2. General framework

In this section we construct a general framework for modeling an insurance liability cash flow. We consider a filtered probability space $(\Omega, {\mathcal G}, {\mathbb G}, P)$ , where ${\mathbb G} \,:\!=\, ({\mathcal G}_t)_{t \geqslant 0}$ , ${\mathcal G} = {\mathcal G}_\infty$ , and ${\mathcal G}_0$ is trivial.

We assume that ${\mathbb G} ={\mathbb F} \vee {\mathbb H}$ , where ${\mathbb F} \,:\!=\, ({\mathcal F}_t)_{t \geqslant 0}$ and ${\mathbb H} \,:\!=\, ({\mathcal H}_t)_{t \geqslant 0}$ are filtrations representing respectively a reference information flow and the internal information flow only available to the insurance company. Hence ${\mathbb G}$ describes the global information flow available to the insurance company. The reference filtration ${\mathbb F}$ typically includes information related to the financial market and to environmental, political, and social indicators. While we do not specify the structure of the reference filtration ${\mathbb F}$ , we assume that the insurance internal filtration ${\mathbb H}$ is generated by a family of marked point processes, representing the times and amounts of losses of the insurance portfolio, as in e.g. [Reference Arjas1], [Reference Jewell22], [Reference Norberg29], and [Reference Norberg30]. The filtrations ${\mathbb F}$ and ${\mathbb H}$ are not supposed be independent. Without loss of generality, we assume that all filtrations satisfy the conditions of completeness and right-continuity. If not otherwise specified, all relations in this paper hold in the P-almost sure (a.s.) sense. For detailed background on marked point processes we refer to e.g. [Reference Last and Brandt25], [Reference Daley and Vere-Jones15], and [Reference Jacobsen20]. In the following we use the classic terminology of non-life insurance (see e.g. [Reference Wuthrich and Merz37] and [Reference Parodi33]), and specify the filtration ${\mathbb H}$ as follows.

We consider an insurance portfolio with n policies. For the ith policy, with $i = 1,\ldots,n$ , the insurance company is typically informed about the accident that occurred at a random time $\tau^i_0$ only after a random delay $\theta^i$ , which may be very long, especially in the case of non-life insurance. Once the accident is reported at time $\tau^i_1$ , where

(2.1) \begin{equation}\tau^i_1 \,:\!= \tau^i_0 + \theta^i,\end{equation}

the accident time $\tau^i_0$ , the reporting delay $\theta^i$ , and the impact size of the accident, described by a nonnegative random variable $X^i_1$ , all become available information. In particular, we assume that for all $i=1,\ldots,n$ , $\tau^i_0 > 0$ P-a.s.

Let ${\mathbb N}_0$ be the set of natural numbers without zero. We describe the ith insurance policy movement by a marked point process $\big(\tau^i_j, \Theta^i_j\big)_{j \in {\mathbb N}_0}$ with two-dimensional nonnegative marks. That is, the sequence $\big({\tau}^i_j\big)_{j \in {\mathbb N}_0}$ is a point process, where

\begin{equation*}{\tau}^i_j\,:\, (\Omega, {\mathcal G}, P) \rightarrow \big(\overline{{\mathbb R}}_+, \mathcal{B}\big(\overline{{\mathbb R}}_+\big)\big), \ \ \ \ j \in {\mathbb N}_0,\end{equation*}

and $\big({\Theta}^i_j\big)_{j \in {\mathbb N}_0}$ is a sequence of two-dimensional nonnegative random variables, with

\begin{equation*}{\Theta}^i_j\,:\, (\Omega, {\mathcal G}, P) \rightarrow \big({{\mathbb R}}^2_+, \mathcal{B}\big({{\mathbb R}}^2_+\big)\big), \ \ \ \ j \in {\mathbb N}_0.\end{equation*}

For every $j \in {\mathbb N}_0$ , the random time $\tau^i_j$ describes the reporting time of the jth event related to the ith policy. The mark components $\Theta^i_j$ describe the reporting delay and the impact size of the corresponding event, respectively, which are known only if the event is reported. More precisely, we set

