Optional stopping theorem

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, with ${\displaystyle \mathbb {N} _{0}}$ denoting the set of natural numbers, including zero.

Let ${\displaystyle X=(X_{t})_{t\in \mathbb {N} _{0}}}$ be a discrete-time martingale and ${\displaystyle \tau }$ a stopping time with values in ${\displaystyle \mathbb {N} _{0}\cup \{\infty \}}$, both with respect to a filtration ${\displaystyle ({\mathcal {F}}_{t})_{t\in \mathbb {N} _{0}}}$. Assume that one of the following three conditions holds:

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

Similarly, if the stochastic process ${\displaystyle X=(X_{t})_{t\in \mathbb {N} _{0}}}$ is a submartingale or a supermartingale and one of the above conditions holds, then $${\displaystyle \mathbb {E} [X_{\tau }]\geq \mathbb {E} [X_{0}]}$$ for a submartingale, and $${\displaystyle \mathbb {E} [X_{\tau }]\leq \mathbb {E} [X_{0}]}$$ for a supermartingale.

Under condition (c) it is possible that ${\displaystyle \tau =\infty }$ happens with positive probability. On this event ${\displaystyle X_{\tau }}$ is defined as the almost surely existing pointwise limit of ${\displaystyle X=(X_{t})_{t\in \mathbb {N} _{0}}}$. See the proof below for details.

Let ${\displaystyle X^{\tau }}$ denote the stopped process, it is also a martingale (or a submartingale or supermartingale, respectively). Under condition (a) or (b), the random variable ${\displaystyle X^{\tau }}$ is well defined. Under condition (c) the stopped process ${\displaystyle X^{\tau }}$ is bounded, hence by Doob's martingale convergence theorem it converges almost surely pointwise to a random variable which we call ${\displaystyle X_{\tau }}$.