Forward transition rates

Abstract

The idea of forward rates stems from interest rate theory. It has natural connotations to transition rates in multi-state models. The generalisation from the forward mortality rate in a survival model to multi-state models is non-trivial and several definitions have been proposed. We establish a theoretical framework for the discussion of forward rates. Furthermore, we provide a novel definition with its own logic and merits and compare it with the proposals in the literature. The definition turns the Kolmogorov forward equations inside out by interchanging the transition probabilities with the transition intensities as the object to be calculated.

Appendix: Proofs

Before we prove Lemma 2.1, we recall the so-called chain rule for conditional independence; see e.g. [14, Proposition 5.8]. Let $\mathcal{G}_{1}$, $\mathcal{G}_{2}$, $\mathcal{K}$ and ℋ be sub-$\sigma$-algebras. The chain rule states that

if and only if

If $\mathcal{G}_{1} \subseteq \mathcal{G}_{2}$, we find that

which is called reduction, and

(A.1)

Proof of Lemma 2.1

Because $X$ is Markovian conditionally on $\mu$, it holds that

(A.2)

Furthermore, by construction, the conditional distribution of $(X_{s})_{s\leq t}$ given $\mathcal{F}^{\mu }_{\infty }$ is $\mathcal{F}_{[0,t]}^{\mu }$-measurable; it is only a function of $\mu$ through $(\mu _{s})_{s \leq t}$, see the properties of the (conditional) transition probabilities $P_{jk}^{\mu }$. It follows from [14, Proposition 5.6] that

Then by reduction,

(A.3)

which proves the first part of the result. Beginning with (A.3), apply (A.1) to obtain

(A.4)

Let $\mathcal{K} = \mathcal{F}_{[0,t]}^{X}$, $\mathcal{H} = \sigma (X _{t}) \vee \mathcal{F}_{[0,t]}^{\mu }$, $\mathcal{G}_{1} = \mathcal{F}_{(t,\infty )}^{\mu }$ and $\mathcal{G}_{2} = \mathcal{F} _{(t,\infty )}^{X}$. Recall that $\mathcal{F}_{\infty }^{\mu }= \mathcal{F}_{[0,t]}^{\mu }\vee \mathcal{F}_{(t,\infty )}^{\mu }$. We might recast (A.4) and (A.2) as

and then it follows from the chain rule that

This exactly reads

which establishes the second part of the result, completing the proof. □

Proof of Theorem 3.2

We first show that there exists a unique solution to (3.5) for any $T \in (t, \infty )$. Fix $T$. In what follows, we suppress $t$ notationally.

With $m_{jj} := - \sum _{k \neq j} m_{jk}$, we can rewrite (3.5) as

$$\begin{aligned} \frac{\partial }{\partial T}{\mathcal{P}}(T) = {\mathcal{P}}(T) {m}(T), \end{aligned}$$

(A.5)

with $\frac{\partial }{\partial T}{\mathcal{P}}$, ${\mathcal{P}}$ and ${m}$ being the corresponding matrices. Because $\mathcal{P}_{jk}(T) = 0$ for $k < j$, it follows that ${\mathcal{P}}(T)$ is an upper triangular matrix with diagonal elements $\mathcal{P}_{jj}(T)$. Now note that

$$\begin{aligned} P_{jj}^{\mu }(T) = \exp \bigg(-\int _{t}^{T} \sum _{k > j} \mu _{jk}(s) \, \mathrm{d}s \bigg) > 0. \end{aligned}$$

Consequently, we have $\mathcal{P}_{jj}(T)>0$ and in particular

$$\begin{aligned} \det {\mathcal{P}}(T) = \prod _{j=0}^{J} \mathcal{P}_{jj}(T) > 0. \end{aligned}$$

This implies that ${\mathcal{P}}(T)$ is invertible with inverse ${\mathcal{P}}^{-1}(T)$, which is also an upper triangular matrix. Hence (A.5) has a unique solution given by

$$\begin{aligned} {m}(T) = {\mathcal{P}}^{-1}(T)\frac{\partial }{\partial T}{\mathcal{P}}(T). \end{aligned}$$

Note that

$$\begin{aligned} \sum _{k \neq j} m_{jk}(T) &= \sum _{k \neq j} \sum _{\ell } \mathcal{P} ^{-1}_{j\ell }(T)\frac{\partial }{\partial T}{\mathcal{P}}_{\ell k}(T) \\ &= \sum _{\ell } \mathcal{P}^{-1}_{j\ell }(T) \frac{\partial }{\partial T} \sum _{k \neq j} {\mathcal{P}}_{\ell k}(T) \\ &= -\sum _{\ell } \mathcal{P}^{-1}_{j\ell }(T) \frac{\partial }{ \partial T} {\mathcal{P}}_{\ell j}(T) = -m_{jj}(T), \end{aligned}$$

according to the definition at the beginning of the proof. This completes the proof of existence and uniqueness.

