Bounded Optional Stopping

The Optional Stopping Theorem is a crucial concept in probability theory, especially concerning martingales. It gives conditions under which the expected value of a martingale at a stopping time is equal to its initial expected value. This module focuses on the bounded-time version of the theorem. We examine a biased random walk that hits absorbing barriers at +A or -B, but is strictly stopped at a deterministic deadline n. This demonstrates that while the position of a biased walk is not a martingale, a drift-compensated process is, and bounded optional stopping always holds for it.

Experimenting with the Module

Open the interactive module to visualize these calculations across different parameters.

1. Select Up probability: Change the increment drift. Notice how the drift per step changes. The expected position E[S_τ] adjusts precisely to maintain the martingale property E[S_τ - drift * τ] = 0.
2. Set Horizon n: As you increase n, the probability of hitting a barrier before the deadline increases, changing E[τ] and E[S_τ], but the compensated expectation remains 0. Set n=0 to see all expectations vanish immediately.
3. Adjust Upper barrier A and Lower magnitude B: Moving the upper absorbing level or the lower absorbing level changes the absorption conditions, which in turn alters the stopped-position chart and the expandable Stopped state-time law table.

The chart illustrates the probability mass function of the position at the stopping time τ. Note that there are peaks at the absorbing boundaries, and some probability mass spread across the interior states if they are reached exactly at the deadline n.