On the forward rate concept in multi-state life insurance

Abstract

Similarly to the notion of modeling credit risk by using forward credit default spread rates, mortality risk in life insurance contracts is nowadays often modeled by using forward mortality (spread) rates. More recently, this concept has also been discussed for more complex life insurances that include multiple lives or intermediate states that correspond to the health status of the insured. For consistency purposes and for technical reasons, most authors assume that the underlying financial and demographic events are stochastically independent.

In the present paper, we study sufficient and necessary conditions under which general transition forward rates are indeed consistent with respect to the relevant insurance claims. This shows the theoretical limitations of the forward rate concept in life insurance. Our study is based on a model where the underlying financial and demographical developments are diffusion processes driven by a multivariate Brownian motion. This allows us to investigate independence properties by analyzing the asymptotic behavior of mixed (conditional) moments. In particular, we obtain that for joint life and disability insurance policies, some specific demographic events need to be dependent in order to ensure consistency.

Appendix

Lemma A.1

Under Assumption  3.3 and with ϵ and (X(t)) _t≥0 as in this assumption, we define for a fixed t with 0≤t≤u≤T ^∗ and for i=1,…,n the processes $\widetilde{X}_{i}(u) := X_{i}(u)-X_{i}(t)$. Then there exist

• (a) a random variable Y with $\mathbb{E}[|Y|^{2(1+\epsilon )}] < \infty$ and

  $$\begin{aligned} Y \ge\bigg| \prod\limits_{i=1}^n X_i(t_i)^{Z_i} \bigg| \mathrm{e} ^{- \sum_{i=1}^n \int_{T_i} X_i(u) \,\mathrm{d}u} \end{aligned}$$

  for all intervals T _i ⊆[0,T ^∗], Z _i ∈{0,1} and t _i ∈[0,T ^∗] for i=1,…,n

and for all 0≤t≤T ^∗ and $x \in\mathbb{R}$ constants C(t,x) such that

• (b) $\mathbb{E}_{\mathrm{Q}}^{(t,x)}[ | \int_{t}^{t+\varDelta }\widetilde{X}_{i}(u) \,\mathrm{d}u |^{k} ] \le C(t,x) \varDelta^{\frac{3}{2} k}$ for all Δ∈[0,T ^∗−t] and $k \in\mathbb{N}$.

• (c) ${\mathbb{E}_{\mathrm{Q}}^{(t,x)}[ \int_{t}^{t+\varDelta }\cdots\int_{t}^{s_{\ell-1}} \int _{t}^{s_{\ell}} { | \alpha_{i}(\tau,X_{i}(\tau)) - \alpha_{i}(\tau,X_{i}(t)) |} \,\mathrm{d} \tau\,\mathrm{d}s_{\ell}\cdots \mathrm{d}s_{1} ]}\hfill \le (\mathbb{E}_{\mathrm{Q}}^{(t,x)}[( \int_{t}^{t+\varDelta}\cdots \int_{t}^{s_{\ell-1}} \int _{t}^{s_{\ell}} { | \alpha_{i}(\tau,X_{i}(\tau)) - \alpha_{i}(\tau,X_{i}(t)) |} \,\mathrm{d} \tau\,\mathrm{d}s_{\ell}\cdots \mathrm{d}s_{1} )^{k}])^{\frac{1}{k}}\hfill \le C(t,x) \varDelta ^{\frac{3}{2} + \ell}$ for all $\varDelta\in[0, T^{*}-t], \ell \in\mathbb{N}_{0}$.

• (d) $\sqrt{\mathbb{E}_{\mathrm{Q}}^{(t,x)}[( \int _{t}^{t+\varDelta}\cdots\int_{t}^{s_{\ell -1}} \int_{t}^{s_{\ell}} \beta_{i}(\tau, X_{i}(\tau)) \,\mathrm {d}W_{i}(\tau) \,\mathrm{d} s_{\ell}\cdots \mathrm{d}s_{1} )^{2}]} \le C(t,x) \varDelta^{\frac{1}{2} + \ell}$ for all $\varDelta\in[0, T^{*}-t], \ell\in\mathbb{N}_{0}$.

