Bounded Optional Stopping

Stop a small random walk at a barrier or deadline and verify the drift-compensated martingale identity exactly.

Bounded Optional Stopping

Concept Overview

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.

Worked Example: Stopping by n = 2

Consider a random walk with p = 0.75, an upper barrier A = 2, lower magnitude B = 1, and horizon n = 2. Let's enumerate the paths to find E[τ] and E[S_τ]. The initial state is S_0 = 0.

Step 1: The walk can move to +1 (prob 0.75) or -1 (prob 0.25). Since B = 1, the walk hitting -1 is absorbed immediately. Stopping time τ = 1 for this path.
Step 2: For the path at +1, it can move to +2 (prob 0.75 * 0.75 = 0.5625) or 0 (prob 0.75 * 0.25 = 0.1875). At +2, it hits A = 2, so it's absorbed at step 2. At 0, the deadline n = 2 is reached, so it also stops at step 2.

We can collect the stopped states and their probabilities:

Stopped Position (S_τ)	Stopping Time (τ)	Probability
-1	1	0.25
2	2	0.5625
0	2	0.1875

Calculating expectations:

E[τ] = 1 * 0.25 + 2 * 0.5625 + 2 * 0.1875 = 0.25 + 1.125 + 0.375 = 1.75
E[S_τ] = (-1) * 0.25 + 2 * 0.5625 + 0 * 0.1875 = -0.25 + 1.125 + 0 = 0.875

The drift per step is (2p - 1) = 2(0.75) - 1 = 0.5. The compensated martingale expectation is E[S_τ - drift * τ] = E[S_τ] - drift * E[τ] = 0.875 - 0.5 * 1.75 = 0.875 - 0.875 = 0. This confirms the bounded optional stopping theorem.

Experimenting with the Module

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

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.
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.
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.

Limitations of the Model

This demonstration focuses purely on bounded stopping times, meaning τ ≤ n is guaranteed. This finite computation does not establish an unbounded optional-stopping claim. An unbounded rule requires the additional hypotheses of an applicable theorem; a future-looking last-visit rule need not even be a stopping time.

Additionally, this model assumes independent increments and constant transition probabilities, which might not accurately represent systems with memory or time-varying dynamics.

References

For a rigorous treatment of martingales and optional stopping, refer to the following external resources:

Ellen Powell's stochastic process notes on stopped martingales and bounded-time Optional Stopping Theorem. Available at: https://www.maths.dur.ac.uk/users/ellen.g.powell/SPnotes.html

Bounded Optional Stopping