Optional sampling theorem

The Optional Sampling Theorem is a probabilistic statement going back to Joseph L. Doob . A popular version of this theorem is that in fair, repetitive play, there is no abandonment strategy that can improve your overall profit.

initial situation

One considers a set of possible points in time and a basic set of possible results. At every point in time there is a σ-algebra on which stands for the level of information at this point in time. Since the available information increases over time, apply to , that is , a filtration is on . In applications there is a probability space and it is . ${\displaystyle T}$ ${\displaystyle \Omega }$ ${\displaystyle t\in T}$ ${\displaystyle {\mathcal {A}}_{t}}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {A}}_{s}\subset {\mathcal {A}}_{t}}$ ${\displaystyle s<t}$ ${\displaystyle ({\mathcal {A}}_{t})_{t\in T}}$ ${\displaystyle \Omega }$ ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle {\mathcal {A}}_{t}\subset {\mathcal {A}}}$

At any point in time there is a measurable random variable , that is, there is an adapted stochastic process that can, for example, represent the payment of a game at the point in time . It is further assumed that there is a martingale ; the defining condition for expresses the fairness of the game: the forecast of the payout at the time under the information available is exactly the observation made at . In particular, the expected value at the point in time agrees with the initial expected value . ${\displaystyle t}$ ${\displaystyle {\mathcal {A}}_{t}}$ ${\displaystyle X_{t}:\Omega \rightarrow \mathbb {R} }$ ${\displaystyle (X_{t})_{t}}$ ${\displaystyle X_{t}}$ ${\displaystyle t}$ ${\displaystyle (X_{t})_{t}}$ ${\displaystyle E(X_{t}|{\mathcal {A}}_{s})=X_{s}}$ ${\displaystyle s<t}$ ${\displaystyle t}$ ${\displaystyle s}$ ${\displaystyle s}$ ${\displaystyle X_{s}}$ ${\displaystyle E(X_{t})}$ ${\displaystyle t}$ ${\displaystyle E(X_{0})}$

A stop time is a figure with . The idea behind this is to terminate the process at the point in time , which then leads to the result , which must be defined appropriately. Whether you stop at the point in time may only depend on the information available up to now , which explains the measurability condition set. ${\displaystyle \tau :\Omega \rightarrow T\cup \{\infty \}}$ ${\displaystyle \{\omega \in \Omega :\,\tau (\omega )\leq t\}\in {\mathcal {A}}_{t}}$ ${\displaystyle \tau (\omega )}$ ${\displaystyle X_{\tau (\omega )}(\omega )}$ ${\displaystyle X_{\infty }}$ ${\displaystyle t}$ ${\displaystyle t}$ ${\displaystyle \tau }$

The question now arises as to whether a better result than can be obtained by choosing a suitable stopping time . The Optional Sampling Theorem states that this is not the case under suitable conditions. ${\displaystyle E(X_{0})}$

Discreet version

If you consider a discrete sequence of points in time, you can model this through . The discrete version of the Optional Sampling Theorem states: ${\displaystyle T=\mathbb {N} }$

• If a filtration and an adapted martingale are on and if a stop time is with , and , then applies ${\displaystyle ({\mathcal {A}}_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (X_{n})_{n}}$ ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle \tau :\Omega \rightarrow \mathbb {N} \cup \{\infty \}}$ ${\displaystyle P(\tau <\infty )=1}$ ${\displaystyle E(|X_{\tau }|)<\infty }$ ${\displaystyle \int _{\{\tau >n\}}|X_{n}|{\rm {d}}P\,{\stackrel {n\to \infty }{\longrightarrow }}\,0}$

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

The technical requirements are met, especially for the realistic case of limited stopping times (you cannot wait forever!). ${\displaystyle \tau }$

The stop strategy of always betting on red in roulette , starting with one euro, doubling the stake each time and canceling the first appearance of red does not meet these technical requirements. Here, however, you have the unrealistic situation of an unlimited stopping time with exponentially growing stakes (at the "end" you win a total of one euro).

