Stopped Martingales at Bounded Stopping Times

What Stopping a Martingale Means

A stopping time chooses a random time using information available up to that time. Stopping a martingale at such a time freezes its value once the time has arrived. The key question here is whether the resulting process still has the defining conditional-expectation property of a martingale. Boundedness matters: it confines the argument to a finite time horizon, where the martingale’s terminal value can control the approximations.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\). A process \(M=(M_t)_{0\le t\le T}\) is a martingale if each \(M_t\) is integrable and \(\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s\) almost surely whenever \(0\le s\le t\le T\). Assume throughout the main result that \(M\) has right-continuous paths and that \(\tau\) is a stopping time satisfying \(0\le\tau\le T\) almost surely. Its stopped process is \(M_{t\wedge\tau}\), where \(t\wedge\tau=\min(t,\tau)\).

The result concerns the whole stopped process, not just the expectation at its terminal time. Its proof will first establish the claim for stopping times taking finitely many deterministic values. The general bounded case then follows by approximating \(\tau\) from above.

The Finite-Valued Argument

Let \(0=u_0<u_1<\cdots<u_N=T\) be a finite deterministic time grid, and set \(X_k=M_{u_k}\). The sequence \((X_k)\) is a discrete-time martingale with respect to \((\mathcal{F}_{u_k})\). A stopping time \(\kappa\) taking values in the grid can be represented by a random index \(K\in\{0,\ldots,N\}\), where \(\{K\le k\}=\{\kappa\le u_k\}\in\mathcal{F}_{u_k}\).

Proof. Each \(Y_k\) is integrable because it takes one of the finitely many integrable values \(X_0,\ldots,X_k\). On the event \(\{K\le k\}\), \(Y_{k+1}=Y_k=X_K\). On \(\{K>k\}\), the stopping index is at least \(k+1\), so \(Y_k=X_k\) and \(Y_{k+1}=X_{k+1}\). The event \(\{K>k\}\) belongs to \(\mathcal{F}_{u_k}\). Therefore \[ \mathbb{E}[Y_{k+1}\mid\mathcal{F}_{u_k}] =\mathbf{1}_{\{K\le k\}}X_K+\mathbf{1}_{\{K>k\}}\mathbb{E}[X_{k+1}\mid\mathcal{F}_{u_k}] =\mathbf{1}_{\{K\le k\}}X_K+\mathbf{1}_{\{K>k\}}X_k =Y_k. \] This proves the martingale property at every grid step. \(\square\)

This lemma also gives the martingale property at deterministic times \(s\le t\) for the process stopped at a grid-valued time. Include \(s,t\), and every possible value of that stopping time in a common finite grid. The lemma applies on the enlarged grid; iterating its conditional-expectation identity, and using the tower property, gives \(\mathbb{E}[M_{t\wedge\tau}\mid\mathcal{F}_s]=M_{s\wedge\tau}\).

Passing from Finite-Valued Times to Bounded Times

For \(T>0\), let \(\delta_n=T2^{-n}\), and round \(\tau\) up to the next grid point: \[ \tau_n=\delta_n\left\lceil\frac{\tau}{\delta_n}\right\rceil. \] Since \(T\) is on every grid and \(\tau\le T\), each \(\tau_n\le T\). Each \(\tau_n\) is a stopping time because, at a grid point \(k\delta_n\), \(\{\tau_n\le k\delta_n\}=\{\tau\le k\delta_n\}\). The nested dyadic grids ensure \(\tau_n\downarrow\tau\).

We need convergence in expectation, not just almost-sure convergence. For each fixed \(s\le T\), the finite-valued argument, applied with terminal time \(T\), gives \[ M_{s\wedge\tau_n}=\mathbb{E}[M_T\mid\mathcal{F}_{s\wedge\tau_n}]. \] The family on the right is uniformly integrable: conditional expectations of a fixed integrable random variable form a uniformly integrable family. Right-continuity of \(M\) and \((s\wedge\tau_n)\downarrow(s\wedge\tau)\) imply \(M_{s\wedge\tau_n}\to M_{s\wedge\tau}\) almost surely. The standard convergence theorem for uniformly integrable families therefore gives convergence in \(L^1\).

Proof of the Theorem. Fix deterministic \(0\le s\le t\le T\). The finite-valued argument gives \(\mathbb{E}[M_{t\wedge\tau_n}\mid\mathcal{F}_s]=M_{s\wedge\tau_n}\). For every \(A\in\mathcal{F}_s\), multiply by \(\mathbf{1}_A\) and take expectations. The \(L^1\) convergence just established lets us pass to the limit on both sides, giving \[ \mathbb{E}[\mathbf{1}_A M_{t\wedge\tau}] =\mathbb{E}[\mathbf{1}_A M_{s\wedge\tau}]. \] Both stopped variables are integrable, also by their \(L^1\) convergence. Since this identity holds for every \(A\in\mathcal{F}_s\), it is exactly the defining property \(\mathbb{E}[M_{t\wedge\tau}\mid\mathcal{F}_s]=M_{s\wedge\tau}\). Taking expectations proves \(\mathbb{E}[M_{t\wedge\tau}]=\mathbb{E}[M_0]\). \(\square\)

If \(T=0\), then \(\tau=0\) almost surely and the theorem is immediate. If the process is defined beyond \(T\), the stopped process is constant after \(T\), so the same martingale identity extends to later deterministic times. The boundedness hypothesis is essential to this argument: it is what supplies the integrable terminal variable \(M_T\) for the approximation. No conclusion for unbounded stopping times is being asserted here.

Worked Finite-Horizon Calculations

Common Pitfalls and Scope

A decision based on future information need not be a stopping time. For example, choosing \(\tau=0\) when \(B_1\ge0\) and \(\tau=1\) otherwise does not give a stopping time: for \(0\le t<1\), the event \(\{\tau\le t\}=\{B_1\ge0\}\) is generally not in \(\mathcal{F}_t\). The stopped-martingale theorem cannot be applied to this rule merely because its values lie in a bounded interval.

A stopped martingale is not automatically constant after a deterministic time unless it has been stopped there. Here it is \(M_{t\wedge\tau}\) that freezes at \(\tau\). The original \(M_t\) continues to evolve. Also, boundedness of \(\tau\) should not be silently dropped: the proof used \(M_T\) to control the approximating stopped values. Results for unbounded stopping times require additional hypotheses and are a separate question.

For any integrable martingale satisfying the theorem’s hypotheses, its stopped values retain the initial expectation. This does not mean that every function of the stopped value has the same expectation as the corresponding function of \(M_0\); the squared-Brownian example shows that \(\mathbb{E}[B_\tau^2]\) can be positive even though \(B_0^2=0\). Instead, apply the theorem to a process that is itself a martingale, such as \(B_t^2-t\), when that is the quantity whose expectation is needed.