Time-Inhomogeneous Kolmogorov Equations

Defining Time-Inhomogeneous Transition Rates

In a time-homogeneous model, the rate of moving from state $i$ to state $j$, ${\mu }_{ij}$, is a constant. In a time-inhomogeneous model, this rate becomes a function of time. In actuarial contexts, this "time" is typically the age of the individual.

We introduce a new notation to capture this dependency. The force of transition or transition intensity from state $i$ to state $j$ at age $x+t$ is denoted by ${\mu }_{x+t}^{ij}$. This represents the instantaneous rate of transition for an individual who was in a starting state at age $x$ and is now age $x+t$.

Let's formalize this. We previously defined the transition probability ${}_t{p}_x^{ij}=P(\text{in state }j\text{ at age }x+t|\text{in state }i\text{ at age }x)$. The transition intensity is defined as the instantaneous rate of change of this probability for a transition out of state $i$.

As the definition shows, the transition intensity is the derivative of the transition probability over an infinitesimally small time interval, evaluated at $t=0$. This leads to the crucial approximation for a small time interval $h$:

This is the building block for the dynamics of the entire system.

The Inhomogeneous Kolmogorov Equations

In the time-homogeneous case, we had the Kolmogorov forward and backward equations, $P^{\prime }(t)=P(t)Q$ and $P^{\prime }(t)=QP(t)$, where $Q$ was a constant matrix of rates. This led to the elegant solution $P(t)=e^{Qt}$.

When the transition rates are time-dependent, the generator matrix itself becomes a function of time. We'll denote it ${\Gamma }_{x+t}$, where its entries are $({\Gamma }_{x+t}{)}_{ij}={\mu }_{x+t}^{ij}$. Because the generator matrix is no longer constant, we cannot simply use the matrix exponential to find the transition probabilities. Instead, we must return to the differential equations themselves.

The video below re-introduces the Kolmogorov equations in their matrix form, which provides a good bridge from the homogeneous case we've already studied.

The core structure of the equations remains, but their solution becomes more complex. Let's look at the formal statement of these equations in the time-inhomogeneous context.

Let's summarize these crucial results. Let ${}_t{P}_x$ be the matrix of transition probabilities ${}_t{p}_x^{ij}$.

• Kolmogorov's Forward Equation:

  $$\frac{d}{dt}({}_t{P}_x)=({}_t{P}_x){\Gamma }_{x+t}$$

  This equation is "forward" because the time derivative $\frac{d}{dt}$ acts on the end of the time interval $[x,x+t]$, while the rates ${\Gamma }_{x+t}$ are also evaluated at the end. It describes how the probabilities evolve by considering the last infinitesimal step of the process.

• Kolmogorov's Backward Equation:

  $$\frac{d}{dx}({}_t{P}_x)=-{\Gamma }_x({}_t{P}_x)$$

  This equation is "backward" because the derivative $\frac{d}{dx}$ acts on the start of the time interval $[x,x+t]$, and the rates ${\Gamma }_x$ are also evaluated at the start. It describes the evolution by considering the first infinitesimal step.

To build some intuition for how these equations are formed, let's consider the backward equation for a single transition probability, ${}_t{p}_{ij}$. We condition on what happens in the first small instant of time. The process can either stay in state $i$ or move to some other state $k$.

The following video provides a nice, non-technical walkthrough of this logic.

An Important Special Case: The Survival Probability

While solving the full system of differential equations for ${}_t{P}_x$ is often complex and requires numerical methods (our next topic), there is a very important special case where we can find a direct analytical solution.

Consider the probability of remaining in a state $i$ for a full duration of $t$ years, which we denote ${}_t{p}_{\overline{ii}}$. If state $i$ is a state that cannot be re-entered once left (like the "Healthy" state in a permanent disability model), then this is the same as the transition probability ${}_t{p}_{ii}$.

The differential equation for this probability is:

Since ${\mu }_{x+t}^{ii}=-{\sum }_{j\ne i}{\mu }_{x+t}^{ij}$, this is a first-order separable ordinary differential equation. With the initial condition ${}_0{p}_x^{ii}=1$, the solution is:

This should look familiar. The term ${\sum }_{j\ne i}{\mu }_{x+s}^{ij}$ is the total force of exit from state $i$. In survival analysis, this is precisely the hazard rate (or force of mortality). The formula above is the fundamental relationship between the survival function and the integrated hazard function.

Let's see this applied in a concrete example.

This example perfectly illustrates the process:

1. Identify the forces of exit from the state of interest (state 0). In this case, they are ${\mu }_{x+t}^{01}$ and ${\mu }_{x+t}^{02}$.
2. Sum them to get the total hazard: ${\mu }_{x+t}^{01}+{\mu }_{x+t}^{02}$.
3. Integrate this total hazard over the desired time interval (from 0 to 10 years).
4. Exponentiate the negative of the result to get the survival probability.

For other probabilities, like ${}_{10}{p}_{30}^{01}$, the resource correctly notes that one must write down the full Kolmogorov differential equation, which generally does not have a simple analytical solution.

Conclusion

In this lesson, we have made the crucial leap from time-homogeneous to time-inhomogeneous Markov models. This allows for a much more realistic representation of processes where risks, like mortality or sickness, evolve with age.

Key Takeaways:

• Time-inhomogeneous models use transition intensities, ${\mu }_{x+t}^{ij}$, which are functions of time or age.
• The governing dynamics are described by Kolmogorov's forward and backward differential equations, where the generator matrix, ${\Gamma }_{x+t}$, is now time-dependent.
• The forward equation, $\frac{d}{dt}({}_t{P}_x)=({}_t{P}_x){\Gamma }_{x+t}$, considers the evolution at the end of the time interval.
• The backward equation, $\frac{d}{dx}({}_t{P}_x)=-{\Gamma }_x({}_t{P}_x)$, considers the evolution from the start of the interval.
• In the special case of calculating the probability of remaining in a state with no re-entry, we can use a direct integral formula: ${}_t{p}_x^{ii}=\exp \left(-{\int }_0^t{\sum }_{j\ne i}{\mu }_{x+s}^{ij}ds\right)$.

We have learned how to define the rates and set up the governing equations. In our next lesson, we will tackle the practical challenge of actually solving these equations using numerical methods to find the transition probabilities.