Expectations of a Supermartingale are Non-increasing

Last updated Nov 1, 2022

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

This follows by An Adapted Process is a Supermartingale iff its negative is a Submartingale and Expectations of a Submartingale are Non-Decreasing and Linearity of Expectation. That is, $$\begin{align*} &\mathbb{E}(-X_{t}) \geq \mathbb{E}(-X_{r})\\ \Rightarrow&\mathbb{E}(X_{t}) \leq \mathbb{E}(X_{r})\\ \end{align*}$$ $\blacksquare$