Calculate state-dependent probabilities for continuous time Markov models.

A disability income contract pays a benefit only while the insured occupies the "disabled" state. To price it you need not just survival, but state-occupancy probabilities at every future moment. Continuous-time Markov machinery delivers them.

Label states $0,1,\dots,n$. Transition intensities $\mu_x^{ij}$ are instantaneous rates from $i$ to $j$. Two probability objects matter:

• ${}_t p_x^{ij}$: occupy $j$ at time $t$, starting in $i$, with any path allowed.
• ${}_t p_x^{\overline{ii}}$: stay continuously in $i$ from time 0 to $t$ (no exits, no returns).

These coincide only when $i$ cannot be re-entered from any other state.

The chance of leaving state $i$ over $(s,s+ds)$ is the sum of all exit intensities. Therefore:

$${}_t p_x^{\overline{ii}} = \exp\!\left(-\int_0^t \mu_{x+s}^{i\bullet}\,ds\right),\qquad \mu_x^{i\bullet}=\sum_{k\neq i}\mu_x^{ik}$$

When intensities are constant, the integral collapses to $\mu^{i\bullet}\cdot t$.

KEY: The overline distinguishes sojourn from occupancy. ${}_t p_x^{11}$ lets the life leave state 1 and return; ${}_t p_x^{\overline{11}}$ does not.

Differentiate by conditioning on the state just before $t+dt$:

$$\frac{d}{dt}\,{}_t p_x^{ij} = \sum_{k\neq j}\bigl({}_t p_x^{ik}\mu_{x+t}^{kj} - {}_t p_x^{ij}\mu_{x+t}^{jk}\bigr)$$

Common mistakes

• Confusing ${}_t p_x^{ii}$ with ${}_t p_x^{\overline{ii}}$. The overline forbids returns. Whenever $i$ is reachable from another state, occupancy strictly exceeds sojourn.
• Forgetting the negative outflow term. Writing only the inflow into $j$ and missing $-{}_t p_x^{ij}\sum_k \mu^{jk}$ produces probabilities that drift above 1.
• Sojourn with the wrong total intensity. Use the sum of every exit from $i$, not just transitions to death. A common slip uses $\mu^{02}$ alone instead of $\mu^{01}+\mu^{02}$.

Bottom line

• Notation: ${}_t p_x^{ij}$ is the probability a life in state $i$ at age $x$ is in state $j$ at age $x+t$. The overline ${}_t p_x^{\overline{ii}}$ requires staying in $i$ throughout.
• Boundary/identity: ${}_0 p_x^{ii}=1$ with zero off-diagonal, and $\sum_j {}_t p_x^{ij}=1$ at every $t$; absorbing states accumulate the departing mass.
• Sojourn formula: ${}_t p_x^{\overline{ii}} = \exp\!\left(-\int_0^t \sum_{k\neq i}\mu_{x+s}^{ik}\,ds\right)$, summing over every exit from $i$.
• First-transition integral: for simple no-return path structures one integral over the first-transition time gives ${}_t p_x^{ij}$ directly, as with permanent disability and ${}_t p_x^{01}$.

Exam shortcut

If the disabled state has no recovery arrow, ${}_t p_x^{11} = {}_t p_x^{\overline{11}}$: use the simple exponential and skip the ODE. DECISION: Constant intensities and at most one returnable state → one integral. Time-varying intensities or two-way transitions → Euler march. Makeham/Gompertz force on a sojourn → closed-form using $(c^t-1)/\ln c$. Run the row-sum audit after each Euler step: confirm $\sum_j {}_t p_x^{ij}=1$. Drift above 1.005 means $h$ is too large.

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Learning objectives

• 1e

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