Optional stopping theorem

In probability theory, optional stopping theorem (or Doob's optional sampling theorem) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to the expected value of its initial value. Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem says that on the average nothing can be gained by stopping to play the game based on the information obtainable so far (i.e., by not looking into the future). Of course, certain conditions are necessary for this result to hold true, in particular doubling strategies have to be excluded.

The optional stopping theorem is an important tool of mathematical finance in the context of the fundamental theorem of asset pricing.

A discrete-time version of the theorem is given below:

Let X = (X_t)_t∈ℕ0 be a discrete-time martingale and τ a stopping time with values in ℕ₀ ∪ {∞}, both with respect to a filtration (F_t)_t∈ℕ0. Assume that one of the following three conditions holds:

Then X_τ is an almost surely well defined random variable and ${\displaystyle \mathbb {E} [X_{\tau }]=\mathbb {E} [X_{0}].}$

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