avewright/proofwiki-math huggingface dataset:
Theorem: Adapted Stochastic Process at Stopping Time is Measurable with respect to Stopped Sigma-Algebra

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 $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let $X_T$ be the adapted stochastic process $\sequence {X_n}_{n \ge 0}$ stopped at $T$.
Let $\FF_T$ be the stopped $\sigma$-algebra associated with $T$.
Then $X_T$ is $\FF_T$-measurable.

Proof:
We have that if $\map T \omega = t$ for $\omega \in \Omega$ and $t \in \Z_{\ge 0}$ then: 
:$\map {X_T} \omega = \map {X_t} \omega$
We aim to show that for each Borel set $A \subseteq \R$:
:$\ds \set {\omega \in \Omega : \map {X_T} \omega \in A} \in \FF_T$
That is, we want to show that:
:$\set {\omega \in \Omega : \map {X_T} \omega \in A} \cap \set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$
for $t \in \Z_{\ge 0}$.
Let $t \in \Z_{\ge 0}$. 
We have that:
:$\ds \set {\omega \in \Omega : \map {X_T} \omega \in A} \cap \set {\omega \in \Omega : \map T \omega \le t} = \bigcup_{s \in \Z_{\ge 0}, \, 0 \le s \le t} \set {\omega \in \Omega : \map T \omega = s} \cap \set {\omega \in \Omega : \map {X_s} \omega \in A}$
Since $T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$, we have:
:$\set {\omega \in \Omega : \map T \omega = s} \in \FF_s$
Since $\sequence {\FF_n}_{n \ge 0}$ is a filtration of $\sigma$-algebra and $s \le t$:
:$\set {\omega \in \Omega : \map T \omega = s} \in \FF_t$
Since $\sequence {X_n}_{n \ge 0}$ is $\sequence {\FF_n}_{n \ge 0}$-adapted, we have that $X_s$ is $\FF_s$-measurable.
So, we have:
:$\set {\omega \in \Omega : \map {X_s} \omega \in A} \in \FF_s$
and so:
:$\set {\omega \in \Omega : \map {X_s} \omega \in A} \in \FF_t$ 
Since $\FF_t$ is closed under finite intersection, we have:
:$\set {\omega \in \Omega : \map T \omega = s} \cap \set {\omega \in \Omega : \map {X_s} \omega \in A} \in \FF_t$
Since $\FF_t$ is closed under finite union, we have:
:$\ds \bigcup_{s \in \Z_{\ge 0}, \, 0 \le s \le t} \set {\omega \in \Omega : \map T \omega = s} \cap \set {\omega \in \Omega : \map {X_s} \omega \in A} \in \FF_t$
and so:
:$\set {\omega \in \Omega : \map {X_T} \omega \in A} \cap \set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$
for all $t \in \Z_{\ge 0}$. 
We therefore have: 
:$\set {\omega \in \Omega : \map {X_T} \omega \in A} \in \FF_T$
So $X_T$ is $\FF_T$-measurable.
{{qed}}
Theorem: Stopped Process is Adapted Stochastic Process

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

Proof:
Let $n \in \Z_{\ge 0}$. 
From Constant Function is Stopping Time, $n$ is a stopping time.
From Pointwise Minimum of Stopping Times is Stopping Time, $n \wedge T$ is a stopping time, where $\wedge$ is the pointwise minimum.
From Adapted Stochastic Process at Stopping Time is Measurable with respect to Stopped Sigma-Algebra, we have: 
:$X_{n \wedge T}$ is $\FF_{n \wedge T}$-measurable
where $\FF_{n \wedge T}$ is the stopped $\sigma$-algebra associated with $n \wedge T$. 
We have by the definition of pointwise minimum: 
:$n \wedge T \le n$
From Stopped Sigma-Algebra preserves Inequality between Stopping Times and Stopped Sigma-Algebra of Constant Stopping Time coincides with Filtration, we have:
:$\FF_{n \wedge T} \subseteq \FF_n$ 
So:
:$X_{n \wedge T}$ is $\FF_n$-measurable.
So:
:$\sequence {X_n^T}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-adapted process.
{{qed}}
Category:Stopped Processes
Category:Adapted Stochastic Processes
r2sjzlcbs0kxyqsfsp4gdpz79rlv47n
Theorem: Primitive of x squared by Cosine of x

:$\ds \int x^2 \cos x \rd x = x^2 \sin x + 2 x \cos x + 2 \sin x + C$

Proof:
From Primitive of $x^2 \cos a x$:
{{:Primitive of x squared by Cosine of a x}}
The result follows on setting $a = 1$.
{{qed}}
Theorem: Adapted Stochastic Process is Supermartingale iff Negative is Submartingale/Discrete Time

Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a discrete-time filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be a discrete-time $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.
Then $\sequence {X_n}_{n \ge 0}$ is a supermartingale {{iff}} $\sequence {-X_n}_{n \ge 0}$ is a submartingale.

