Optional stopping theorem

The Optional Stopping Theorem is a mathematical theorem about Martingales , a special class of stochastic processes , and can therefore be assigned to probability theory . The sentence goes back to Joseph L. Doob and has far-reaching implications for the existence of game strategies which are advantageous for the player and which are based on the player leaving the game.

Framework

There is a stochastic process that formalizes the player's capital. This process can now be either ${\displaystyle X=(X_{n})_{n\in \mathbb {N} }}$

• be a martingale , which is a fair game,
• be a super martingale, which equates to a losing game for the player or
• being a submartingale , which is a beneficial game for the player.

The exit strategy is mathematically equivalent to a stop time that indicates when the game is left. ${\displaystyle \tau }$

The game combined with the exit strategy result in the stopped process , which then gives the long-term development when using the exit strategy. ${\displaystyle X^{\tau }=(X_{\min(n,\tau )})_{n\in \mathbb {N} }}$ ${\displaystyle \tau }$

The question now arises whether the process classes described above can be changed by choosing a suitable stopping time . In the interest of the player, a stopping time would be that would turn a martingale into a submartingale after stopping or turn a super martingale into a (sub) martingale . ${\displaystyle \tau }$ ${\displaystyle X}$ ${\displaystyle X^{\tau }}$ ${\displaystyle X}$ ${\displaystyle X^{\tau }}$

The sentence answers this question negatively: there is no stopping time, so the stopped process is in a different class than the original process.

statement

It is shorthand . Given a filtration and a stop time . Denote the σ-algebra of the past of the stopping time and define the filtration ${\displaystyle \min(n,\tau )=\tau \wedge n}$ ${\displaystyle \mathbb {F} =({\mathcal {F}}_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \tau }$ ${\displaystyle {\mathcal {F}}_{\tau \wedge n}}$ ${\displaystyle \tau \wedge n}$

${\displaystyle \mathbb {F} ^{\tau }:=({\mathcal {F}}_{\tau \wedge n})_{n\in \mathbb {N} }}$.

Then:

If a (sub- / super) martingale is related , the stopped process is also a (sub- / super) martingale both related and related . ${\displaystyle X}$ ${\displaystyle \mathbb {F} }$ ${\displaystyle X^{\tau }}$ ${\displaystyle \mathbb {F} }$ ${\displaystyle \mathbb {F} ^{\tau }}$

The following also applies:

Is a martingale so is ${\displaystyle X}$

${\displaystyle \operatorname {E} (X_{\tau \wedge n})=\operatorname {E} (X_{0})}$.

Additionally, either

• the stop time is limited, d. H. there is one with almost certainly, or ${\displaystyle \tau }$ ${\displaystyle c\in \mathbb {N} }$ ${\displaystyle \tau \leq c}$
• the stopping time is almost certainly finite and can be integrated equally , ${\displaystyle (X_{\tau \wedge n})_{n\in \mathbb {N} }}$

so is also

${\displaystyle \operatorname {E} (X_{\tau })=\operatorname {E} (X_{0})}$.

The two statements above also apply to submartingals if the equal sign is replaced by a . They also apply to supermartingales if the equal sign is replaced by a . ${\displaystyle \geq }$ ${\displaystyle \leq }$

The statement is not always formulated to the same extent in the literature. Sometimes only the stability property of (sub / super) martingales under the stopped process is referred to as the optional stopping theorem.

Derivation

The main statement is derived by means of the martingale transformation , one then sets . It follows from this that , and corresponding to the martingale transformation, this is again a (sub- / super) martingale. The detailed description can be found in the article on the martingale transformation as an example. ${\displaystyle H_{n}:=\mathbf {1} _{\{\tau \geq n\}}}$ ${\displaystyle H\cdot X=X^{\tau }}$

Relationship to the Optional Sampling Theorem

The main difference between the Optional Stopping Theorem and the Optional Sampling Theorem is that with the Optional Stopping Theorem the stopped process is examined, whereas with the Optional Sampling Theorem the sampled random variables are examined ${\displaystyle X^{\tau }}$

${\displaystyle X_{\tau }:=X_{\tau (\omega )}(\omega )}$

can be examined for different stop times.

There is an overlap between the stopped process and , for example, with almost certainly finite stop times ${\displaystyle X_{\tau }}$

${\displaystyle \lim _{n\to \infty }X_{\tau \wedge n}=X_{\tau }}$ pretty sure

applies. Therefore, the second part of the above statement is also referred to as a special case of the Optional Sampling Theorem. This provides for two stop times with the σ-algebra of σ-past and a martingale the statement ${\displaystyle \sigma ,\tau }$ ${\displaystyle \sigma \leq \tau }$ ${\displaystyle {\mathcal {F}}_{\sigma }}$ ${\displaystyle X}$

${\displaystyle X_{\sigma }=\operatorname {E} (X_{\tau }|{\mathcal {F}}_{\sigma })}$

and thus after forming the expected value

${\displaystyle \operatorname {E} (X_{\tau })=\operatorname {E} (X_{\sigma })}$.

If you set the stop time here , then this is exactly the above statement. ${\displaystyle \sigma =0}$

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• Christian Hesse: Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , doi : 10.1007 / 978-3-663-01244-3 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .

Individual evidence

1. ↑ Klenke: Probability Theory. 2013, p. 214.
2. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 317.