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.
Assume that one of the following three conditions holds: Then Xτ is an almost surely well defined random variable and
On this event Xτ is defined as the almost surely existing pointwise limit of (Xt)t∈
Let Xτ denote the stopped process, it is also a martingale (or a submartingale or supermartingale, respectively).
0, where By the monotone convergence theorem If condition (a) holds, then this series only has a finite number of non-zero terms, hence M is integrable.
If condition (b) holds, then we continue by inserting a conditional expectation and using that the event {τ > s} is known at time s (note that τ is assumed to be a stopping time with respect to the filtration), hence where a representation of the expected value of non-negative integer-valued random variables is used for the last equality.
Therefore, under any one of the three conditions in the theorem, the stopped process is dominated by an integrable random variable M. Since the stopped process Xτ converges almost surely to Xτ, the dominated convergence theorem implies By the martingale property of the stopped process, hence Similarly, if X is a submartingale or supermartingale, respectively, change the equality in the last two formulas to the appropriate inequality.