OptionalStoppingTheorem - crowlogic/arb4j GitHub Wiki

The optional stopping theorem is an important result in probability theory related to stochastic processes, particularly in the context of martingales. The theorem provides conditions under which the expectation of a martingale at a stopping time remains unchanged. In other words, it states when it is possible to take the expected value of a martingale at a stopping time and get the same result as taking the expected value at the starting time.

Suppose $(M_t, \mathcal{F}_t)$ is a martingale or a submartingale (respectively, a supermartingale) with respect to a filtration $\mathcal{F}_t$, and $\tau$ is a stopping time with respect to the same filtration. The optional stopping theorem states that under certain conditions on the stopping time, the following relation holds:

• If $M_t$ is a martingale, then $\mathbb{E}[M_{\tau}] = \mathbb{E}[M_0]$.
• If $M_t$ is a submartingale, then $\mathbb{E}[M_{\tau}] \geq \mathbb{E}[M_0]$.
• If $M_t$ is a supermartingale, then $\mathbb{E}[M_{\tau}] \leq \mathbb{E}[M_0]$.

There are several versions of the optional stopping theorem, which differ in the specific conditions they impose on the stopping time. Some common conditions include:

• Bounded stopping time: $\tau \leq T$ for some constant $T$.
• Bounded increments: $|M_{t+1} - M_t| \leq C$ for some constant $C$ and all $t$.
• Integrability condition: $\mathbb{E}[\tau] < \infty$.

The optional stopping theorem has a wide range of applications in probability theory, including gambling strategies, financial mathematics, and statistical estimation.