Proof:
Since $\sequence {X_n}_{n \ge 0}$ is a discrete-time $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process:
:$X_n$ is $\FF_n$-measurable for each $n \in \N$.
From Pointwise Scalar Multiple of Measurable Function is Measurable:
:$-X_n$ is $\FF_n$-measurable for each $n \in \N$.
So $\sequence {-X_n}_{n \ge 0}$ is a discrete-time $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.
We then just need to check conditional expectations.
From Conditional Expectation is Linear, we have: 
:$\expect {X_{n + 1} \mid \FF_n} \le X_n$ almost surely {{iff}} $\expect {-X_{n + 1} \mid \FF_n} \ge -X_n$ almost surely.
That is: 
:$\sequence {X_n}_{n \ge 0}$ is a supermartingale {{iff}} $\sequence {-X_n}_{n \ge 0}$ is a submartingale.
{{qed}}
Theorem: Stopped Submartingale is Submartingale

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 $\sequence {\FF_n}_{n \ge 0}$-submartingale.
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 $\sequence {\FF_n}_{n \ge 0}$-submartingale.

Proof:
From Adapted Stochastic Process is Supermartingale iff Negative is Submartingale, we have:
:$\sequence {-X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale.
From Stopped Supermartingale is Supermartingale, we have:
:$\sequence {-X_n^T}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale.
From Adapted Stochastic Process is Supermartingale iff Negative is Submartingale, we have:
:$\sequence {X_n^T}$ is a $\sequence {\FF_n}_{n \ge 0}$-submartingale.
{{qed}}
Category:Submartingales
Category:Stopped Processes
Category:Stopped Submartingale is Submartingale
3amido3brikcxqneerk60f8b8jsqdnc
Theorem: Stopped Martingale is Martingale

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 $\sequence {\FF_n}_{n \ge 0}$-martingale.
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 $\sequence {\FF_n}_{n \ge 0}$-martingale.

Proof:
From Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale:
:$\sequence {X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale and $\sequence {\FF_n}_{n \ge 0}$-submartingale.
From Stopped Supermartingale is Supermartingale and Stopped Submartingale is Submartingale:
:$\sequence {X_n^T}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale and $\sequence {\FF_n}_{n \ge 0}$-submartingale.
From Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale:
:$\sequence {X_n^T}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-martingale.
{{qed}}
Category:Stopped Processes
Category:Martingales
Category:Stopped Martingale is Martingale
avat9ktvmy53u1q9yjisou03azou7pm
Theorem: Equivalence of Definitions of Martingale in Discrete Time

Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \mathop \ge 0}$ be an adapted stochastic process.
{{TFAE|def = Martingale in Discrete Time|view = martingale in discrete time}}

Proof:
=== Definition 1 implies Definition 2 ===
Suppose that:
:$(1): \quad$ $X_n$ is integrable for each $n \in \Z_{\ge 0}$
:$(2): \quad \forall n \in \Z_{\ge 0}: \expect {X_{n + 1} \mid \FF_n} = X_n$.
We prove that:
:$\forall n \in \Z_{\ge 0}, \, \forall m \ge n: \expect {X_m \mid \FF_n} = X_n$.
Fix $n \in \Z_{\ge 0}$.
We induct on $m$. 
For all $m \in \Z_{\ge 0}$ with $m \ge n$, let $\map P m$ be the proposition:
:$\expect {X_m \mid \FF_n} = X_n$ almost surely.
Theorem: Equivalence of Definitions of Submartingale in Discrete Time

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 adapted stochastic process.
{{TFAE|def = Submartingale/Discrete Time|submartingale in discrete time}}

