Expectations of a Martingale are Constant

Last updated Nov 1, 2022

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

This follows by Expectations of a Submartingale are Non-Decreasing and Expectations of a Supermartingale are Non-increasing. Recall that A Martingale is a Supermartingale and a Submartingale , so $\forall r \leq t \in T$ $$\begin{align*} \mathbb{E}(X_{t}) \geq \mathbb{E}(X_{r})\\ \mathbb{E}(X_{t}) \leq \mathbb{E}(X_{r})\\ \end{align*}$$ Thus, $\mathbb{E}(X_{t}) = \mathbb{E}(X_{r})$, but now order of $r,t$ no longer matter (we can switch them), so we have the result $\forall r, t \in T$.

$\blacksquare$