Expectations of a Submartingale are Non-Decreasing

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a Probability Space. Let $(X_{t}){t \in T}$ be a Submartingale wrt Filtration $\mathcal{F}{*} := (\mathcal{B}{t}){t \in T}$ on $\Omega$. Then $\forall r \leq t \in T$, $\mathbb{E}(X_{r}) \leq \mathbb{E}(X_{t})$.

This follows by Smoothing and Expectation is Non-Decreasing: $$\begin{align*} &\mathbb{E}(X_{t}|\mathcal{B}{r}) \geq X{r}\\ \Rightarrow&\mathbb{E}(X_{t}) = \mathbb{E}(\mathbb{E}(X_{t}|\mathcal{B}{r})) \geq \mathbb{E}(X{r}) \end{align*}$$ $\blacksquare$