• (e) For all Δ∈[0,T ^∗−t], let T _i be an interval with T _i ⊆[t,t+Δ], Z _i ∈{0,1} and t _i ∈[t,t+Δ] for all i=1,…,n. Then

  $$\begin{aligned} &C(t,x) \varDelta^{\frac{9}{2}} \\ &\quad\ge \mathbb{E}_\mathrm{Q}^{(t,x)}\bigg[\bigg| \mathrm {e}^{- \sum_{i=1}^n \int_{T_i} \widetilde{X}_i(u) \,\mathrm{d}u} \prod_{i=1}^n X_i(t_i)^{Z_i} \\ & \qquad{}- \bigg( \prod_{i=1}^n X_i(t_i)^{Z_i} \bigg) \bigg( 1 - \sum _{i=1}^n \int_{T_i} \widetilde{X}_i(u) \,\mathrm{d}u + \frac{1}{2} \bigg(\sum _{i=1}^n \int_{T_i} \widetilde{X}_i(u) \,\mathrm{d}u \bigg)^2 \bigg) \bigg|\bigg]. \end{aligned}$$

• (f) $| \frac{1}{\mathbb{E}_{\mathrm{Q}}^{(t,x)}[\mathrm{e}^{-\int _{t}^{t+\varDelta}\widetilde{X}_{i}(\tau) \,\mathrm{d}\tau } ]} -1 - \mathbb{E}_{\mathrm{Q}}^{(t,x)}[\int_{t}^{t+\varDelta }\widetilde{X}_{i}(\tau) \,\mathrm{d}\tau ]\hfill + \frac{1}{2} \mathbb{E}_{\mathrm{Q}}^{(t,x)}[ ( \int _{t}^{t+\varDelta}\widetilde{X}_{i}(\tau) \,\mathrm{d}\tau)^{2} ] - ( \mathbb{E}_{\mathrm{Q}}^{(t,x)}[\int_{t}^{t+\varDelta }\widetilde{X}_{i}(\tau) \,\mathrm{d}\tau] )^{2} |\hfill \le C(t,x) \varDelta^{\frac{9}{2}}$ for sufficiently small Δ>0.

Proof

In the following, we write C(t,x)<∞ for any constant.

(a) A direct consequence from Hölder’s inequality for sums is the inequality

$$\begin{aligned} (|a_1|+ \cdots+ |a_n|)^p \le (|a_1|^p + \cdots+ |a_n|^p)n^{p-1} \quad \mbox{for } p\ge1 . \end{aligned}$$

(A.1)

For each i=1,…,n, we have with (A.1) and the SDE (3.1) that

$$\begin{aligned} |X_i(t_i)|^k &\le\sup_{t\in[0,T^*]} |X_i(t_i)|^k \\ &\le3^{k-1} \bigg( \sup_{t\in[0,T^*]} \bigg| \int_0^t \beta _i(u,X_i(u)) \,\mathrm{d}W_i(u) \bigg|^k \\ & \quad{}+ |X_i(0)|^k + \sup_{t\in[0,T^*]}\bigg| \int_0^t \alpha _i(u,X_i(u)) \,\mathrm{d}u\bigg|^k \bigg) . \end{aligned}$$

With this estimation, by applying Hölder’s inequality to the α-term and Doob’s inequality to the β-term, we get for k≥2 that

$$\begin{aligned} \mathbb{E}\bigg[ \sup_{t\in[0,T^*]} |X_i(t)|^k \bigg] &\le 3^{k-1} \bigg( \Big( \frac{k}{k-1}\Big)^k \mathbb{E}\bigg[ \bigg| \int_0^{T^*} \beta _i(u,X_i(u)) \,\mathrm{d}W_i(u) \bigg|^k \bigg] \\ & \quad {}+ |X_i(0)|^k + (T^*)^{k-1} \int_0^{T^*} \mathbb{E}\big[\big| \alpha_i\big(u,X_i(u)\big) \big|^k \,\mathrm{d}u \big] \bigg) < \infty, \end{aligned}$$

since X _i (0) is deterministic. With Assumption 3.1(iii) for α _i which is also true for the CIR process, we get

$$\begin{aligned} \int_0^{T^*} \mathbb{E}\big[ \big|\alpha_i\big(X_i(u),u\big)\big|^k \big] \,\mathrm{d}u &\le\int_0^{T^*} \mathbb{E}\big[ K_2^k |1+X_i(u)|^{\frac{k}{2}}\big] \,\mathrm{d}u \\ &\le\int_0^{T^*} K_2^k 2^{\frac{k}{2} -1} \big( 1 + \mathbb{E}[|X_i(u)|^k ]\big) \,\mathrm{d}u, \end{aligned}$$

