Actuarial Mathematics for Modelling · Valuing cashflows contingent on multiple transition events

Transition Intensities and Kolmogorov Forward Equations

Updated 11 October 2026 · Fact-checked

A transition intensity μ_ij(x) is the instantaneous rate of moving from state i to state j at age x. Kolmogorov forward equations give the rate of change of each transition probability as inflow minus outflow. You solve them exactly when the structure is simple, or numerically with Euler's method.

Understand Transition Intensities and Kolmogorov Forward Equations

A multi-state model has a set of states, such as Healthy, Sick and Dead. A person is in one state at each time. The model is Markov if the future depends only on the current state and age, not on the past path.

The transition intensity from state i to state j (i ≠ j) at age x is μ_ij(x) = lim as h→0+ of h p_x^ij ÷ h. Here h p_x^ij is the probability of being in state j at age x+h, given you are in state i at age x. For small h, h p_x^ij ≈ h × μ_ij(x). So an intensity is a rate, not a probability. It can exceed 1. The force of mortality is just the special case of the intensity from Alive to Dead in a two-state model. In a larger model, each state has its own intensities out of it.

Transition probabilities come from intensities. The probability of staying in state i continuously from x to x+t is exp(−∫ Σ_{j≠i} μ_ij(x+s) ds), with the integral from 0 to t. This is the waiting time result. Note that t p_x^ii (in state i at time t) can be larger than this if the model lets you leave and come back.

For general probabilities, think of a small interval from t to t+dt. To be in state j at t+dt, you were either in j and stayed, or in another state k and moved to j. This gives the Kolmogorov forward equation: the rate of change of t p_x^ij equals the probability flowing in to j minus the probability flowing out of j, each weighted by the intensity at age x+t. The starting state i is fixed. The equations are linear differential equations in the probabilities.

To solve them, use the initial condition 0 p_x^ii = 1 and 0 p_x^ij = 0 for j ≠ i. Work exactly if the structure allows, for example by using the waiting time result first and then integrating. Otherwise use Euler's method with a small step h. In the exam the working, the notation and a final check that the probabilities sum to 1 earn the marks.

Key rules to remember

Definition of transition intensity: μ_ij(x) = lim (h→0+) [ h p_x^ij ÷ h ], for i ≠ j; Assumes a Markov model with probabilities that are differentiable in time.
Small-interval approximation: h p_x^ij = h × μ_ij(x) + o(h), for i ≠ j; o(h) is a term that goes to zero faster than h. Use it to build the equations.
Staying probability (waiting time): t p_x^īi = exp( −∫₀ᵗ Σ_{j≠i} μ_ij(x+s) ds ); Probability of remaining continuously in state i from x to x+t. It equals t p_x^ii only if you cannot return to i after leaving.
Kolmogorov forward equation: d/dt [ t p_x^ij ] = Σ_{k≠j} ( t p_x^ik × μ_kj(x+t) − t p_x^ij × μ_jk(x+t) ); Start state i is fixed. Inflow to j minus outflow from j.
Euler step: (t+h) p_x^ij ≈ t p_x^ij + h × d/dt [ t p_x^ij ]; Evaluate the derivative at time t using the current probabilities and intensities at age x+t.
Probability check: Σ_j t p_x^ij = 1; The derivatives across all j must sum to zero. Use this to check your working.

How to solve Transition Intensities and Kolmogorov Forward Equations questions

Use this method for any question that gives intensities and asks for probabilities or their rates of change.

1Draw the state diagram. Label every arrow with its intensity μ_ij(x+t). Mark which transitions are impossible, such as Dead to any other state.
2Fix the starting state i and the initial conditions: 0 p^ii = 1 and all other 0 p^ij = 0.
3For each target state j, list the arrows into j and out of j.
4Write the forward equation: sum of (probability in source state × intensity into j) minus (probability in j × total intensity out of j).
5If the question wants an exact answer, first solve for the staying probability using the exponential waiting time formula. Then substitute it into the next equation and integrate.
6If the question wants a numerical answer, apply Euler's method. Calculate all derivatives at time t, then update every probability at the same time.
7Check that the derivatives sum to zero and the updated probabilities sum to 1.
8State your assumptions (Markov, time step used) and give the final answer with the right notation and enough decimal places.

Quickest way: Inflow minus outflow in one table

When to use it: Use this for Euler questions or when asked for a derivative at a given time with given probabilities.

For each state, write the total outflow intensity as the sum of the arrows leaving it.
Derivative of state j = Σ (source probability × intensity in) − (probability in j × total outflow intensity).
Compute the Dead state derivative last, or get it as minus the sum of the others.
Multiply each derivative by h and add it to the current probability.
Add the new probabilities. If they do not sum to 1, find the arithmetic slip before moving on.

Common mistakes in Transition Intensities and Kolmogorov Forward Equations

Treating μ_ij(x) as a probability and capping it at 1.
The notation looks like a probability and the name is similar to force of mortality.
Fix: Remember that it is a rate per year. Only h × μ_ij(x) is approximately a probability, and only for small h.
Using the outflow intensity of the wrong state in the negative term.
Students copy the pattern of the inflow term and use μ_kj instead of μ_jk.
Fix: Outflow is always from state j, so the negative term is t p^ij × μ_jk. Check the arrow direction on your diagram.
Updating the probabilities one at a time within an Euler step.
Students use a new value for one state while computing the next derivative.
Fix: Compute all derivatives at time t from the old values. Only then update all the probabilities.
Using the staying-probability formula as t p^ii when recovery is possible.
The exponential waiting time result is memorised without its condition.
Fix: The exponential gives the probability of remaining continuously in i. If a return to i is possible, solve the forward equation for t p^ii instead.
Forgetting zero intensities, such as Dead to anything, or leaving out the Dead state.
Students focus on living states and lose the probability sum check.
Fix: Always include Dead and set its outflow intensities to zero. Use the sum to 1 as a check.
Applying the age shift wrongly, using μ(x) rather than μ(x+t) when intensities depend on age.
Constant-intensity examples hide the age dependence.
Fix: In the equation at time t, always evaluate intensities at age x+t.

