Martingale Isometry for Variance Derivation

Why Predictable Variation Gives a Variance

An event count fluctuates around its compensator: the part of its cumulative expected count determined by the event intensity and the information available over time. Subtracting that compensator from the count produces a martingale. A predictable integral against this martingale is therefore a centered accumulation of weighted event-count fluctuations. The martingale isometry turns its variance into an expectation involving the squared weights and the process's predictable variation.

This result is useful because it gives a direct route from a specified intensity model to the variance of an event-history quantity. It also clarifies why the weights must be predictable: they may use information available just before each time point, but not the future event increments they are weighting.

Here, a process is predictable if its value at time \(t\) is determined by information available immediately before \(t\). A risk-set indicator and a weight determined by baseline covariates are common examples. Throughout, the basic formula assumes unit jumps of \(N\), no simultaneous multiple jumps within this counting process, and a continuous compensator. These are the usual conditions for a counting process with an intensity.

The Isometry and Its Conditions

The martingale isometry states that a predictable integral has second moment equal to the expected predictable variation of that integral. For a square-integrable integrand over \([0,\tau]\), the result is

A sufficient condition for this finite-variance statement is \(E[\int_0^\tau H(u)^2\,dA(u)]<\infty\). Under the martingale assumptions above, the integral has mean zero. Its variance is therefore the displayed second moment. For an intensity expressed as \(A(t)=\int_0^t \lambda(u)\,du\), the same result can be written as

The right side is an expectation: the predictable variation \(\int H^2\,dA\) is generally random when \(H\) or \(A\) is random. It is not correct to treat one observed value of that variation as automatically equal to the sampling variance.

Why the Formula Holds

The core argument is that martingale increments over disjoint intervals are orthogonal in expectation, and the conditional second moment of a martingale increment is governed by its predictable variation. To see the structure, first take a simple predictable integrand that is constant on successive intervals:

The corresponding integral is \(\sum_k \xi_k\{M(t_{k+1})-M(t_k)\}\). For \(j<k\), the earlier increment and \(\xi_j\) are known by time \(t_k\). Since the conditional mean of the later martingale increment given \(\mathcal{F}_{t_k}\) is zero, the expectation of the cross-product of these two terms is zero. Thus only the squared increments contribute to the second moment. For each interval, the martingale square identity gives

Multiplying by the \(\mathcal{F}_{t_k}\)-measurable \(\xi_k^2\), taking expectations, and summing yields

where \(\Delta_k M=M(t_{k+1})-M(t_k)\). The expression on the right is the expected integral of \(H^2\) against \(\langle M\rangle\). General predictable square-integrable integrands are obtained by approximating them with simple predictable integrands in this squared-integral norm. The equality extends by that approximation, giving the isometry.

For a simple unit-jump counting process with continuous \(A\), each jump of \(M=N-A\) is a unit jump at an event time. Its realized quadratic variation is \([M](t)=N(t)\), while its predictable variation is \(\langle M\rangle(t)=A(t)\). For the weighted integral, the corresponding predictable variation is \(\int_0^t H(u)^2\,dA(u)\). This explains why the variance uses the compensator rather than the realized event count.

Risk-Set Weights and Random Predictable Weights

In event-history applications, the intensity often has the multiplicative form \(\lambda_N(t)=Y(t)\lambda(t)\), where \(Y(t)\) is a predictable at-risk process and \(\lambda(t)\) is an event rate per at-risk unit. A common weight is \(H(t)=1/Y(t)\), used only while \(Y(t)>0\). The isometry then reduces the variance calculation to

with the integrand defined as zero when the process is not being integrated and \(Y(u)=0\). This calculation requires that the risk-set process be predictable and that the expected integral be finite. It does not require the risk set to be deterministic.

A predictable weight need not be fixed in advance. It may depend on observed history, provided its value at each time is determined by information just before that time. The isometry then averages over the randomness in both the event process and the predictable weight. Conditioning on baseline information can make that averaging especially transparent.

Multiple Event Types and Covariance

For several compensated counting processes \(M_j=N_j-A_j\), the isometry also describes covariances. If \(H_j\) are predictable and the integrals are square-integrable, then

Here \(\langle M_j,M_k\rangle\) is the predictable covariation, the predictable compensator of the joint quadratic variation. For distinct event types that cannot jump simultaneously, with continuous compensators, this cross predictable variation is zero. Their martingale integrals are then uncorrelated, though that statement alone does not assert independence. If simultaneous jumps are possible, the cross term must be considered rather than discarded.

Checks, Failure Modes, and Alternatives

Before using the isometry, verify that the process being compensated is a martingale under the stated information and intensity model; that the weight is predictable; and that the expected weighted predictable variation is finite. For a risk-set weight, also check that the weight is only used where the risk set is positive. A formula may be algebraically correct but still describe the wrong variance if any of these conditions fails.

A useful distinction is between predictable variation and realized quadratic variation. For a simple counting-process martingale, the realized quadratic variation of the unweighted martingale is the observed count \(N(\tau)\), whereas the predictable variation is its compensator \(A(\tau)\). For a weighted integral, the realized quadratic variation is \(\int H^2\,dN\); the predictable variation is \(\int H^2\,dA\). The former is observable from the path, while the isometry expresses the sampling variance as the expectation of the latter. In some settings the realized quantity can serve as a data-based estimate of variation, but it is not the identity that derives the variance.

Another route in simple models is to calculate the variance directly from a known count distribution or independent increments. The isometry is more general: it handles predictable weights and random risk sets without requiring a closed-form distribution for the full count. It is still model-dependent, because the compensator must correctly represent the event intensity. If that intensity or the filtration is misspecified, the resulting variance need not describe the actual sampling variation.

Common warning signs include a weight that uses whether an event occurs at the same time it is applied, a zero risk set in a denominator, or an infinite or undefined expected value of \(\int H^2\,dA\). In these cases the standard finite-variance result cannot simply be invoked. If the expected variation is infinite, the integral may still be defined under weaker local conditions, but this formula does not establish that it has a finite variance.