Counting-process notation separates an observed event history into a predictable compensator and an unpredictable martingale residual. It is not merely shorthand: predictability identifies which covariates may legitimately enter a hazard, brackets provide variance formulas, and martingale limit theorems supply the asymptotics of nonparametric and Cox estimators.
Filtrations, counting processes, and predictability
Definition. A filtration \((\mathcal F_t)_{t\ge0}\) is an increasing family of \(\sigma\)-fields satisfying the usual conditions. An adapted counting process \(N\) is càdlàg, integer-valued, nondecreasing, starts at zero, and has unit jumps unless ties or recurrent events are allowed. Its history is \[ \mathcal F_t^N=\sigma\{N(u):0\le u\le t\}, \] augmented in the usual way.
Definition. A process \(H\) is predictable if it is measurable for the predictable \(\sigma\)-field generated by adapted left-continuous processes. Informally, \(H(t)\) is determined immediately before time \(t\). A left-continuous risk indicator and a baseline covariate are predictable; \(1\{\Delta N(t)=1\}\) is not.
This distinction prevents using knowledge of the event occurring at \(t\) to predict that same event. Time-dependent covariates in hazard models must be predictable relative to the chosen filtration.
Compensators and intensities
Theorem (Doob-Meyer decomposition). Let \(X\) be a càdlàg submartingale of class \(D\). Then there is a unique decomposition \[ X(t)=X(0)+M(t)+A(t), \] where \(M\) is a càdlàg uniformly integrable martingale with \(M(0)=0\), and \(A\) is predictable, integrable, increasing, and satisfies \(A(0)=0\). For a counting process \(N\), \(A\) is its compensator. If \[ A(t)=\int_0^t\lambda(u)\,\mathrm{d}u, \] then the predictable nonnegative process \(\lambda\) is an intensity.
Proof outline. On a finite partition, subtract from each submartingale increment its conditional expectation given the preceding \(\sigma\)-field. The accumulated conditional expectations form a discrete predictable increasing process. Uniform integrability and weak compactness yield limits as partitions refine; right-continuous modification produces \(M\) and \(A\). If two decompositions exist, their difference is simultaneously a predictable finite-variation process and a martingale, hence is indistinguishably constant. This establishes uniqueness; existence in continuous time is the limiting part of the argument.
Proposition (multiplicative intensity model). Let \[ N_i(t)=1\{X_i\le t,\Delta_i=1\},\qquad Y_i(t)=1\{A_i\le t\le X_i\}. \] If subject \(i\), conditional on \(\mathcal F_{t-}\), has event hazard \(\alpha_i(t)\) whenever \(Y_i(t)=1\), and censoring is noninformative in this filtration, then \[ M_i(t)=N_i(t)-\int_0^tY_i(u)\alpha_i(u)\,\mathrm{d}u \] is a local martingale.
Proof. For bounded predictable simple \(H\), \[ E\int H\,\mathrm{d}N_i =E\int H(u)Y_i(u)\alpha_i(u)\,\mathrm{d}u \] by the defining conditional-hazard relation. Thus the signed random measure \(\mathrm{d}N_i-Y_i\alpha_i\,\mathrm{d}u\) integrates every bounded predictable process to mean zero. Localization handles an unbounded horizon or intensity and establishes the local-martingale property.
Example (one event). With no censoring and deterministic hazard \(\lambda\), \[ N(t)=1\{T\le t\},\qquad M(t)=N(t)-\int_0^{t\wedge T}\lambda(u)\,\mathrm{d}u. \] For \(s<t\), conditioning on \(T>s\) gives \[ E\{N(t)-N(s)\mid\mathcal F_s\} =P(s<T\le t\mid T>s) =E\left\{\int_s^{t\wedge T}\lambda(u)\,\mathrm{d}u\mid\mathcal F_s\right\}, \] which verifies the martingale increment relation directly.
Optional and predictable variation
Definition. For a locally square-integrable martingale \(M\), its optional quadratic variation \([M]\) is characterized by \(M^2-[M]\) being a local martingale. Its predictable quadratic variation \(\langle M\rangle\) is the compensator of \([M]\), so \(M^2-\langle M\rangle\) is a local martingale.
Proposition. If \(N\) has continuous compensator \(A(t)=\int_0^t\lambda(u)\,\mathrm{d}u\) and unit jumps, then for \(M=N-A\), \[ [M](t)=N(t),\qquad \langle M\rangle(t)=A(t). \] For distinct subjects whose counting processes have no simultaneous jumps and are conditionally independent, \(\langle M_i,M_j\rangle=0\).
Proof. Since \(A\) is continuous and of finite variation, \(\Delta M=\Delta N\), and the quadratic variation of \(M\) is \(\sum_{u\le t}(\Delta N(u))^2=N(t)\). The compensator of \(N\) is \(A\), hence \(\langle M\rangle=A\). No common jumps gives \([M_i,M_j]=0\), and its compensator is therefore zero.
