##  [Optional Stopping Theorem](/optional-stopping-theorem-0) 

 Definition

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

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

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

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

Consequence

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

 

 

 

 

## Reversal

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

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

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

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.