Theorem: Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Supermartingale

Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-supermartingale.
Let $S$ and $T$ be bounded stopping times with respect to $\sequence {\FF_n}_{n \ge 0}$ and $S \le T$. 
Let $\FF_S$ be the stopped $\sigma$-algebra associated with $S$. 
Let $X_T$ and $X_S$ be $X$ at the stopping times $T$ and $S$.
Then: 
:$\expect {X_T \mid \FF_S} \le X_S$ almost surely.

Proof:
From Adapted Stochastic Process at Stopping Time is Measurable with respect to Stopped Sigma-Algebra:
:$X_S$ is $\FF_S$-measurable.
Then from Conditional Expectation of Measurable Random Variable:
:$\expect {X_S \mid \FF_S} = X_S$ almost surely.
Also from Doob's Optional Stopping Theorem: Discrete Time: Supermartingale:
:$X_T$ and $X_S$ are integrable.
Then from Conditional Expectation is Linear, we have:
:$\expect {X_T \mid \FF_S} \le X_S$ almost surely.
{{iff}}:
:$\expect {X_T - X_S \mid \FF_S} \le 0$ almost surely.
From Condition for Conditional Expectation to be Almost Surely Non-Negative, we aim to show that:
:$\expect {\paren {X_T - X_S} \cdot \chi_A} \le 0$ for each $A \in \FF_S$.
Suppose that $t \in \Z_{\ge 0}$ is such that: 
:$\map T \omega \le t$ for all $\omega \in \Omega$.
Then we have: 
{{begin-eqn}}
{{eqn	| l = X_T
		| r = X_S + \sum_{k \mathop = S}^{T - 1} \paren {X_{k + 1} - X_k}
}}
{{eqn	| r = X_S + \sum_{k \mathop = 0}^\infty \paren {X_{k + 1} - X_k} \cdot \chi_{S \le k \le T - 1}
}}
{{eqn	| r = X_S + \sum_{k \mathop = 0}^\infty \paren {X_{k + 1} - X_k} \cdot \chi_{S \le k < T}
		| c = since $T$ is $\Z_{\ge 0}$-valued, so $k \le T - 1$ {{iff}} $k < T$
}}
{{end-eqn}}
Since $T \le t$, we have: 
:$\ds X_S + \sum_{k \mathop = 0}^\infty \paren {X_{k + 1} - X_k} \cdot \chi_{S \le k < T} = X_S + \sum_{k \mathop = 0}^{t - 1} \paren {X_{k + 1} - X_k} \cdot \chi_{S \le k < T}$
Then for $A \in \FF_S$ we have by Expectation is Linear: 
:$\ds \expect {X_T \cdot \chi_A} = \expect {X_S \cdot \chi_A} + \sum_{k \mathop = 0}^{t - 1} \expect {\paren {X_{k + 1} - X_k} \cdot \chi_{S \le k < T} \cdot \chi_A}$
By Characteristic Function of Intersection, we have: 
:$\chi_{S \le k < T} \cdot \chi_A = \chi_{\set {S \le k < T} \cap A}$
Since $A \in \FF_S$, we have: 
:$\set {S \le k} \cap A \in \FF_k$
We also have:
:$\set {T > k} = \set {T \le k - 1}^c \in \FF_{k - 1}$
Since $\sequence {\FF_n}_{n \ge 0}$ is a filtration:
:$\FF_{k - 1} \subseteq \FF_k$ 
So:
:$\set {S \le k < T} \cap A = \paren {\set {S \le k} \cap A} \cap \set {T > k} \in \FF_k$
Since $\sequence {X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale, we have: 
:$\expect {X_{k + 1} \mid \FF_k} \le X_k$
and so, from Condition for Conditional Expectation to be Almost Surely Non-Negative:
:$\expect {\paren {X_{k + 1} - X_k} \cdot \chi_A} \le 0$ 
for each $A \in \FF_k$.
So, we have:
:$\expect {\paren {X_{k + 1} - X_k} \cdot \chi_{S \le k < T} \cdot \chi_A} \le 0$
So:
:$\expect {X_T \cdot \chi_A} \le \expect {X_S \cdot \chi_A}$
that is:
:$\expect {\paren {X_T - X_S} \cdot \chi_A} \le 0$
for each $A \in \FF_S$, as required.
{{qed}}
Category:Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time
Category:Supermartingales
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