and this term is finite since analogously to Proposition 3.2, $\mathbb{E}[ |X_{i}(u)|^{k}]$ is finite. By adding and subtracting an α-term, using (A.1) and by the same arguments as above, we obtain

$$\begin{aligned} &\mathbb{E}\bigg[ \bigg| \int_0^{T^*} \beta_i(u,X_i(u)) \,\mathrm {d}W_i(u) \bigg|^k \bigg] \\ &\quad \le2^{k-1} \bigg( \mathbb{E}[ | X_i(T^*) |^k ] + \mathbb{E}\bigg[ \bigg| \int _0^{T^*} \alpha_i\big(u,X_i(u)\big) \,\mathrm{d}u \bigg|^k \bigg] \bigg) < \infty. \end{aligned}$$

In total, we get with Assumption 3.3(i) that

$$\begin{aligned} \bigg| \prod\limits_{i=1}^n X_i(t_i)^{Z_i} \bigg| \mathrm{e}^{- \sum_{i=1}^n \int_{T_i} \widetilde{X}_i(u) \,\mathrm{d}u} \le\prod\limits_{i=1}^n \Big(\sup_{t \in[0,T^*]} |X_i(t)| \Big)^{Z_i} W =: Y . \end{aligned}$$

This random variable Y is integrable, since with Hölder’s inequality for $p=\frac{1+\epsilon}{\epsilon}$,

$$\begin{aligned} \mathbb{E}[|Y|] &\le\bigg( \mathbb{E}\bigg[ \bigg( \prod_{i=1}^n \Big(\sup_{t \in [0,T^*]} |X_i(t)| \Big)^{Z_i} \bigg)^{\frac{1+\epsilon}{\epsilon }}\bigg] \bigg)^{\frac{\epsilon}{1+\epsilon}} W^{1+\epsilon} \\ & \le\prod_{i=1}^n \bigg( \mathbb{E}\bigg[ \bigg( \Big(\sup_{t \in[0,T^*]} |X_i(t)| \Big)^{Z_i} \bigg)^{\frac{1+\epsilon}{\epsilon}n}\bigg]\bigg)^{\frac{\epsilon}{n(1+\epsilon)}} W^{1+\epsilon} < \infty \end{aligned}$$

by assumption and by the first part, where we distinguish the cases Z _i =0 and Z _i =1.

(b) First of all, we consider the case k>1. With Hölder’s inequality for p=k and Proposition 3.2, it follows that

$$\begin{aligned} \mathbb{E}_\mathrm{Q}^{(t,x)}\bigg[ \bigg| \int_t^{t+\varDelta }\widetilde{X}_i(u) \,\mathrm{d}u \bigg|^k \bigg] &\le\mathbb{E}_\mathrm{Q}^{(t,x)} \bigg[ \bigg( \int _t^{t+\varDelta}1 \,\mathrm{d}u \bigg)^{k-1} \bigg] \mathbb{E} _\mathrm{Q}^{(t,x)}\bigg[ \int_t^{t+\varDelta}| \widetilde{X}_i(u) |^k \,\mathrm{d}u \bigg] \\ & \le\varDelta^{k-1} \int_t^{t+\varDelta}C(t,x) \varDelta^{\frac{k}{2}}\,\mathrm{d}u = C(t,x) \varDelta^{\frac{3}{2} k} . \end{aligned}$$

The case k=1 also follows from the calculation above by skipping the second step.

(c) The first step is a direct consequence from Hölder’s inequality with parameter p=k. By applying ℓ+1 times Hölder’s inequality with p=k and using Assumption 3.1(ii), we get

