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Concept
Optional stopping theorem
Summary
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. Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem says 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.
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 = (Xt)t∈0 be a discrete-time martingale and τ a stopping time with values in 0 ∪ {∞}, both with respect to a filtration (Ft)t∈0. Assume that one of the following three conditions holds:
() The stopping time τ is almost surely bounded, i.e., there exists a constant c ∈ such that τ ≤ c a.s.
() The stopping time τ has finite expectation and the conditional expectations of the absolute value of the martingale increments are almost surely bounded, more precisely, and there exists a constant c such that almost surely on the event {τ > t} for all t ∈ 0.
() There exists a constant c such that Xt∧τ ≤ c a.s. for all t ∈ 0 where ∧ denotes the minimum operator.
Then Xτ is an almost surely well defined random variable and
Similarly, if the stochastic process X = (Xt)t∈0 is a submartingale or a supermartingale and one of the above conditions holds, then
for a submartingale, and
for a supermartingale.
In the previous paragraphs, 0
denotes the set of natural integers, including zero.
Under condition () it is possible that τ = ∞ happens with positive probability. On this event Xτ is defined as the almost surely existing pointwise limit of (Xt)t∈0 , see the proof below for details.
Official source
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