(2.2) \begin{equation}\tau^i_1 \ \ \ \ \text{with mark} \ \ \ \ \Theta^i_1 = \big(\theta^i, X^i_1\big),\end{equation}

and

(2.3) \begin{equation}\tau^i_{j + 1} = \tau^i_1 + \tilde{\tau}^i_j \ \ \ \ \text{with mark} \ \ \ \ \Theta^i_{j+1} = \Big(0, {X}^i_{j + 1}\Big) \,:\!=\, \Big(0, \tilde{X}^i_j\Big),\end{equation}

for $j \geqslant 1$ , where $\Big(\tilde{\tau}^i_j, \tilde{X}^i_{j}\Big)_{j \in {\mathbb N}_0}$ is an auxiliary marked point process, which describes updating and development after the first reporting at ${\tau}^i_1$ . For the sake of simplicity, we assume here that only the first reporting delay is different from zero, since in this paper we focus on modeling the first accident times $\tau^i_0$ and their relation to the reference filtration. However, our setting can be easily generalized by considering nonzero random delays in (2.3). We assume that the marked point process $\Big(\tilde{\tau}^i_j, \tilde{X}^i_{j}\Big)_{j \in {\mathbb N}_0}$ is simple, i.e.

\begin{equation*}\lim_{j \rightarrow \infty}{\tilde{\tau}}^i_j = \infty,\end{equation*}

and ${\tilde{\tau}}^i_j < {\tilde{\tau}}^i_{j+1}$ , if ${\tilde{\tau}}^i_j < \infty$ , and satisfies the following integrability condition:

(2.4) \begin{equation} \mathbb E\Bigg[\sum_{j=1}^\infty \textbf{1}_{\left\{ \tilde{\tau}^i_j \leqslant t \right\}} \tilde{X}^i_{j} \Bigg] < \infty \ \ \ \ \text{for all } t \geqslant 0,\end{equation}

for $i=1,\ldots,n$ . In particular, the random times $\big(\tau^i_j\big)_{j \in {\mathbb N}_0}$ are strictly ordered in case of finite value:

(2.5) \begin{equation}\begin{array}{l}\tau_1^i < \tau_2^i < \cdots < \tau_j^i < \tau_{j+1}^i < \cdots, \ \ \ i=1,\ldots,n.\end{array}\end{equation}

Note that we may have $\infty = \tau^i_j = \tau^i_{j+1} = \ldots$ ; in such a case, an infinite value stands for an event which never happens. For the sake of simplicity we assume also the following.

We emphasize that the above assumptions cover sufficient generality. The homogeneity assumptions can be satisfied by appropriately subdividing the insurance portfolio into homogeneous groups of claims. The independence assumptions reflect the fact that reporting delays $\theta^i$ , occurrences, and sizes of losses after the first reporting time, described by $\Big(\tilde{\tau}^i_j, \tilde{X}^i_j\Big)_{j \in {\mathbb N}_0}$ , are typically idiosyncratic factors which are independent of each other and independent of the reference information. However, we introduce a dependence structure by modeling ${\mathbb F}$ -progressively measurable occurrence intensities of the accidents, as we will present in (2.13) and (2.14). This will reflect the assumption that the occurrence intensity of accidents can be deduced from the reference information flow represented by ${\mathbb F}$ , while further updates of accident events $\Big(\tilde{\tau}^i_j, \tilde{X}^i_j\Big)_{j \in {\mathbb N}_0}$ are typically insurance-portfolio-specific and are not available as third-party or reference information. We assume furthermore that the distribution of delay variables $\theta^i$ , $i=1,\ldots,n$ , has the following structure.

Assumption 2. The common cumulative distribution function $G\,:\, [0,+\infty) \rightarrow [0,1]$ of $\theta^i$ , $i=1,\ldots,n$ , assigns probability $\alpha_0$ at 0 and has a density function g for $x>0$ , i.e.,

(2.6) \begin{equation}G(x) = \alpha_0 + \int_0^{x} g(y) \textrm{d} y, \ \ \ \ x \in {\mathbb R}_+.\end{equation}

According to the above assumption, the delays are nonnegative and may have a mixed distribution. In this way, we cover both the case of life insurance with $\theta^i \equiv 0$ , i.e. $g = 0$ , and the case of non-life insurance with non-null delays.