$$\begin{aligned} &\bigg(\mathbb{E}_\mathrm{Q}^{(t,x)}\bigg[\Big( \underbrace{\int _t^{t+\varDelta}\cdots \int_t^{s_{\ell-1}}}_{\ell\,\mathrm{times}} \int_t^{s_\ell} \underbrace{ \left| \alpha_i\big(\tau,X_i(\tau)\big) - \alpha _i\big(\tau ,X_i(t)\big) \right|}_{\le K_1 | \widetilde{X}_i(\tau)|} \,\mathrm {d}\tau\,\mathrm{d} s_\ell\cdots \mathrm{d}s_1 \Big)^k\bigg]\bigg)^{\frac{1}{k}} \\ &\quad \le\Big( \varDelta^{(k-1)(\ell+1)} \int_t^{t+\varDelta}\cdots \int _t^{s_{\ell-1}} \int_t^{s_\ell} K_1^k \mathbb{E}_\mathrm {Q}^{(t,x)}\big[\big( | \widetilde{X}_i(\tau)| \big)^k \big] \,\mathrm{d}\tau\,\mathrm {d}s_\ell\cdots \mathrm{d}s_1 \Big)^{\frac{1}{k}} \\ &\quad \le\big(K_1^k C(t,x) \varDelta^{k\ell+ k\frac{3}{2}}\big)^{\frac{1}{k}}, \end{aligned}$$

using that $\mathbb{E}_{\mathrm{Q}}^{(t,x)} [( |\widetilde {X}_{i}(\tau)|)^{k}] \le C(t,x) \varDelta^{\frac{k}{2}}$.

(d) By applying ℓ times Hölder’s inequality with p=2, the Itô isometry, Assumption 3.1(iii) and Proposition 3.2, we can show that the initial term is smaller than or equal to

$$\begin{aligned} &\sqrt{\varDelta^\ell\int_t^{t+\varDelta}\cdots\int_t^{s_\ell} \mathbb{E}_\mathrm{Q}^{(t,x)} \bigg[ \bigg( \int_t^{s_\ell} \beta_i\big(\tau, X_i(\tau)\big) \, \mathrm{d}W_i(\tau ) \bigg)^2 \bigg] \,\mathrm{d}s_\ell\cdots \mathrm{d}s_1} \\ & \quad \le\sqrt{\varDelta^\ell\int_t^{t+\varDelta}\cdots\int _t^{t+\varDelta}\int_t^s C(t,x) \,\mathrm{d} \tau\,\mathrm{d}s_\ell\cdots \mathrm{d}s_1} = \sqrt{C(t,x) \varDelta^{2\ell+1}} . \end{aligned}$$

(e) By a Taylor expansion, we have $| \mathrm{e}^{-x} -1 + x - \frac{1}{2} x^{2} | \le\frac{1}{6} |x|^{3} \mathrm {e}^{-\xi}$ for ξ between 0 and x. Applying this approximation to $x = \sum _{i=1}^{n} \int_{T_{i}} \widetilde{X}_{i}(u) \,\mathrm{d}u$ and using Hölder’s inequality with p=q=2 gives the inequality

$$\begin{aligned} &\mathbb{E}_\mathrm{Q}^{(t,x)} \bigg[\bigg| \mathrm{e}^{- \sum _{i=1}^n \int_{T_i} \widetilde {X}_i(u) \,\mathrm{d}u} \prod_{i=1}^n X_i(t_i)^{Z_i} \\ & \qquad{}-\bigg( \prod_{i=1}^n X_i(t_i)^{Z_i}\bigg) \bigg( 1 - \sum_{i=1}^n \int_{T_i} \widetilde{X}_i(u) \,\mathrm{d}u + \frac{1}{2} \bigg(\sum _{i=1}^n \int _{T_i} \widetilde{X}_i(u) \,\mathrm{d}u \bigg)^2 \bigg) \bigg|\bigg] \\ &\quad\le\frac{1}{6} \bigg( \mathbb{E}_\mathrm{Q}^{(t,x)} \bigg[ \bigg| \prod_{i=1}^n X_i(t_i)^{2 Z_i} \bigg| \bigg] \mathbb{E}_\mathrm{Q}^{(t,x)} \bigg[ \bigg| \bigg( \sum_{i=1}^n \int_{T_i} \widetilde{X}_i(u) \,\mathrm{d}u \bigg)^6 \mathrm{e}^{- 2\xi} \bigg| \bigg] \bigg)^{\frac{1}{2}} . \end{aligned}$$

