Continuous-time martingale theory adds two issues absent from a finite discrete grid: one must choose path-regular versions, and many natural stochastic integrals are martingales only after stopping. Localization permits bounded arguments to be reused without pretending that local and true martingales have the same expectation behavior.
Martingales and regular versions
Definition. An adapted process \(M=(M_t)_{t\ge0}\) is a martingale if \(M_t\in L^1\) and \[ \mathbb E[M_t\mid\mathcal F_s]=M_s \qquad(0\le s\le t). \] Conditional inequalities define submartingales and supermartingales.
Theorem (Doob regularization). Under the usual conditions on the filtration, every martingale whose mean function \(t\mapsto\mathbb EM_t\) is right-continuous has a cadlag modification. In particular, martingales are normally identified with their cadlag versions.
Proof outline. Restrict the process to dyadic rational times. The discrete upcrossing inequality shows that, simultaneously for every rational interval \([a,b]\), only finitely many upcrossings occur near each time. Therefore left and right limits exist along dyadic rationals outside one null set. Define the right-limit process there. Right continuity of the filtration makes the limit adapted, while conditional Fatou and right continuity of expectations identify it with the original variables and preserve the martingale property.
Localization
Definition. An adapted cadlag process \(M\) is a local martingale if there are stopping times \(\tau_n\uparrow\infty\) almost surely such that \[ M_t^{\tau_n}:=M_{t\wedge\tau_n} \] is a martingale for every \(n\). The sequence \((\tau_n)\) is a localizing sequence.
Proposition. If \(M\) is a continuous local martingale, then \[ \tau_n=\inf\{t\ge0:|M_t|\ge n\}\wedge n \] localizes \(M\) after intersecting with any original localizing sequence. The stopped processes are bounded on compact time intervals.
Proof. The hitting times are stopping times by continuity. If \(\sigma_n\) is an original localizing sequence, then \(\rho_n=\sigma_n\wedge\tau_n\) still increases to infinity. A stopped martingale stopped again remains a martingale, and \(|M_{t\wedge\rho_n}|\le n\) up to the harmless terminal continuity value.
A local martingale identity may be used safely after first inserting \(\tau_n\). Removing the stopping requires a separate convergence argument; this is exactly where strict local martingales arise.
Nonnegative local martingales
Theorem. Every nonnegative local martingale is a supermartingale. Hence \[ \mathbb EM_t\le\mathbb EM_s \qquad(s\le t). \]
Proof. Let \(\tau_n\) localize \(M\). Conditional Fatou gives \[ \mathbb E[M_t\mid\mathcal F_s] = \mathbb E[\liminf_nM_{t\wedge\tau_n}\mid\mathcal F_s] \le\liminf_n \mathbb E[M_{t\wedge\tau_n}\mid\mathcal F_s]. \] Optional sampling for the stopped martingale gives the last conditional expectation as \(M_{s\wedge\tau_n}\), whose limit is \(M_s\). Thus \(\mathbb E[M_t\mid\mathcal F_s]\le M_s\).
Definition. A local martingale that is not a martingale is a strict local martingale. For a nonnegative local martingale, strictness is visible through loss of mean: \[ \mathbb EM_t<\mathbb EM_0 \] at some time.
Example (inverse Bessel process). Let \(R_t\) be a three-dimensional Bessel process started from \(r>0\). Ito's formula shows that \[ M_t=\frac1{R_t} \] is a positive local martingale. Three-dimensional transience implies \(R_t\to\infty\) in probability, and the transition density calculation gives \(\mathbb E(1/R_t)<1/r\) for \(t>0\). Thus \(M\) is strict. The example warns that zero local drift does not alone preserve expectation.
Class \(D\) and closed martingales
Definition. An adapted process \(X\) is of class \(D\) on \([0,T]\) if \[ \{X_\tau:\tau\le T\text{ is a stopping time}\} \] is uniformly integrable.
Theorem. A cadlag local martingale of class \(D\) on \([0,T]\) is a uniformly integrable martingale there. Moreover, the following are equivalent for a cadlag martingale:
- \(M\) is of class \(D\);
- there is \(M_T\in L^1\) such that \[ M_t=\mathbb E[M_T\mid\mathcal F_t],\qquad0\le t\le T. \]
Proof. Let \(\tau_n\) localize \(M\). For \(s<t\le T\), \[ \mathbb E[M_{t\wedge\tau_n}\mid\mathcal F_s] =M_{s\wedge\tau_n}. \] Both stopped variables belong to the class-\(D\) family, so uniform integrability and almost-sure convergence give \(L^1\)-convergence as \(n\to\infty\). Passing to the limit proves the true martingale identity. Optional sampling then gives \(M_t=\mathbb E[M_T\mid\mathcal F_t]\). Conversely, conditional expectations of one \(L^1\) variable over arbitrary stopping-time sigma-fields are uniformly integrable, so the representation implies class \(D\).
Proposition (useful sufficient conditions). A local martingale \(M\) is a true martingale on \([0,T]\) if either:
- \(\sup_{t\le T}\mathbb E|M_t|^p<\infty\) for some \(p>1\); or
- \(M\) is square-integrable after localization and \(\mathbb E\langle M\rangle_T<\infty\).
Proof. The first condition gives uniform integrability by the \(L^p\) criterion. Under the second, the martingale isometry gives \[ \mathbb E|M_{t\wedge\tau_n}-M_0|^2 =\mathbb E\langle M\rangle_{t\wedge\tau_n} \le\mathbb E\langle M\rangle_T, \] uniformly in \(n,t\). Thus the localized family is uniformly integrable, and the preceding theorem removes localization.