Worked examples

Example 1

A three-state model has states H (healthy), S (sick) and D (dead). There is no recovery. The constant intensities per year are μ_HS = 0.05, μ_HD = 0.01 and μ_SD = 0.10. A life is in H at time 0. (a) Find t p^HH exactly. (b) Use Euler's method with step h = 0.5 to estimate 1 p^HH, 1 p^HS and 1 p^HD.

Show the solution

(a) With no recovery, the only way to be in H at time t is to stay continuously. The total outflow intensity from H is 0.05 + 0.01 = 0.06.
So t p^HH = exp(−0.06 t).
(b) Forward equations from H, with the starting state fixed: d/dt p^HH = −0.06 p^HH. d/dt p^HS = 0.05 p^HH − 0.10 p^HS. d/dt p^HD = 0.01 p^HH + 0.10 p^HS.
Initial values at t = 0: p^HH = 1, p^HS = 0, p^HD = 0.
Step 1 derivatives: HH = −0.06, HS = 0.05, HD = 0.01. At t = 0.5: p^HH = 1 − 0.03 = 0.97, p^HS = 0.025, p^HD = 0.005. Sum = 1.
Step 2 derivatives at t = 0.5: HH = −0.06 × 0.97 = −0.0582. HS = 0.05 × 0.97 − 0.10 × 0.025 = 0.0485 − 0.0025 = 0.046. HD = 0.01 × 0.97 + 0.10 × 0.025 = 0.0097 + 0.0025 = 0.0122.
At t = 1: p^HH = 0.97 − 0.0291 = 0.9409. p^HS = 0.025 + 0.023 = 0.048. p^HD = 0.005 + 0.0061 = 0.0111.
Check: 0.9409 + 0.048 + 0.0111 = 1.0000. The exact 1 p^HH is exp(−0.06) = 0.9418, so Euler is close.

Answer: (a) t p^HH = exp(−0.06 t). (b) 1 p^HH ≈ 0.9409, 1 p^HS ≈ 0.0480, 1 p^HD ≈ 0.0111.

Example 2

In a model with states H, S and D, the constant intensities per year are μ_HS = 0.08, μ_HD = 0.01, μ_SH = 0.15 and μ_SD = 0.04. A life starts in H. At some time t, t p^HH = 0.80, t p^HS = 0.15 and t p^HD = 0.05. (a) Write the forward equation for t p^HS and find its value. (b) Estimate the probabilities at time t + 0.1 using one Euler step.

Show the solution

(a) Into S: from H at rate 0.08. Out of S: to H at 0.15 and to D at 0.04. Dead has no transition into S, so that term is zero.
d/dt t p^HS = t p^HH × μ_HS − t p^HS × (μ_SH + μ_SD) = 0.80 × 0.08 − 0.15 × 0.19.
= 0.064 − 0.0285 = 0.0355.
(b) Other derivatives. For H: d/dt t p^HH = −t p^HH × (0.08 + 0.01) + t p^HS × 0.15 = −0.072 + 0.0225 = −0.0495.
For D: d/dt t p^HD = t p^HH × 0.01 + t p^HS × 0.04 = 0.008 + 0.006 = 0.014.
Check: −0.0495 + 0.0355 + 0.014 = 0.
Euler with h = 0.1: p^HH = 0.80 − 0.00495 = 0.79505. p^HS = 0.15 + 0.00355 = 0.15355. p^HD = 0.05 + 0.0014 = 0.0514.
Sum: 0.79505 + 0.15355 + 0.0514 = 1.0000.

Answer: (a) d/dt t p^HS = 0.0355 per year. (b) At t + 0.1: p^HH ≈ 0.7951, p^HS ≈ 0.1536, p^HD ≈ 0.0514.

Exam tips

Always draw the state diagram first. Most lost marks come from a missing or wrongly directed arrow.
Show the general forward equation in standard notation before you substitute numbers. Method marks are given for it.
In Euler questions, set out a small table of t, probabilities and derivatives. Check that the probabilities sum to 1 at each step.
For MCQs, check whether the question asks for a rate (intensity, derivative) or a probability. The units tell you.
In computer-based questions, code the same forward equations with a loop over time steps, and state the step size and the assumptions.

Practice questions from Valuing cashflows contingent on multiple transition events

Transition Intensities and Kolmogorov Forward Equations: frequently asked questions

What is the difference between force of mortality and transition intensity?

The force of mortality is the transition intensity from Alive to Dead in a two-state model. A transition intensity is the general version for any pair of different states in a multi-state model. Both are instantaneous rates, not probabilities.

Why are they called forward equations?

They fix the starting state i and the starting age x, and differentiate with respect to the end time. You move forward from a known start. The Kolmogorov backward equations instead differentiate with respect to the starting time.

How do I solve Kolmogorov forward equations numerically with Euler's method?

Start from the initial probabilities. At each step, work out every derivative from the current probabilities, then add h times each derivative to its probability. Smaller steps give better accuracy, and the probabilities should still sum to 1.

Do I need the Markov assumption?

Yes. The forward equations depend on intensities that look only at the current state and age. If the past path mattered, the same intensity would not describe the future from a given state.