Proof:
=== Definition 1 implies Definition 2 ===
Suppose that:
:$(1): \quad$ $X_n$ is integrable for each $n \in \Z_{\ge 0}$
:$(2): \quad \forall n \in \Z_{\ge 0}: \expect {X_{n + 1} \mid \FF_n} \ge X_n$.
We prove that:
:$\forall n \in \Z_{\ge 0}, \, \forall m \ge n: \expect {X_m \mid \FF_n} \ge X_n$.
Fix $n \in \Z_{\ge 0}$.
We induct on $m$. 
For all $m \in \Z_{\ge 0}$ with $m \ge n$, let $\map P m$ be the proposition:
:$\expect {X_m \mid \FF_n} \ge X_n$ almost surely.
Theorem: Equivalence of Definitions of Supermartingale in Discrete Time

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 adapted stochastic process.
{{TFAE|def = Submartingale/Discrete Time|supermartingale in discrete time}}

Proof:
=== Definition 1 implies Definition 2 ===
Suppose that:
:$(1): \quad$ $X_n$ is integrable for each $n \in \Z_{\ge 0}$
:$(2): \quad \forall n \in \Z_{\ge 0}: \expect {X_{n + 1} \mid \FF_n} \le X_n$.
We prove that:
:$\forall n \in \Z_{\ge 0}, \, \forall m \ge n: \expect {X_m \mid \FF_n} \le X_n$.
Fix $n \in \Z_{\ge 0}$.
We induct on $m$. 
For all $m \in \Z_{\ge 0}$ with $m \ge n$, let $\map P m$ be the proposition:
:$\expect {X_m \mid \FF_n} \ge X_n$ almost surely.
Theorem: Expected Value of Supermartingale Less Than or Equal To Initial Expected Value

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:
:$\expect {X_n} \le \expect {X_0}$
{{qed}}
Category:Supermartingales
59zb3rgvts3vtabipnumlbtydq6m49x
Theorem: Expected Value of Submartingale Greater Than or Equal To Initial Expected Value

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:
:$\expect {X_n} \ge \expect {X_0}$
{{qed}}
Category:Submartingales
iep04jiij4bd4zcedsnqtmrgea3s4cc
Theorem: Primitive of x by Square of Logarithm of x

:$\ds \int x \ln^2 x \rd x = \dfrac {x^2 \ln^2 x} 2 - \dfrac {x^2 \ln x} 2 + \dfrac {x^2} 4 + C$

Proof:
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
      | r = x \ln x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = x \cdot \dfrac \d {\d x} \ln x + \ln x \dfrac \d {\d x} x
      | c = Product Rule for Derivatives
}}
{{eqn | r = x \cdot \frac 1 x + \ln x \cdot 1
      | c = Derivative of Natural Logarithm, Derivative of Identity Function
}}
{{eqn | r = 1 + \ln x
      | c = simplifying
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \ln x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = x \ln x - x
      | c = Primitive of $\ln x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \ln^2 x \rd x
      | r = \int \paren {x \ln x} \paren {\ln x} \rd x
      | c = 
}}
{{eqn | r = \paren {x \ln x} \paren {x \ln x - x} - \int \paren {x \ln x - x} \paren {1 + \ln x} \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = x^2 \ln^2 x - x^2 \ln x - \int x \ln x \rd x + \int x \rd x - \int x \ln^2 x \rd x + \int x \ln x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | ll= \leadsto
      | l = 2 \int x \ln^2 x \rd x
      | r = x^2 \ln^2 x - x^2 \ln x  + \int x \rd x + C
      | c = simplifying
}}
{{eqn | r = x^2 \ln^2 x - x^2 \ln x + \dfrac {x^2} 2 + C
      | c = Primitive of $x$
}}
{{eqn | ll= \leadsto
      | l = \int x \ln^2 x \rd x
      | r = \dfrac {x^2 \ln^2 x} 2 - \dfrac {x^2 \ln x} 2 + \dfrac {x^2} 4 + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x squared by Exponential of a x/Examples/x squared by e^-x

