Expected Value of Supermartingale Less Than or Equal To Initial Expected Value

Theorem

Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.

Let $\sequence {X_n}_{n \ge 0}$ be a supermartingale.

Then:

$\expect {X_n} \le \expect {X_0}$

for each $n \in \Z_{\ge 0}$.

Proof

From Definition 2 of a discrete time supermartingale, we have:

$\expect {X_n \mid \FF_0} \le X_0$ almost surely.

So from Expectation is Monotone:

$\expect {\expect {X_n \mid \FF_0} } \le \expect {X_0}$

From Expectation of Conditional Expectation, we have:

$\expect {\expect {X_n \mid \FF_0} } \le \expect {X_n}$

So:

$\expect {X_n} \le \expect {X_0}$

$\blacksquare$