For every $i= 1, \ldots, n$ , we define the marked cumulative process $N^i$ by

\begin{equation*}N^i(t,B) (\omega) \,:\!=\, \sum_{j=1}^\infty \textbf{1}_{\big\{ \tau^i_j(\omega) \leqslant t \big\}} \textbf{1}_{\big\{ \Theta^i_j(\omega) \in B \big\}}, \end{equation*}

for every $t \geqslant 0$ , $B \in \mathcal{B}\big({\mathbb R}^2_+\big)$ , $\omega \in \Omega$ . The process $(\textbf{N}^i_t)_{t \geqslant 0}$ defined by

\begin{equation*}{\textbf{N}^i_t}\,:\!=\, N^i\big(t,{\mathbb R}^2_+\big) = \sum_{j=1}^\infty \textbf{1}_{\big\{ \tau^i_j \leqslant t \big\}}, \ \ \ \ t \geqslant 0,\end{equation*}

is called the ground process associated to the marked point process. At any time $t \geqslant 0$ , the random variable ${\textbf{N}^i_t}$ counts the number of occurrences of $\tau^i_j$ up to time t. In the literature, the name ‘marked point process’ sometimes refers to the process $N^i$ . Indeed, there is a unique correspondence between the marked point process $\Big(\tau^i_j, \Theta^i_j\Big)_{j \in {\mathbb N}_0}$ and its marked cumulative process $N^i$ . More precisely, $\tau^i_j$ can be uniquely defined by $N^i$ as

(2.7) \begin{equation}\big\{ \tau^i_j \leqslant t \big\} = \big\{ {\textbf{N}^i_t} \geqslant j \big\},\end{equation}

for all $t \geqslant 0$ , and $\Theta^i_j$ can be uniquely defined by $N^i$ as

(2.8) \begin{equation}\big\{\Theta^i_j \in B \big\} = \bigcup_{K^{\prime}=1}^\infty \bigcap_{K=K^{\prime}}^\infty \bigcup_{k=1}^\infty \Big\{{\textbf{N}^i_{(k-1)/2^K}} = n-1, N^i\big(k/2^K, B\big) - N^i\big((k-1)/2^K, B\big) = 1\Big\},\end{equation}

for all $B \in \mathcal{B}\big({\mathbb R}^2_+\big)$ . See Equations (2.8), (2.9) of [Reference Jacobsen20] and Lemma 2.2.2 of [Reference Last and Brandt25].

We consider the filtrations ${{\mathbb H}}^{i,1} \,:\!=\, \big({{\mathcal H}}^{i,1}_t\big)_{t \geqslant 0}$ with

\begin{equation*}{{\mathcal H}}^{i,1}_t \,:\!=\, \sigma \bigg( \textbf{1}_{\left\{ \tau^i_1 \leqslant s \right\}} \textbf{1}_{\big\{ \big(\theta^i, X^i_1\big) \in B \big\}}, 0 \leqslant s \leqslant t, \text{ for all } B \in \mathcal{B}\Big({\mathbb R}^2_+\Big) \bigg),\end{equation*}

for all $t \geqslant 0$ , and ${{\mathbb H}}^{i,j} \,:\!=\, \big({{\mathcal H}}^{i,j}_t\big)_{t \geqslant 0}$ , $j >1$ , with

\begin{equation*}{{\mathcal H}}^{i,j}_t \,:\!=\, \sigma \bigg( \textbf{1}_{\big\{ \tau^i_j \leqslant s \big\}} \textbf{1}_{\big\{ X^i_j \in B \big\}}, 0 \leqslant s \leqslant t, \text{ for all } B \in \mathcal{B}({\mathbb R}_+) \bigg),\end{equation*}

for all $t \geqslant 0$ . It holds that

\begin{equation*}{{\mathcal H}}^{i,j}_\infty = \sigma\big(\tau^i_j\big) \vee \sigma\big( X^i_j\big) \ \ \ \ \text{for } j > 1.\end{equation*}

In particular, in view of (2.1) we have

(2.9) \begin{equation} {{\mathcal H}}^{i,1}_\infty = \sigma\big(\tau^i_1\big) \vee \sigma\big(\big(\theta^i, X^i_1\big)\big) = \sigma\big(\tau^i_0\big) \vee \sigma\big(\big(\theta^i, X^i_1\big)\big).\end{equation}