:$\ds \int x^2 e^{-x} \rd x = -e^{-x} \paren {x^2 + 2 x + 2} + C$

Proof:
From Primitive of $x^2 e^{a x}$:
{{:Primitive of x squared by Exponential of a x}}
So:
{{begin-eqn}}
{{eqn | l = \int x^2 e^{-x} \rd x
      | r = \frac {e^{-x} } {-1} \paren {x^2 - \frac {2 x} {-1} + \frac 2 {\paren {-1}^2} } + C
      | c = Primitive of $x^2 e^{a x}$: setting $a = -1$
}}
{{eqn | r = -e^{-x} \paren {x^2 + 2 x + 2} + C
      | c = 
}}
{{end-eqn}}
{{qed}}
Category:Primitives involving Cosine Function
dlxd4bnnuvuou9mocvhsvhxpkh1gfkw
Theorem: Doob's Optional Stopping Theorem/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 $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let:
:$\map {X_T} \omega = \map {X_{\map T \omega} } \omega \map {\chi_{\set {\omega \in \Omega : \map T \omega < \infty} } } \omega$
for each $\omega \in \Omega$.
Suppose one of the following conditions holds: 
:$(1): \quad$ $T$ is bounded
:$(2): \quad$ $T$ is finite almost surely, and there exists an integrable random variable $Y$ with $\size {X_n} \le Y$ for $n \in \Z_{\ge 0}$
:$(3): \quad$ $T$ is integrable, and there exists a real number $M > 0$ such that for each $n \in \Z_{\ge 0}$ we have $\size {X_{n + 1} - X_n} \le M$ almost surely.
Then $X_T$ is integrable and:
:$\expect {X_T} \le \expect {X_0}$

Proof:
We first show that if $T$ is finite almost surely, then: 
:$\map {X_{n \wedge T} } \omega \to \map {X_T} \omega$
for almost all $\omega \in \Omega$.
Let $\omega \in \Omega$ be such that $\map T \omega = s < \infty$. 
Then for $n \ge s$, we have: 
:$\map {X_{n \wedge T} } \omega = \map {X_s} \omega$
From Constant Sequence in Topological Space Converges, we have:
:$\map {X_{n \wedge T} } \omega \to \map {X_s} \omega = \map {X_T} \omega$
for all $\omega \in \Omega$ such that $\map T \omega < \infty$.
Note that since $T$ is finite almost surely, we have that: 
:$\map {X_{n \wedge T} } \omega \to \map {X_T} \omega$
for almost all $\omega \in \Omega$, as required.
So:
:$X_{n \wedge T} \to X_T$
almost surely. 
We have from Stopped Supermartingale is Supermartingale: Corollary:
:$X_{n \wedge T}$ is integrable for each $n \in \N$
and:
:$\expect {X_{n \wedge T} } \le \expect {X_0}$
So from Expectation is Linear, and since $X_0$ is integrable, this is equivalent to:
:$\expect {X_{n \wedge T} - X_0} \le 0$
We aim to show: 
:$\ds \expect {X_T - X_0} = \lim_{n \mathop \to \infty} \expect {X_{n \wedge T} - X_0}$
in each of the three cases.
We then have half the result from Limits Preserve Inequalities.
We will then have to establish that $X_T$ is integrable.
Theorem: Doob's Optional Stopping Theorem/Discrete Time/Submartingale

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 $\sequence {\FF_n}_{n \ge 0}$-submartingale.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let:
:$\map {X_T} \omega = \map {X_{\map T \omega} } \omega \map {\chi_{\set {\omega \in \Omega : \map T \omega < \infty} } } \omega$
for each $\omega \in \Omega$.
Suppose one of the following conditions holds: 
:$(1) \quad$ $T$ is bounded
:$(2) \quad$ $T$ is finite almost surely, and there exists an integrable random variable $Y$ with $\size {X_n} \le Y$ for $n \in \Z_{\ge 0}$
:$(3) \quad$ $T$ is integrable, and there exists a real number $M > 0$ such that for each $n \in \Z_{\ge 0}$ we have $\size {X_{n + 1} - X_n} \le M$ almost surely.
Then $X_T$ is integrable and:
:$\expect {X_T} \ge \expect {X_0}$

Proof:
From Adapted Stochastic Process is Supermartingale iff Negative is Submartingale:
:$\sequence {-X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale.
Theorem: Doob's Optional Stopping Theorem/Discrete Time/Martingale