To complete the proof, we have to show that the solution is $\mathcal{F}_{t}^{\mu }$-measurable (as a function of $T$). This follows immediately by e.g. Cramér’s rule because the entries of ${\mathcal{P}}$ are $\mathcal{F}_{t}^{\mu }$-measurable.

If $T \mapsto \frac{\partial }{\partial T}\mathcal{P}(t,T)$ is assumed to be continuous, it follows by similar arguments and an application of e.g. Cramér’s rule that the solution is also continuous. □

Forward transition rates in the survival model with surrender and free policy We consider the state-wise forward transition rates given by (4.2). Because $\psi$ and $\sigma$ are deterministic, the only non-trivial derivations are related to $m_{13}$ and $m_{14}$. By setting $\sigma =0$ and using symmetry, the derivation of $m_{14}$ follows from the derivation of $m_{13}$. Hence it suffices to derive $m_{13}$. On $\{X_{t} = 1\}$, it holds that

and

Consequently,

$$\begin{aligned} m_{13}(t,T) &= \sigma (T) + \frac{ \mathbb{E} [ e^{-\int _{(t,T]} ( \eta (s) + \rho (s)) \, \mathrm{d}s} \rho (T) \, | \mathcal{F}^{ \eta ,\rho }_{t}] }{ \mathbb{E} [ e^{-\int _{(t,T]} (\eta (s) + \rho (s)) \, \mathrm{d}s} | \mathcal{F}^{\eta ,\rho }_{t}] } \end{aligned}$$

on $\{X_{t} = 1\}$. Let now $C$ be defined by

$$\begin{aligned} C(t,T) = \int _{(t,T]} e^{-\int _{(t,s]} \psi (u) \, \mathrm{d}u} \psi (s) e^{-\int _{(s,T]} \sigma (u) \, \mathrm{d}u} \, \mathrm{d}s. \end{aligned}$$

Note that on $\{X_{t} = 0\}$,

Thus whenever $\psi$ is strictly positive on a subset of $(t,T]$ with non-zero Lebesgue measure, it holds on $\{X_{t} = 0\}$ that

$$\begin{aligned} m_{13}(t,T) &= \frac{ \mathbb{E} [ e^{-\int _{(t,T]} (\eta (s) + \rho (s)) \, \mathrm{d}s} (\rho (T) + \sigma (T)) \, | \mathcal{F} ^{\eta ,\rho }_{t}] }{ \mathbb{E} [ e^{-\int _{(t,T]} (\eta (s) + \rho (s)) \, \mathrm{d}s} \, | \mathcal{F}^{\eta ,\rho }_{t}] } \\ &= \sigma (T) + \frac{ \mathbb{E} [ e^{-\int _{(t,T]} (\eta (s) + \rho (s)) \, \mathrm{d}s} \rho (T) \, | \mathcal{F}^{\eta ,\rho }_{t}] }{ \mathbb{E} [ e^{-\int _{(t,T]} (\eta (s) + \rho (s)) \, \mathrm{d}s} | \mathcal{F}^{\eta ,\rho }_{t}] }, \end{aligned}$$

as the terms involving $C(t,T)$ cancel. To conclude, this shows that

$$\begin{aligned} m_{13}(t,T) = \sigma (T) + \frac{ \mathbb{E} [ e^{-\int _{(t,T]} ( \eta (s) + \rho (s)) \, \mathrm{d}s} \rho (T) \, | \mathcal{F}^{ \eta ,\rho }_{t}] }{ \mathbb{E} [ e^{-\int _{(t,T]} (\eta (s) + \rho (s)) \, \mathrm{d}s} | \mathcal{F}^{\eta ,\rho }_{t}] } \end{aligned}$$

is an $\mathcal{F}_{t}^{\eta ,\rho }$-measurable version of the state-wise forward transition rates. □