Stochastic Integration Against Counting Processes

Why Integrate Against an Event Process?

A counting process records how many events have occurred by each time. In event-history work, however, the events need not contribute equally to a summary. A predictable weight can select particular times, give greater importance to some events, or represent a covariate known immediately before an event. Integrating that weight against a counting process provides a precise way to add its contributions over the observed events.

The corresponding integral against the compensator answers a different but related question. It accumulates the same predictable weights against the process's conditional event rate, rather than against the events that happened on the observed path. These two forms—the event sum and the cumulative-rate integral—are the central calculations in this tutorial.

Predictability is what makes the weighting rule appropriate for event-history analysis: the weight at an event time must be determined by information available just before that event, rather than by information revealed by the event itself or afterward. A left-continuous, adapted process is a common way to ensure predictability. For such a process, its value at \(T_k\) is the value in force immediately before the event. More generally, \(H(T_k)\) denotes the predictable value used by the stochastic integral.

For a counting process with finitely many events on every bounded time interval, the event-sum definition is intuitive and directly calculable. The more general construction begins with simple predictable processes and extends by suitable approximation. The next derivation shows why the event-sum formula follows from that construction.

From Simple Processes to Event Sums

A simple predictable process is constant on time intervals, with each interval's value already known at its start. For example, on a partition \(0=s_0<s_1<\cdots<s_m\), write \(H(u)=\sum_{j=0}^{m-1}\xi_j\,1_{(s_j,s_{j+1}]}(u)\), where \(\xi_j\) is known at time \(s_j\). The integral on \((0,t]\) is defined by multiplying each interval's value by the number of events in the interval, truncated at \(t\).

Each difference in braces counts events in \((s_j,s_{j+1}]\) up to time \(t\). Thus an event in that interval receives weight \(\xi_j\), the value fixed at the interval's start. Rewriting the right-hand side by grouping contributions by event gives the sum of the predictable weight at each event. Predictable integrands that are not simple are handled by approximating them with simple predictable processes and taking an appropriate limit; under the usual local boundedness and finite-activity conditions, this agrees with the pathwise event sum.

This is a pathwise identity: it describes the realized integral for the observed event history. It does not claim that a weighted event count equals its model-based expectation. For example, a weight of zero removes an event's contribution, while a negative weight subtracts from the total. The integral need not be an integer or even be nonnegative.

Integration Against a Compensator

Let \(A(t)\) be the compensator of \(N(t)\): the predictable increasing process that accounts for the event process's cumulative conditional rate. When the compensator has an intensity \(\lambda(t)\), its cumulative form is \(A(t)=\int_0^t\lambda(s)\,ds\). Here \(\lambda(s)\) is the predictable event rate, measured in expected events per unit time, conditional on the available history. Integration against \(A\) accumulates the weight against this rate.

For a simple process with value \(\xi_j\) on \((s_j,s_{j+1}]\), integration against the compensator is defined in parallel with integration against \(N\): $$ \int_0^t H(u)\,dA(u) = \sum_j \xi_j\left\{A(t\wedge s_{j+1})-A(t\wedge s_j)\right\}. $$ When \(A\) is absolutely continuous, the increment of \(A\) on each interval is the integral of \(\lambda\) over that interval. This gives the cumulative-rate expression above. Thus the two integrals have matching interval structure, but one counts realized events and the other accumulates conditional rate.

The compensator property also links these calculations: for a nonnegative predictable \(H\), or an integrable predictable \(H\) for which the expectations exist, \(E\{\int_0^t H(s)\,dN(s)\}=E\{\int_0^t H(s)\,dA(s)\}\). This is an expectation identity, not a pathwise equality. On a particular path, the event sum can be larger or smaller than the cumulative-rate integral.

Predictability and Integrability Conditions

The calculations above rely on assumptions that should be checked before applying the formulas. A useful standard setup is a simple, nonexplosive counting process with \(N(0)=0\), so that there are finitely many unit jumps on every bounded time interval. The integrand \(H\) is predictable, and its pathwise event sum is finite on the time interval of interest. Local boundedness of \(H\) is a common sufficient condition when \(N\) has finitely many events there.

For integration against \(A\), the Stieltjes integral must be defined and finite on the interval. If \(A(t)=\int_0^t\lambda(s)\,ds\), a common sufficient condition is that \(\int_0^t |H(s)|\lambda(s)\,ds\) is finite. To use the expectation identity, stronger integrability may be needed: for example, require the expected absolute integral against \(A\) to be finite. Nonnegative predictable \(H\) permits the identity in the extended nonnegative sense, even if the expectations are infinite.

In applications, at-risk indicators and covariates are frequent integrands. An at-risk indicator must turn off when observation ends, and a time-varying covariate used as a weight must be based on its predictable, pre-event value. These conditions do not establish that a chosen intensity model is scientifically correct; they specify what is required for the integral calculations and compensator interpretation to be valid.

Common Pitfalls and Practical Checks

• Using a weight that looks into the future. If an event's occurrence determines the weight assigned to that same event, the weight may not be predictable. Use information available just before the event, or justify a different model and construction.
• Replacing the event sum with the rate integral. The former uses the observed jumps; the latter uses the compensator. They are not interchangeable on an individual path.
• Dropping jump sizes. The formula \(\sum H(T_k)\) relies on unit jumps. For a process with a jump of size 3, that jump contributes \(3H(T_k)\).
• Ignoring interval endpoints. The convention \((s_j,s_{j+1}]\) assigns an event at \(s_{j+1}\) to that interval. Use a consistent convention so boundary events are neither omitted nor counted twice.
• Reporting a rate integral without checking finiteness. A large or unbounded weight can make \(\int |H|\lambda\,ds\) infinite even on a finite time interval. Check the relevant pathwise or expectation condition.

A practical diagnostic is to compare the event times with the intervals on which \(H\) changes. Every event should receive exactly one weight, and that weight should be supported by information available before the event. Separately, check that the intensity is expressed per unit of the same time scale used in the integration. A rate per day integrated over hours, for example, requires a conversion before the numerical result has the intended units.

The main alternative to stochastic integration is to report an unweighted count, \(N(t)\), or to calculate a cumulative intensity, \(A(t)\), without weights. Those summaries are appropriate when every event should count equally or when the target is the model's accumulated rate. Stochastic integrals extend both summaries by allowing predictable weights, but they do not remove the need to define the target carefully. A weighted event total, a weighted cumulative rate, and an unweighted event count answer different questions.

The central limitation is that these integrals summarize a specified process and weight; they do not by themselves validate the event-rate model, identify a causal effect, or resolve complications such as observation error. Their value is algebraic and conceptual: they make explicit how event-level contributions and conditional rates are accumulated. The later treatment of martingale transforms builds on this integration framework, but here the focus is the construction and calculation of the integrals themselves.