A result in martingale theory stating that, under specified integrability or boundedness conditions on a stopping time τ and a martingale (M_t), the expectation at the stopping time equals the initial expectation: E[M_τ] = E[M_0].

Principle

A martingale's fair‑game property is preserved under stopping provided the stopping rule does not allow uncontrolled accumulation of variance or unbounded waiting with insufficient integrability.

Demonstration

If M_t is a martingale and τ is almost surely bounded by a constant T, then by optional sampling one has E[M_τ]=E[M_0]; e.g., a fair random walk stopped at a bounded stopping time has zero expected net gain.

Misapplication

Applying the theorem without verifying conditions (e.g., using an unbounded stopping time with unbounded increments) can lead to false conclusions—classical counterexamples show E[M_τ] may differ from E[M_0].

Consequence

Provides justification for many 'fair game' stopping arguments and is a foundational tool for proving identities and inequalities involving stopped processes.

Reversal

If integrability/boundedness conditions fail, optional stopping can fail and stopping can systematically change expectation, so stopping may introduce bias rather than preserve fairness.

Boundary

Requires a filtration, adapted martingale, and a stopping time; admissible conditions include bounded τ, uniform integrability of M_{t∧τ}, or integrable domination—cases outside these are excluded.

Semantic Tension

Tension between almost‑sure stopping and expectation preservation: almost‑sure finiteness of τ alone does not guarantee E[M_τ]=E[M_0] without additional integrability control.

Synthesis

The optional stopping theorem asserts that a martingale's expected value is preserved at a stopping time when appropriate boundedness or integrability conditions hold, formalizing when 'stopping' a fair game remains fair.