The following tightening for limited stopping times is also referred to as Optional Sampling Theorem:

• If a filtration and an adapted submartingale are on and if there are limited stop times, then the following applies ${\displaystyle ({\mathcal {A}}_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (X_{n})_{n}}$ ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle \sigma ,\tau :\Omega \rightarrow \mathbb {N} }$ ${\displaystyle \sigma \leq \tau }$

${\displaystyle X_{\sigma }\,\leq \,E(X_{\tau }|{\mathcal {A}}_{\sigma })}$.

It is the so-called σ-algebra of σ-past . If one sets special , then it is safe and it follows and after the application of the expected value . In the case of martingales this argument can also be applied to, and one obtains the statement of the first mentioned theorem for limited stop times. ${\displaystyle {\mathcal {A}}_{\sigma }}$ ${\displaystyle \sigma =0}$ ${\displaystyle \sigma \leq \tau }$ ${\displaystyle X_{0}\leq E(X_{\tau }|{\mathcal {A}}_{0})}$ ${\displaystyle E(X_{0})\leq E(X_{\tau })}$ ${\displaystyle (-X_{n})_{n}}$

• If a filtration and an adapted martingale are on and if there are limited stop times, then the following applies ${\displaystyle ({\mathcal {A}}_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (X_{n})_{n}}$ ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle \sigma ,\tau :\Omega \rightarrow \mathbb {N} }$ ${\displaystyle \sigma \leq \tau }$

${\displaystyle X_{\sigma }\,=\,E(X_{\tau }|{\mathcal {A}}_{\sigma })}$.

This follows immediately from the above inequality, because if a martingale is, then and are submartingales. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle -X}$

Continuous version

In the continuous-time case, which is modeled by, further technical prerequisites must be set that allow the proof to be traced back to the discrete case. Analogous to the discrete case, the following two sentences apply, which are also referred to as Optional Sampling Theorem. ${\displaystyle T=[0,\infty )}$

• If a filtration and an adapted martingale with right-hand continuous paths are on and if a stop time with , and , then applies ${\displaystyle ({\mathcal {A}}_{t})_{t\in [0,\infty )}}$ ${\displaystyle (X_{t})_{t}}$ ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle \tau :\Omega \rightarrow [0,\infty ]}$ ${\displaystyle P(\tau <\infty )=1}$ ${\displaystyle E(|X_{\tau }|)<\infty }$ ${\displaystyle \int _{\{\tau >t\}}|X_{t}|{\rm {d}}P\,{\stackrel {t\to \infty }{\longrightarrow }}\,0}$

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

• If a filtration and an adapted submartingale with continuous paths on the right side are on and limited stop times are also , then the following applies ${\displaystyle ({\mathcal {A}}_{t})_{t\in [0,\infty )}}$ ${\displaystyle (X_{t})_{t}}$ ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle \sigma ,\tau :\Omega \rightarrow [0,\infty )}$ ${\displaystyle \sigma \leq \tau }$ ${\displaystyle X_{\sigma }\,\leq \,E(X_{\tau }|{\mathcal {A}}_{\sigma })}$.

• If a filtration and an adapted martingale with continuous paths on the right are on and limited stop times are also , then the following applies ${\displaystyle ({\mathcal {A}}_{t})_{t\in [0,\infty )}}$ ${\displaystyle (X_{t})_{t}}$ ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle \sigma ,\tau :\Omega \rightarrow [0,\infty )}$ ${\displaystyle \sigma \leq \tau }$ ${\displaystyle X_{\sigma }\,=\,E(X_{\tau }|{\mathcal {A}}_{\sigma })}$.

See also

• Optional stopping theorem

swell

• Albrecht Irle : Finanzmathematik , Teubner-Verlag (2003), ISBN 3-519-12640-0