Expected Value of Submartingale Greater 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 submartingale.

Then:

$\expect {X_n} \ge \expect {X_0}$

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

Proof

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

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

So from Expectation is Monotone:

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

From Expectation of Conditional Expectation, we have:

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

So:

$\expect {X_n} \ge \expect {X_0}$

$\blacksquare$