Optional sampling compares a martingale at random observation times. Stopping preserves the local martingale property automatically, but preserving expectations at an unbounded terminal time requires a genuine limiting condition. Most apparent “gambling paradoxes” are failures of uniform integrability or integrable stopping time assumptions.
Stopped processes
Definition. For a process \(X_n\) and a stopping time \(\tau\), the stopped process is \[ X_n^\tau=X_{n\wedge\tau}. \]
Proposition. If \(M_n\) is a martingale and \(\tau\) is a stopping time, then \(M_n^\tau\) is a martingale.
Proof. Its increment is \[ M_{n+1}^\tau-M_n^\tau =1_{\{\tau>n\}}(M_{n+1}-M_n). \] The event \(\{\tau>n\}\) belongs to \(\mathcal F_n\), so \[ \mathbb E[M_{n+1}^\tau-M_n^\tau\mid\mathcal F_n] = 1_{\{\tau>n\}} \mathbb E[M_{n+1}-M_n\mid\mathcal F_n]=0. \] Integrability follows because \(M_n^\tau=\sum_{k=0}^{n-1}M_k1_{\{\tau=k\}}+M_n1_{\{\tau\ge n\}}\).
Bounded optional sampling
Theorem (optional sampling). Let \(M\) be a martingale and let \(\sigma\le\tau\) be bounded stopping times. Then \[ \mathbb E[M_\tau\mid\mathcal F_\sigma]=M_\sigma \quad\text{almost surely}, \qquad \mathbb EM_\tau=\mathbb EM_\sigma=\mathbb EM_0. \] For a submartingale, the conditional equality is replaced by \(\ge\).
Proof. Choose \(N\) with \(\tau\le N\). For \(A\in\mathcal F_\sigma\), the definition of \(\mathcal F_\sigma\) implies \(A\cap\{\sigma=k\}\in\mathcal F_k\). Decompose \[ M_\tau-M_\sigma =\sum_{j=1}^N1_{\{\sigma<j\le\tau\}}\Delta M_j. \] The indicator \(1_A1_{\{\sigma<j\le\tau\}}\) is \(\mathcal F_{j-1}\)-measurable: by time \(j-1\) one knows whether \(\sigma<j\) and \(\tau\ge j\). Hence \[ \mathbb E[ 1_A1_{\{\sigma<j\le\tau\}}\Delta M_j]=0. \] Summation gives \(\mathbb E[1_AM_\tau]=\mathbb E[1_AM_\sigma]\), the defining identity for the conditional expectation. For a submartingale, each conditional increment has nonnegative expectation.
Unbounded stopping times
Theorem (uniformly integrable stopping). Let \(M\) be a martingale and let \(\tau<\infty\) almost surely. If \[ \{M_{n\wedge\tau}:n\ge0\} \quad\text{is uniformly integrable}, \] then \(M_\tau\in L^1\) and \[ \mathbb EM_\tau=\mathbb EM_0. \] More generally, if \(\sigma\le\tau\) and the stopped family is uniformly integrable, then \(\mathbb E[M_\tau\mid\mathcal F_\sigma]=M_\sigma\).
Proof. Since \(\tau<\infty\), \(M_{n\wedge\tau}\to M_\tau\) almost surely. Uniform integrability and Vitali's theorem upgrade this to \(L^1\)-convergence. Bounded optional stopping gives \(\mathbb EM_{n\wedge\tau}=\mathbb EM_0\); pass to the limit. For the conditional assertion, multiply by \(1_A\), \(A\in\mathcal F_\sigma\), apply bounded optional sampling to \(\sigma\wedge n\le\tau\wedge n\), and pass to the limit in \(L^1\).
Proposition (bounded increments and integrable time). Suppose \(|M_n-M_{n-1}|\le C\) almost surely and \(\mathbb E\tau<\infty\). Then \(\{M_{n\wedge\tau}\}\) is uniformly integrable and \(\mathbb EM_\tau=\mathbb EM_0\).
Proof. Pathwise, \[ |M_{n\wedge\tau}| \le |M_0|+C(n\wedge\tau) \le |M_0|+C\tau. \] The right side is integrable and independent of \(n\), so the stopped family is dominated by one \(L^1\) variable and is uniformly integrable.
Gambler's ruin
Example. Let \(S_n=i+\sum_{k=1}^n\xi_k\), where \(P(\xi_k=\pm1)=1/2\), and stop at \[ \tau=\inf\{n:S_n\in\{0,N\}\}, \qquad0<i<N. \] Because \(S_{n\wedge\tau}\in[0,N]\), the stopped martingale is uniformly integrable. Therefore \[ i=\mathbb ES_\tau =N\,P(S_\tau=N), \] so \[ P_i(S_\tau=N)=\frac{i}{N}. \]
Proposition (expected ruin time). In the same model, \[ \mathbb E_i\tau=i(N-i). \]
Proof. Since \(\mathbb E[\xi_{n+1}^2]=1\), the process \(S_n^2-n\) is a martingale. Apply bounded optional stopping to \(\tau\wedge m\): \[ \mathbb E S_{\tau\wedge m}^2-\mathbb E(\tau\wedge m)=i^2. \] The stopped positions are bounded by \(N\), and \(\tau\wedge m\uparrow\tau\). Dominated convergence on the first term and monotone convergence on the second give \[ \mathbb E\tau=\mathbb ES_\tau^2-i^2 =N^2P(S_\tau=N)-i^2 =Ni-i^2. \]
Why hypotheses cannot be dropped
Example (failure at an unbounded time). Let \(S_n\) be a simple symmetric random walk starting at zero and \[ \tau=\inf\{n\ge0:S_n=1\}. \] Recurrence gives \(\tau<\infty\) almost surely, yet \(S_\tau=1\), so \[ \mathbb ES_\tau=1\ne0=\mathbb ES_0. \] There is no contradiction: \(S_{n\wedge\tau}\to1\) almost surely but the stopped family is not uniformly integrable. Rare paths with a large negative position carry the missing expectation.
A reliable optional-stopping argument therefore has two separate parts: first apply the theorem to bounded truncations \(\tau\wedge n\); then justify passage to \(\tau\) by domination, uniform integrability, or another explicit convergence theorem.