Lagmental Vicfred

Optional Stopping Preserves Expectation under Safe Hypotheses by Vicfred

tags: mathematics, martingales, branching processes, optional stopping

A bounded stopping time applied to an integrable martingale has the same expected value as the start. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Definitions first

A process \((M_n,\mathcal F_n)\) is a martingale when \(\mathbf E[M_{n+1}\mid\mathcal F_n]=M_n\). Its baseline \(M_0\) models fair evolution relative to the information currently available.

$$ \tau\le N\ \text{a stopping time} $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ \mathbf E[M_\tau]=\mathbf E[M_0] $$

A small case in full

A worked instance is useful here because it exposes every index that the compressed statement hides.

$$ M_\tau=M_0+\sum_{k=1}^{N}(M_k-M_{k-1})\mathbf1_{\{\tau\ge k\}},\qquad\mathbf E[(M_k-M_{k-1})\mathbf1_{\{\tau\ge k\}}]=0 $$

The reusable statement

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad \tau\le N\ \text{a stopping time},\\[5pt] \mathsf{C}\;&:\quad \mathbf E[M_\tau]=\mathbf E[M_0]. \end{aligned} $$

A nearby false statement

Optional stopping is not valid for every stopping time. Boundedness, integrability, or uniform-integrability conditions prevent hidden mass from escaping at infinity.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \mathbf E[M_\tau]=\mathbf E[M_0] \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Thu 19 October 2023. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.