Expected Value of Supermartingale Less Than or Equal To Initial Expected Value
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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:
- $\expect {X_n} \le \expect {X_0}$
$\blacksquare$