Stopped Supermartingale is Supermartingale

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 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.

Corollary

$\expect {X_n^T} \le \expect {X_0}$ for each $n \in \Z_{\ge 0}$.

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:

\(\ds \expect {X_{n + 1}^T \mid \FF_n}\)

\(=\)

\(\ds \expect {\sum_{k \mathop = 0}^n X_k \chi_{\set {\omega \in \Omega : \map T \omega = k} } \mid \FF_n} + \expect {X_{n + 1} \chi_{\set {\omega \in \Omega : \map T \omega \ge n + 1} } \mid \FF_n}\)

Conditional Expectation is Linear

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 0}^n X_k \chi_{\set {\omega \in \Omega : \map T \omega = k} } + \expect {X_{n + 1} \chi_{\set {\omega \in \Omega : \map T \omega \ge n + 1} } \mid \FF_n}\)

Conditional Expectation of Measurable Random Variable

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 0}^n X_k \chi_{\set {\omega \in \Omega : \map T \omega = k} } + \chi_{\set {\omega \in \Omega : \map T \omega \ge n + 1} } \expect {X_{n + 1} \mid \FF_n}\)

Rule for Extracting Random Variable from Conditional Expectation of Product

\(\ds \)

\(\le\)

\(\ds \sum_{k \mathop = 0}^n X_k \chi_{\set {\omega \in \Omega : \map T \omega = k} } + \chi_{\set {\omega \in \Omega : \map T \omega \ge n + 1} } X_n\)

Definition of Supermartingale

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 0}^{n - 1} X_k \chi_{\set {\omega \in \Omega : \map T \omega = k} } + \chi_{\set {\omega \in \Omega : \map T \omega = n} } X_n + \chi_{\set {\omega \in \Omega : \map T \omega \ge n + 1} } X_n\)

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 0}^{n - 1} X_k \chi_{\set {\omega \in \Omega : \map T \omega = k} } + \chi_{\set {\omega \in \Omega : \map T \omega \ge n} } X_n\)

Characteristic Function of Disjoint Union

\(\ds \)

\(=\)

\(\ds X_n^T\)

Definition of Stopped Process

almost surely.

So $\sequence {X_n^T}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale.

$\blacksquare$

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