n} \geq a}a \leq 1_{X_{n} \geq a}X_{n} \leq (X_{n})^{+}$$ and if $a < 0$ $$(X_{n} - a)^{+} = 1_{X_{n} \geq a}X_{n} -1_{X_{n} \geq a}a = (X_{n})^{+} + 1_{0 > X_{n} \geq a}X_{n} -1_{X_{n} \geq a}a \leq (X_{n})^{+} + |a|.$$

So $\forall n \geq 1$ $$\begin{align*} (b - a) \mathbb{E}(U_{n}[a, b]) &\leq \mathbb{E}(X_{n} - a)^{+} - \mathbb{E}(X_{1} - a)^{+}\\ &\leq \mathbb{E}(X_{n})^{+} + |a|\\ &\leq M + |a| \end{align*}$$ for $\infty > M \geq \sup\limits_{n \geq 1} \mathbb{E}(X_{n})^{+}$. Therefore, we have that $\mathbb{E}(U_{n}[a,b]) \leq \frac{M + |a|}{b-a}$. Since $U_{n}[a,b]$ is Non-Decreasing Function as $n \to \infty$, it converges (possibly to $\infty$), say to $U[a,b]$. By Monotone Convergence Theorem, we have that $$\frac{M + |a|}{b-a} \geq \lim\limits_{n \to \infty} \mathbb{E}(U_{n}[a,b]) = \mathbb{E}(U[a,b]).$$ Therefore, $U[a,b]$ must be Almost Surely finite (otherewise $\mathbb{E}(U[a,b]) = \infty$). This means that $$\mathbb{P}[\liminf\limits_{n \to \infty} X_{n} \leq a < b \leq \limsup\limits_{n \to \infty} X_{n}] = 0,$$ since that Event occurs If and Only If $U[a,b] = \infty$.

Therefore $$\bigcup\limits_{a, b \in \mathbb{Q}} [\liminf\limits_{n \to \infty} X_{n} \leq a < b \leq \limsup\limits_{n \to \infty} X_{n}]$$ is a Null Set. But then we must have that $\liminf\limits_{n \to \infty} X_{n} = \limsup\limits_{n \to \infty} X_{n} =: X$ Almost Surely.

Finally, we show that $X \in L^{1}(\mathcal{B})$. Recall that A Submartingale has uniformly Bounded First Moment iff it has uniformly Bounded Positive Moment, so $\infty > \sup\limits_{n \geq 1} \mathbb{E}|X_{n}|$. By Fatou’s Lemma, we have that $$\mathbb{E}|X| \leq \liminf\limits_{n \to \infty} \mathbb{E}|X_{n}| \leq \sup\limits_{n \geq 1} \mathbb{E}|X_{n}| < \infty.$$ so $X \in L^{1}(\mathcal{B})$. $\blacksquare$

1. This holds for Martingales as well since a Martingale is a Submartingale. A corresponding result holds for Supermartingales since An Adapted Process is a Supermartingale iff its negative is a Submartingale.