Why a Short-Interval Approximation Is Useful
In a continuous-time state model, a transition intensity describes the instantaneous rate of movement from one state to another. Calculations often require a probability over a short, but nonzero, interval. The key connection is that the probability of a particular transition is approximately its intensity at the start of the interval multiplied by the interval length.
For example, an actuarial model might distinguish a person who is active from one who is disabled. If the person is active at time \(t\), the transition intensity from active to disabled determines the leading-order probability of that transition during the next short interval. The approximation is local: it uses the intensity at the interval's starting time and is most informative when the interval is sufficiently short.
The conditioning on \(X(t)=i\) matters: the approximation concerns a person or policy in state \(i\) at time \(t\), not an arbitrary member of the original population. In a Markov state model, given the current state and time, the future transition behaviour does not depend on the earlier path. The short-interval argument below uses the transition intensity conditional on the current state.
From the Limit to the First-Order Term
The definition of intensity says that the ratio \(p_{ij}(t,h)/h\) approaches \(\mu_{ij}(t)\) as \(h\) decreases to zero. Rewriting that limit gives the approximation. To make the reasoning precise, let the difference between the ratio and its limit be \(\epsilon(h)\). Then \(\epsilon(h)\to 0\), and
The term \(h\epsilon(h)\) is written \(o(h)\): its ratio to \(h\) tends to zero as \(h\downarrow0\). This establishes the first-order expansion
This is a limit statement, not a claim that the omitted term is always zero. The approximation keeps the term proportional to \(h\) and discards terms that become negligible relative to \(h\). In a regular finite-state jump model with locally bounded transition intensities, the probability of two or more jumps within an interval of length \(h\) is of smaller order than \(h\). Thus the leading term describes the transition from \(i\) to \(j\); paths involving multiple jumps do not change that first-order term.
Worked Applications
Each example applies the same method: identify the state at the start, use the intensity for the specified transition at that time, and multiply by the short interval length. A probability approximation must be nonnegative and, for a sufficiently short interval, should be small when the intensity-times-interval product is small.
Worked Example: Active to Disabled Over a Short Interval
Consider a hypothetical disability-income model. At time \(t\), a policyholder is active. The transition intensity from active to disabled at that time is \(0.08\) per year. Estimate the probability of being disabled at the end of the next \(0.01\) years, conditional on being active now. State the first-order approximation and identify a common mistake to avoid.
Method. Use the transition intensity for the specified transition, active to disabled, at the start of the interval. The interval length is \(h=0.01\) years.
The first-order product is $$ \mu_{ij}(t)h=0.08(0.01)=0.0008. $$ Therefore, $$ P\bigl(X(t+0.01)=\text{disabled}\mid X(t)=\text{active}\bigr) \approx 0.0008. $$
As a percentage, this is approximately \(0.08\%\). The result is a short-interval approximation; no rounding beyond the displayed value is needed because the multiplication is exact for the first-order term.
Check. The estimate is between zero and one, and is small for the short interval given. Common mistake avoided: using an intensity for a transition out of the disabled state would answer a different conditional question. The starting state and the requested destination must match the intensity used.
Worked Example: Using the Intensity at the Start Time
In a hypothetical long-term care model, the intensity of transition from independent to needing care varies with time according to \(\mu_{ij}(t)=0.03+0.01t\), where \(t\) is measured in years from the model's origin. A person is independent at \(t=4\). Estimate the probability of being in the care state at \(t=4.005\), conditional on being independent at \(t=4\). Give the answer as a probability and a percentage.
Method. First evaluate the intensity at the interval's starting time \(t=4\). Then multiply by \(h=4.005-4=0.005\) years.
The starting intensity is $$ \mu_{ij}(4)=0.03+0.01(4)=0.03+0.04=0.07. $$ The first-order probability is $$ \mu_{ij}(4)h=0.07(0.005)=0.00035. $$ Thus the requested conditional probability is approximately \(0.00035\), or \(0.035\%\).
Check. The product uses the intensity at \(t=4\), the time at which the person is known to be independent. Common mistake avoided: evaluating the intensity at \(t=4.005\) instead would use the ending time rather than the starting time in the local first-order approximation.