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 $\sequence {\FF_n}_{n \ge 0}$-martingale.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let:
:$\map {X_T} \omega = \map {X_{\map T \omega} } \omega \map {\chi_{\set {\omega \in \Omega : \map T \omega < \infty} } } \omega$
for each $\omega \in \Omega$.
Suppose one of the following conditions holds: 
:$(1): \quad T$ is bounded
:$(2): \quad T$ is finite almost surely, and there exists an integrable random variable $Y$ with $\size {X_n} \le Y$ for $n \in \Z_{\ge 0}$
:$(3): \quad T$ is integrable, and there exists a real number $M > 0$ such that for each $n \in \Z_{\ge 0}$ we have $\size {X_{n + 1} - X_n} \le M$ almost surely.
Then $X_T$ is integrable and in particular:
:$\expect {X_T} = \expect {X_0}$

Proof:
From Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale:
:$\sequence {X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale and $\sequence {\FF_n}_{n \ge 0}$-submartingale.
From Doob's Optional Stopping Theorem: Discrete Time: Supermartingale and Doob's Optional Stopping Theorem: Discrete Time: Submartingale:
:$\expect {X_T} \le \expect {X_0}$
and:
:$\expect {X_T} \ge \expect {X_0}$ 
respectively, with $X_T$ integrable.
So we have:
:$\expect {X_T} = \expect {X_0}$
in any of the three cases.
{{qed}}
Theorem: Least Time at which Discrete-Time Adapted Stochastic Process equals or exceeds Real Number is Stopping Time

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 discrete-time $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.
Let $\lambda \in \R$. 
Let:
:$T = \inf \set {n \in \Z_{\ge 0} : X_n \ge \lambda}$
Then $T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.

Proof:
Note that for $t \in \Z_{\ge 0}$ and $\omega \in \Omega$, we have: 
:$\inf \set {n \in \Z_{\ge 0} : \map {X_n} \omega \ge \lambda} \le t$
{{iff}}:
:$\map {X_s} \omega \ge \lambda$ for some $s \le t$.
That is, we have: 
:$\ds \set {\omega \in \Omega : \inf \set {n \in \Z_{\ge 0} : \map {X_n} \omega \ge \lambda} \le t} = \bigcup_{s \in \Z_{\ge 0}, \, 0 \le s \le t} \set {\omega \in \Omega : \map {X_s} \omega \ge \lambda}$
Since $\sequence {X_n}_{n \ge 0}$ is $\sequence {\FF_n}_{n \ge 0}$-adapted, we have:
:$\set {\omega \in \Omega : \map {X_s} \omega \ge \lambda} \in \FF_s$ for each $0 \le s \le t$.
Since $\sequence {\FF_n}_{n \ge 0}$ is a filtration, we have: 
:$\FF_s \subseteq \FF_t$
So:
:$\set {\omega \in \Omega : \map {X_s} \omega \ge \lambda} \in \FF_t$ for each $s \in \Z_{\ge 0}$ with $0 \le s \le t$.
So since $\FF_t$ is closed under finite union, we conclude:
:$\set {\omega \in \Omega : \inf \set {n \in \Z_{\ge 0} : \map {X_n} \omega \ge \lambda} \le t} \in \FF_t$
for each $t \in \Z_{\ge 0}$.
So $T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
{{qed}}
Category:Adapted Stochastic Processes
Category:Stopping Times
lu57u6grronehl98ky4dkjd79e2kz8e
Theorem: Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Submartingale

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 $\sequence {\FF_n}_{n \ge 0}$-submartingale.
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: 
:$\expect {X_T \mid \FF_S} \ge X_S$ almost surely.

Proof:
From Adapted Stochastic Process is Supermartingale iff Negative is Submartingale:
:$\sequence {-X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale. 
From Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time: Discrete Time: Supermartingale:
:$\expect {-X_T \mid \FF_S} \le -X_S$ almost surely.
From Expectation is Linear:
:$\expect {X_T \mid \FF_S} \ge X_S$ almost surely.
{{qed}}
Category:Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time
Category:Submartingales
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Theorem: Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Martingale

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 $\sequence {\FF_n}_{n \ge 0}$-martingale.
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: 
:$\expect {X_T \mid \FF_S} = X_S$ almost surely.

Proof:
From Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale:
:$\sequence {X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale and $\sequence {\FF_n}_{n \ge 0}$-submartingale.
From Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time: Discrete Time: Supermartingale and Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time: Discrete Time: Submartingale, we have:
:$\expect {X_T \mid \FF_S} \le X_S$ almost surely and $\expect {X_t \mid \FF_S} \ge X_S$ almost surely
respectively. 
So:
:$\expect {X_T \mid \FF_S} = X_S$ almost surely.
{{qed}}
Category:Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time
Category:Martingales
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