Start-Time Dependence Under Nonhomogeneous Rates

Why the Start Time Matters

In a time-homogeneous model, the transition probabilities over an interval depend on its length, not on when the interval begins. A one-year interval has the same transition matrix whether it starts at time 0 or time 5. That conclusion need not hold when intensities change with calendar time. Two intervals can have the same duration but expose the insured to different rates along the way.

The key comparison is therefore not just “How long is the interval?” but also “What rates apply during it?” This distinction matters when interpreting a transition matrix over a specified calendar period. A probability conditional on being in a state at time 2 is not automatically the same as the probability conditional on being in that state at time 0, even if both probabilities refer to the next year.

A transition intensity \(\lambda_{ij}(t)\) is an instantaneous rate, per unit of time, for a transition from state \(i\) to state \(j\) at calendar time \(t\). Informally, conditional on being in state \(i\) at time \(t\), the probability of that transition in the next short interval of length \(h\) is approximately \(\lambda_{ij}(t)h\). The rate can change while the process moves through an interval.

For the formulas and examples below, time is measured in years and intensities are rates per year. The examples specify the rates throughout the time periods considered. When comparing intervals, each probability is conditional on being in the stated starting state at the beginning of its own interval. It is not an unconditional probability for a population observed from some earlier time.

One-Way Transitions: Integrate the Rate Over the Interval

Consider a model with two states, \(A\) and \(B\), in which transition from \(A\) to \(B\) is possible but return from \(B\) to \(A\) is not. Suppose the intensity from \(A\) to \(B\) at time \(u\) is \(\lambda(u)\). The probability of remaining in \(A\) throughout the interval from \(s\) to \(t\) is determined by the accumulated intensity over that interval.

The accumulated intensity \(\int_s^t \lambda(u)\,du\) is sometimes called the integrated hazard. It is the area under the rate curve between the two calendar times. Equal interval lengths do not imply equal areas when the rate changes over time.

Why the Integrated-Intensity Formula Holds

For the one-way model, let \(S(s,t)\) be the probability of remaining in \(A\) from time \(s\) through time \(t\), given that the process is in \(A\) at \(s\). Over a short additional interval of length \(h\), the conditional probability of leaving \(A\) is \(\lambda(t)h+o(h)\), where \(o(h)\) is a term whose ratio to \(h\) tends to zero as \(h\) tends to zero. Therefore the probability of surviving that short interval is \(1-\lambda(t)h+o(h)\).

Conditioning on survival through time \(t\), the survival probability to \(t+h\) is consequently

Subtract \(S(s,t)\), divide by \(h\), and let \(h\) tend to zero. This gives the differential relation

Separating variables and integrating from \(s\) to \(t\) yields

The transition probability is the complementary event, so it is \(1-S(s,t)\). This derivation also explains the role of the start time: it determines the lower limit of the integral, and thus which part of the intensity function contributes.

Competing Destinations Also Depend on the Start Time

If a state has several possible destinations, leaving the state is not the same event as moving to one particular destination. The intensity for each destination contributes to the total rate of leaving. To find the probability of a particular destination, account both for its own intensity and for the possibility that another transition occurs first.

A useful special case occurs when the cause-specific intensities share the same time-varying factor. If \(\lambda_k(u)=c_k r(u)\), where the constants \(c_k\) are nonnegative and sum to 1, then the total intensity is \(r(u)\). Conditional on leaving during the interval, each destination has a constant share \(c_k\) of the total intensity. Its probability over \([s,t]\), starting in the original state, is \(c_k\left(1-\exp(-\int_s^t r(u)\,du)\right)\). This special-case formula relies on the constant shares; it is not a general formula for arbitrary competing intensities.

Direction of Change and a Common Exam Trap

When an intensity increases throughout time, later intervals of a fixed length will have a greater integrated intensity, and therefore a greater one-way transition probability. If the intensity decreases, the direction reverses. If the intensity varies irregularly, neither direction can be inferred from the interval start alone; the relevant comparison is between the integrated intensities. For multiple destinations, comparing a particular transition probability can also require comparing the cause-specific rates, not only the total rate.

A common shortcut is to multiply the intensity at the interval’s start by the interval length. That calculation is exact only when the intensity is constant over the interval. For example, in the disability model above, using the starting rate for \([0,1]\) gives \(0.02\) as the integrated intensity, whereas the correct integral is \(0.025\). The resulting transition probability, \(1-e^{-0.02}=0.0198\), is too small. For \([2,3]\), the same shortcut gives \(0.04\), rather than \(0.045\), and produces \(1-e^{-0.04}=0.0392\), also too small. Integrating the changing rate avoids both errors.

A further distinction is important when reading transition probabilities: the probability of being in a destination state at the end of an interval is not always the probability of ever visiting that state during the interval. In the one-way examples, once the process enters the destination it cannot return, so those events coincide. If return transitions or multiple routes are possible, an endpoint probability must account for all paths. The one-way integrated-intensity shortcut should not be applied to such a model without justification.