Worked Example: Finding an Interval Length for a Target Approximation
A hypothetical rehabilitation model assigns a transition intensity of \(0.12\) per year from recovering to recovered at the current time. While the person is in the recovering state, find the short interval length \(h\), in years, for which the first-order approximation to the conditional probability of reaching recovered is \(0.0018\). Then check the result by substitution.
Method. Set the first-order expression \(\mu_{ij}(t)h\) equal to the target probability and solve for \(h\): $$ 0.12h=0.0018, \qquad h=\frac{0.0018}{0.12}=0.015\text{ years}. $$
Substituting the interval back into the approximation gives $$ \mu_{ij}(t)h=0.12(0.015)=0.0018. $$ The estimated conditional probability is therefore \(0.0018\), or \(0.18\%\), over \(0.015\) years.
Check. The recovered interval is positive, and substituting it reproduces the target first-order probability. Common mistake avoided: dividing the interval by the intensity would invert the relationship. The approximation is intensity multiplied by time, so solving for time requires dividing the target probability by the intensity.
Conditions and Common Traps
The approximation is grounded in a limiting argument and should be used with that scope in mind. Its main assumptions and practical checks are:
- The starting state is specified. The conditional probability begins with \(X(t)=i\), so use the intensity for a transition out of \(i\).
- The transition is specified. For a transition to a particular state \(j\), use \(\mu_{ij}(t)\), not an intensity for a different destination.
- The interval is short. The first-order term is justified as \(h\downarrow0\). A larger interval may require methods that account for changes in state and time over the interval.
- The intensity is evaluated at the start time. If it varies with time, the local approximation uses its value at \(t\), not automatically its value at \(t+h\).
- The product is a probability approximation. It should be nonnegative and, in ordinary applications, small for a sufficiently short interval. A result above one signals that the interval is not suitable for this first-order calculation or that the inputs have been mishandled.
A frequent exam error is to treat the first-order expression as though it were exact for any interval. The definition only guarantees that the error divided by \(h\) tends to zero as the interval shrinks. Another error is to substitute an unconditional probability for the conditional probability in the definition. If the model has several possible starting states, the probability of a transition in the population also depends on the probability of occupying its starting state; \(\mu_{ij}(t)h\) alone is conditional on already being in state \(i\).
Write down the current state \(i\) and destination state \(j\), with \(i\ne j\).
Use \(t\) for the beginning and \(h\) for the short interval length.
Find \(\mu_{ij}(t)\), using its value at the start time if it depends on time.
Calculate \(\mu_{ij}(t)h\), then check that the result is a plausible small probability and label it as approximate.
The result is foundational because it translates the local rate information in a continuous-time state model into a probability over a short interval. It also clarifies why the intensity is not itself a probability: it is the coefficient of the interval length in the first-order probability expansion. This local connection supports later calculations involving transition probabilities over longer periods, where the changing state and accumulated transition opportunities must be handled explicitly.
Check Your Understanding
For each problem, use the first-order approximation and keep the conditioning and starting time explicit.
- A hypothetical policy is in a healthy state at time \(t\). The intensity from healthy to disabled is \(0.05\) per year. Estimate the conditional probability of being disabled after \(0.004\) years. (Answer: approximately \(0.0002\), or \(0.02\%\))
- For a hypothetical care model, \(\mu_{ij}(t)=0.02+0.005t\), with \(t\) in years. A person is independent at \(t=6\). Estimate the probability of being in care at \(t=6.01\), conditional on being independent at \(t=6\). (Answer: approximately \(0.0005\), or \(0.05\%\))
- A transition from recovering to recovered has current intensity \(0.09\) per year. What interval length gives a first-order probability estimate of \(0.0009\)? (Answer: \(0.01\) years)
- At \(t=2\), a transition from active to disabled has intensity \(0.04\) per year, and the proposed interval is \(0.02\) years. Calculate the first-order estimate and state whether it is conditional on the person being active at time \(t=2\). (Answer: approximately \(0.0008\); yes, conditional on being active at \(t=2\))
- A time-varying intensity is \(\mu_{ij}(t)=0.06+0.02t\). At \(t=3\), estimate the transition probability over the next \(0.002\) years. A candidate instead evaluates the intensity at \(t=3.002\). Give the correct estimate and identify the candidate's error. (Answer: approximately \(0.00024\); the intensity should be evaluated at the starting time \(t=3\), not the ending time)