We apply the generalization of Hölder’s inequality (cf. Corollary 2.11.5 in [4]) to the first factor with $p_{i}=\frac{1}{n}$ and get that the expression from above is smaller than or equal to

$$\begin{aligned} & \frac{1}{6} \bigg( \prod_{i=1}^n \big( {\mathbb{E}_\mathrm{Q}^{(t,x)} \big[ |X_i(t_i)|^{2 n Z_i} \big]} \big)^{\frac{1}{n}} \mathbb{E}_\mathrm{Q}^{(t,x)} \bigg[\bigg| \bigg( \sum_{i=1}^n \int_{T_i} \widetilde{X}_i(u) \,\mathrm{d}u \bigg)^6 \mathrm{e}^{- 2\xi} \bigg| \bigg] \bigg)^{\frac{1}{2}} , \end{aligned}$$

where the first expression is bounded by (3.3). In the following, we use the inequality |e^−ξ|≤1+e^−2x for ξ between 0 and x and get that the expression is again smaller than or equal to

$$\begin{aligned} &\frac{\sqrt{C(t,x)}}{6} \bigg( \mathbb{E}_\mathrm{Q}^{(t,x)} \bigg[\bigg| \bigg( \sum _{i=1}^n \int_{T_i} \widetilde{X}_i(u) \,\mathrm{d}u \bigg)^6 \bigg|\bigg] \\ & \qquad{}+ \mathbb{E}_\mathrm{Q}^{(t,x)} \bigg[\bigg| \bigg( \sum_{i=1}^n \int_{T_i} \widetilde{X}_i(u) \,\mathrm{d}u \bigg)^6 \mathrm{e}^{- 2 \sum _{i=1}^n \int_{T_i} \widetilde{X}_i(u) \,\mathrm{d}u} \bigg|\bigg] \bigg)^{\frac{1}{2}} \\ &\quad\le\frac{\sqrt{C(t,x)}}{6} \Big( C(t,x) \varDelta^9 + C(t,x) \varDelta^9 \big(K(t,x)\big)^{\frac{1}{1+\epsilon}} \Big)^{\frac{1}{2}} , \end{aligned}$$

where we apply Hölder’s inequality to the second summand with the two parameters $p= \epsilon_{0}^{-1}(1+\epsilon_{0})$ and q=1+ϵ ₀, where ϵ ₀ is chosen such that 0<ϵ ₀≤ϵ and $6 \frac{1+\epsilon_{0}}{\epsilon_{0}} \in\mathbb{N}$. The moments of the first two expected values are estimated by applying (b), and for the third expected value we plug in the definition of $\widetilde {X}_{i}(u)$, take X _i (t) out of the expected value and get by Assumption 3.3(i) that this expected value is finite.

(f) The Taylor approximation of second order gives us that

$$\begin{aligned} \mathrm{e}^{-\int_t^{t+\varDelta} \widetilde{X}_i(\tau) \,\mathrm {d}\tau} &= 1 - \int_t^{t+\varDelta}\widetilde{X}_i(\tau) \,\mathrm{d}\tau+ \frac{1}{2} \Big( - \int_t^{t+\varDelta}\widetilde {X}_i(\tau) \,\mathrm{d}\tau\Big)^2 \\ &\quad{}+\frac{1}{6} \Big( - \int_t^{t+\varDelta}\widetilde{X}_i(\tau ) \,\mathrm{d}\tau\Big)^3 \mathrm{e}^{\xi_1}, \end{aligned}$$

where ξ ₁ is between 0 and $-\int_{t}^{t+\varDelta}\widetilde {X}_{i}(\tau) \,\mathrm{d}\tau$. Again, we use the Taylor approximation of second order and obtain

$$\begin{aligned} \frac{1}{1+y} = 1- y + y^2 - \frac{y^3}{(1+\xi_2)^4} , \quad |y| < 1, \end{aligned}$$

where ξ ₂ is between 0 and y. We define

$$\begin{aligned} y:\!&= - \mathbb{E}_\mathrm{Q}^{(t,x)} \bigg[\int_t^{t+\varDelta }\widetilde{X}_i(\tau) \,\mathrm{d}\tau\bigg] + \mathbb{E}_\mathrm{Q}^{(t,x)} \bigg[ \frac{1}{2} \bigg( - \int _t^{t+\varDelta}\widetilde{X}_i(\tau) \,\mathrm{d} \tau\bigg)^2\bigg] \\ &\quad{}+ \mathbb{E}_\mathrm{Q}^{(t,x)} \bigg[ \frac{1}{6} \bigg( - \int_t^{t+\varDelta}\widetilde{X}_i(\tau) \,\mathrm{d}\tau\bigg)^3 \mathrm{e}^{\xi_1} \bigg] \end{aligned}$$

and get in total that

$$\begin{aligned} \frac{1}{\mathbb{E}_\mathrm{Q}^{(t,x)}[\mathrm{e}^{-\int _t^{t+\varDelta}\widetilde{X}_i(\tau) \,\mathrm{d}\tau} ]} = 1 - y + y^2 - \frac{y^3}{(1+\xi_2)^4} . \end{aligned}$$

