Theorem: Stopped Supermartingale is 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 $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let $\sequence {X_n^T}_{n \ge 0}$ be the stopped process.
Then $\sequence {X_n^T}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale.

Proof:
By Stopped Process is Adapted Stochastic Process, $\sequence {X_n^T}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.
From Integrable Adapted Stochastic Process at Stopping Time is Integrable:
:$X_n^T$ is integrable for each $n \in \Z_{\ge 0}$.
Note that by definition we have for $\omega \in \Omega$ and $n \in \Z_{\ge 0}$:
:$\map {X_n^T} \omega = \map {X_n} \omega$ if $n \le \map T \omega$
and:
:$\map {X_n^T} \omega = \map {X_t} \omega$ if $n > \map T \omega = t$
So we can write: 
:$\ds \map {X_{n + 1}^T} \omega = \sum_{k \mathop = 0}^n \map {X_k} \omega \map {\chi_{\set {\omega \in \Omega : \map T \omega = k} } } \omega + \map {X_{n + 1} } \omega \map {\chi_{\set {\omega \in \Omega : \map T \omega \ge n + 1} } } \omega$
for each $\omega \in \Omega$.
Since $T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$ we have:
:$\set {\omega \in \Omega : \map T \omega = k} \in \FF_k$
for each $k \in \Z_{\ge 0}$ with $0 \le k \le n$.
So, since $\sequence {\FF_n}_{n \ge 0}$ is a filtration, we have: 
:$\set {\omega \in \Omega : \map T \omega = k} \in \FF_n$
Then, from Characteristic Function Measurable iff Set Measurable:
:$\chi_{\set {\omega \in \Omega : \map T \omega = k} }$ is $\FF_n$-measurable. 
Since $\sequence {X_n}_{n \ge 0}$ is $\sequence {\FF_n}_{n \ge 0}$-adapted we have that:
:$X_k$ is $\FF_k$-measurable
so:
:$X_k$ is $\FF_n$-measurable.
Then, from Pointwise Product of Measurable Functions is Measurable, we have:
:$X_k \chi_{\set {\omega \in \Omega : \map T \omega = k} }$ is $\FF_n$-measurable.
So, from Pointwise Sum of Measurable Functions is Measurable: General Result:
:$\ds \sum_{k \mathop = 0}^n X_k \chi_{\set {\omega \in \Omega : \map T \omega = k} }$ is $\FF_n$-measurable.
Finally, note that since $T$ is a stopping time with respect to $\sequence {\FF_n}$ and: 
:$\set {\omega \in \Omega : \map T \omega \ge n + 1}^c = \set {\omega \in \Omega : \map T \omega \le n}$
we have that:
:$\set {\omega \in \Omega : \map T \omega \ge n + 1} \in \FF_n$
since $\sigma$-algebras are closed under relative complement. 
From Characteristic Function Measurable iff Set Measurable, we have: 
:$\chi_{\set {\omega \in \Omega : \map T \omega \ge n + 1} }$ is $\FF_n$-measurable.
We can now calculate:
{{begin-eqn}}
{{eqn	| l = \expect {X_{n + 1}^T \mid \FF_n}
		| r = \expect {\sum_{k \mathop = 0}^n X_k \chi_{\set {\omega \in \Omega : \map T \omega = k} } \mid \FF_n} + \expect {X_{n + 1} \chi_{\set {\omega \in \Omega : \map T \omega \ge n + 1} } \mid \FF_n}
		| c = Conditional Expectation is Linear
}}
{{eqn	| r = \sum_{k \mathop = 0}^n X_k \chi_{\set {\omega \in \Omega : \map T \omega = k} } + \expect {X_{n + 1} \chi_{\set {\omega \in \Omega : \map T \omega \ge n + 1} } \mid \FF_n}
		| c = Conditional Expectation of Measurable Random Variable
}}
{{eqn	| r = \sum_{k \mathop = 0}^n X_k \chi_{\set {\omega \in \Omega : \map T \omega = k} } + \chi_{\set {\omega \in \Omega : \map T \omega \ge n + 1} } \expect {X_{n + 1} \mid \FF_n}
		| c = Rule for Extracting Random Variable from Conditional Expectation of Product
}}
{{eqn	| o = \le
		| r = \sum_{k \mathop = 0}^n X_k \chi_{\set {\omega \in \Omega : \map T \omega = k} } + \chi_{\set {\omega \in \Omega : \map T \omega \ge n + 1} } X_n
		| c = {{Defof|Supermartingale}}
}}
{{eqn	| r = \sum_{k \mathop = 0}^{n - 1} X_k \chi_{\set {\omega \in \Omega : \map T \omega = k} } + \chi_{\set {\omega \in \Omega : \map T \omega = n} } X_n + \chi_{\set {\omega \in \Omega : \map T \omega \ge n + 1} } X_n
}}
{{eqn	| r = \sum_{k \mathop = 0}^{n - 1} X_k \chi_{\set {\omega \in \Omega : \map T \omega = k} } + \chi_{\set {\omega \in \Omega : \map T \omega \ge n} } X_n
		| c = Characteristic Function of Disjoint Union
}}
{{eqn	| r = X_n^T
		| c = {{Defof|Stopped Process}}
}}
{{end-eqn}}
almost surely.
So $\sequence {X_n^T}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale.
{{qed}}
{{improve|The notation is particularly opaque. Recommend (if no better way of presenting it can be found) at least to use \mathop tags to force spacing into the subscript expressions, which should improve readability.}}
Category:Supermartingales
Category:Stopped Processes
Category:Stopped Supermartingale is Supermartingale
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