Stochastic integration against event martingales
Theorem (martingale isometry). Let \(M\) be locally square-integrable and \(H\) predictable with \[ E\int_0^tH(u)^2\,\mathrm{d}\langle M\rangle(u)<\infty. \] Then \(H\mathbin{\boldsymbol\cdot}M(t)=\int_0^tH\,\mathrm{d}M\) is a square-integrable martingale and \[ \left\langle H\mathbin{\boldsymbol\cdot}M\right\rangle(t) =\int_0^tH(u)^2\,\mathrm{d}\langle M\rangle(u), \qquad E\left(H\mathbin{\boldsymbol\cdot}M(t)\right)^2 =E\int_0^tH^2\,\mathrm{d}\langle M\rangle. \]
Proof. For \(H=\sum_kH_k1_{(t_k,t_{k+1}]}\) with \(H_k\in\mathcal F_{t_k}\), the integral is a sum of martingale increments. Cross terms have expectation zero by iterated conditioning, while each diagonal term equals the corresponding increment of \(\langle M\rangle\). Approximate a general predictable \(H\) in \(L^2(P\otimes\mathrm{d}\langle M\rangle)\) by simple processes and use completeness.
Corollary. In the multiplicative intensity model, \[ \int_0^tH_i(u)\,\mathrm{d}M_i(u) \] has predictable variation \[ \int_0^tH_i(u)^2Y_i(u)\alpha_i(u)\,\mathrm{d}u. \] Consequently, sums of covariate-weighted residual increments have variance obtained by replacing the unobserved compensator with event increments. This is the origin of observed-information and sandwich estimators.
Likelihood for a multiplicative intensity
Theorem. Relative to a reference intensity, the log-likelihood contribution of a counting process with predictable intensity \(\lambda_\theta(t)\) is, up to a parameter-free term, \[ \ell_t(\theta) =\int_0^t\log\lambda_\theta(u)\,\mathrm{d}N(u) -\int_0^t\lambda_\theta(u)\,\mathrm{d}u. \] For independent subjects, the full log-likelihood is the sum over subjects.
Proof. On a fine partition, an interval containing a jump contributes approximately \(\lambda_\theta(t_k)\Delta t\), while an interval with no jump contributes \(1-\lambda_\theta(t_k)\Delta t\). Taking logarithms, summing, and sending the mesh to zero gives the two stochastic integrals. A rigorous derivation follows from the likelihood-ratio formula for point processes.
Martingale laws of large numbers and central limit theorems
Theorem (Lenglart inequality). If \(M\) is a locally square-integrable martingale and \(\eta,\delta>0\), then \[ P\left(\sup_{s\le t}|M(s)|\ge\eta\right) \le \frac{\delta}{\eta^2} +P\{\langle M\rangle(t)>\delta\}. \] Thus \(\langle M_n\rangle(t)\xrightarrow{P}0\) implies \(\sup_{s\le t}|M_n(s)|\xrightarrow{P}0\).
Proof. Stop at \(\tau=\inf\{s:|M(s)|\ge\eta\}\) and at the first time the bracket exceeds \(\delta\). Optional sampling applied to \(M^2-\langle M\rangle\) bounds \(\eta^2P(\tau\le t,\langle M\rangle(t)\le\delta)\) by \(\delta\). Add the probability that the bracket exceeds \(\delta\).
Theorem (Rebolledo martingale FCLT). Let \(M_n\) be \(d\)-dimensional locally square-integrable martingales. Suppose on \([0,\tau]\):
- \(\langle M_n\rangle(t)\xrightarrow{P}V(t)\) uniformly, where \(V\) is deterministic and continuous;
- for every \(\varepsilon>0\), the predictable contribution of jumps larger than \(\varepsilon\) converges to zero.
Then \[ M_n\rightsquigarrow G\quad\text{in }D([0,\tau],\mathbb R^d), \] where \(G\) is the continuous centered Gaussian martingale satisfying \(\operatorname{Cov}\{G(s),G(t)\}=V(s\wedge t)\).
Proof outline. Bracket convergence and Lenglart's inequality imply tightness through a stopping-time criterion. The jump condition makes every subsequential limit continuous. For each \(a\in\mathbb R^d\), the exponential-martingale expansion identifies \(a^\top M_n\) with a Gaussian martingale of variance \(a^\top Va\). Cramér-Wold identifies all finite-dimensional laws, and uniqueness of the Gaussian martingale problem identifies the process limit.
On later pages, every Cox score is written as \[ n^{-1/2}\sum_{i=1}^n\int_0^\tau \{Z_i(t)-\bar Z(t)\}\,\mathrm{d}M_i(t). \] The bracket determines its information matrix, and the martingale FCLT turns it into the Gaussian term in the estimator expansion.