By using (b) and analogously to the proof of (e), we get

$$\begin{aligned} \bigg| \mathbb{E}_\mathrm{Q}^{(t,x)}\bigg[\bigg(\int _t^{t+\varDelta}\widetilde{X}_i(\tau) \,\mathrm{d}\tau \bigg)^3 \mathrm{e}^{\xi_1} \bigg] \bigg| \le C(t,x) \varDelta ^{\frac{9}{2}}. \end{aligned}$$

Analogously, we can show that $|y| \le C(t,x) \varDelta^{\frac{3}{2}}$, and so we can make y arbitrarily small by choosing a small Δ. We get that |1+ξ ₂|⁻⁴≤16 for all $| \xi_{2} | \le|y| \le \frac{1}{2}$ and obtain the claim. □

Proposition A.2

For 0≤t≤T≤T ^∗, let the mapping F as defined in Sect. 2 have a continuous derivative $f=\frac{\mathrm{d}}{\mathrm{d}T} F$ in the second argument and let there be integrable random variables W and $\widetilde{W}$ with

$$\begin{aligned} | F(t,T,r,m) | \le W \quad\textit{and} \quad \bigg| \frac{\mathrm{d}}{\mathrm{d}T} F(t,T,r,m) \bigg| \le\widetilde{W} . \end{aligned}$$

Given that $\mathbb{E}_{\mathrm{Q}}[F(t,T,r,m) | \mathcal{F}_{t}] = F(t,T,\rho,\mu)$, we get

$$\begin{aligned} \mathbb{E}_\mathrm{Q}\bigg[ \frac{\mathrm {d}}{\mathrm{d}T} F(t,T,r,m) \bigg| \mathcal{F}_{t}\bigg] = \frac{\mathrm{d}}{\mathrm{d}T} F(t,T,\rho,\mu) \quad \textit{a.s.} \end{aligned}$$

Proof

We have

$$\begin{aligned} F(t,T,r,m) = F(t,t,r,m) + \int_t^T \frac{d}{d u} F(t,u,r,m) \,\mathrm {d}u . \end{aligned}$$

From the definition of conditional expectations, we conclude for all $A \in\mathcal{F}_{t}$ that

$$\begin{aligned} &\int_A \mathbb{E}_\mathrm{Q}[ F(t,T,r,m) | \mathcal{F}_{t}] \, \mathrm{d}\mathrm{Q}= \int_A F(t,T,r,m) \,\mathrm{d}\mathrm{Q} \\ &\quad= \int_A \bigg( F(t,t,r,m) + \int_t^T \frac{\mathrm {d}}{\mathrm{d}u} F(t,u,r,m) \,\mathrm{d}u \bigg) \,\mathrm{d}\mathrm{Q} \\ &\quad= \int_A \mathbb{E}_\mathrm{Q}[F(t,t,r,m) | \mathcal{F}_{t}] \,\mathrm{d}\mathrm{Q}+ \int_A \int_t^T \mathbb{E}_\mathrm {Q}\bigg[ \frac{\mathrm{d}}{\mathrm{d}u} F(t,u,r,m) \bigg| \mathcal {F}_{t}\bigg] \,\mathrm{d}u \,\mathrm{d}\mathrm{Q}. \end{aligned}$$

The second summand of the last line can be obtained by exchanging the order of integration with Fubini’s theorem, applying again the definition of conditional expectations and using again Fubini’s theorem. This is where we need the existence of the integrable upper bounds W and $\widetilde{W}$. In total, we get for all T≥t that

$$\begin{aligned} F(t,t,r,m) + \int_t^T \mathbb{E}_\mathrm{Q}\bigg[ \frac{\mathrm {d}}{\mathrm{d}u} F(t,u,r,m) \bigg| \mathcal{F}_{t}\bigg] \,\mathrm{d}u & = \mathbb{E}_\mathrm{Q}[ F(t,T,r,m) | \mathcal{F}_{t}] \\ & = F(t,T,\rho,\mu) \\ & = F(t,t,\rho,\mu) + \int_t^T f(t,u,\rho,\mu) . \end{aligned}$$

Since f is continuous, the claim holds. □