Optional stopping theorem

Not to be confused with Optimal stopping.

In probability theory, the optional stopping theorem (or sometimes Doob's optional sampling theorem, for American probabilist Joseph Doob) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value.

The concept can be understood through the following key principles:

  • Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem implies that, on average, nothing can be gained by stopping play based on the information obtainable so far (i.e., without looking into the future).
  • Certain conditions are necessary for this result to hold true. In particular, the theorem applies to doubling strategies and illustrates mathematically why such strategies cannot guarantee a profit with finite resources.
  • The optional stopping theorem is an important tool of mathematical finance in the context of the fundamental theorem of asset pricing, helping to evaluate the expected returns of various pricing models.
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