Let ${\mathbb H}^i \,:\!=\, \big({\mathcal H}^i_t\big)_{t \geqslant 0}$ be the natural filtration of the marked cumulative process $N^i$ , that is, for all $t \geqslant 0$ ,

\begin{equation*}{\mathcal H}^i_t = \sigma \Big(N^i(s, B), 0 \leqslant s \leqslant t, \text{ for all } B \in \mathcal{B}\Big({\mathbb R}^2_+\Big)\Big).\end{equation*}

The internal information flow of the insurance company is described by the filtration ${\mathbb H} \,:\!=\, ({\mathcal H}_t)_{t \geqslant 0}$ , where

(2.10) \begin{equation}{\mathcal H}_t \,:\!=\, {\mathcal H}^1_t \vee \ldots \vee {\mathcal H}^n_t, \ \ \ \ t \geqslant 0.\end{equation}

Similarly, for $i=1,\ldots,n$ , we call $\tilde{N}^i$ the corresponding marked cumulative processes associated to the marked point processes $\Big(\tilde{\tau}^i_j, \tilde{X}^i_{j}\Big)_{j \in {\mathbb N}_0}$ and $\tilde{{\mathbb H}}^i$ the corresponding filtration, respectively. Similarly, all other notation associated to these last processes will be marked by the symbol ‘ $\sim$ ’.

Proof. Clearly, we have

\begin{equation*}{\mathcal H}^i_t \subseteq \bigvee_{j \in {\mathbb N}_0} {\mathcal H}^{i,j}_t.\end{equation*}

For the other inclusion, it is sufficient to show that for all $0 \leqslant s \leqslant t$ and $B \in \mathcal{B}\big({\mathbb R}^2_+\big)$ ,

\begin{equation*}\big\{\tau^i_j \leqslant s\big\} \cap \big\{\Theta^i_j \in B\big\} \in {\mathcal H}^i_t.\end{equation*}

This follows directly from (2.7) and (2.8).

We now introduce the following notation, which is useful in the sequel. For $i = 1,\ldots,n$ , $j \in {\mathbb N}_0$ , we define

\begin{equation*}{\mathbb H}^{i,\leqslant\, j} \,:\!=\, \bigvee_{k \leqslant j} {\mathbb H}^{i,k}, \ \ \ \ {\mathbb H}^{i,\geqslant j} \,:\!=\, \bigvee_{k \geqslant j} {\mathbb H}^{i,k},\end{equation*}

and similarly for ${\mathbb H}^{i,>j}$ and ${\mathbb H}^{i,< j}$ . In particular, in the case of $j = 1$ , we set ${\mathcal H}^{i,< 1}_t \,:\!=\, \{ \emptyset, \Omega \}$ for every $t \geqslant 0$ . The following corollary is a direct consequence of Lemma 1.

Corollary 1. For every $i = 1,\ldots,n$ , $j \in {\mathbb N}_0$ , we have

\begin{equation*}{\mathbb H}^i = {\mathbb H}^{i,\leqslant\, j} \vee {\mathbb H}^{i,> j} = {\mathbb H}^{i,< j} \vee {\mathbb H}^{i,\geqslant j}.\end{equation*}

Similarly to the reduced-form setting for credit risk and life insurance, we now model the accident times $\tau^i_0$ , $i = 1,\ldots,n$ , and their relation to the reference filtration in the following way. We assume that random times $\big(\tau^i_j\big)_{j \in {\mathbb N}}$ , $i = 1,\ldots,n$ , are not ${\mathbb F}$ -stopping times, i.e. for every i and j, there is t such that $\big\{\tau^i_j \leqslant t\big\} \notin {\mathcal F}_t$ . As in Section 9.1.2 of [Reference Bielecki and Rutkowski11], we set that accident times $\tau^i_0$ , $i = 1,\ldots,n$ , are such that for $t \in [0,\infty]$ and $s \in [0,t] \cap [0,\infty)$ ,

(2.11) \begin{equation}\mathbb P\Big(\tau^i_0 > s | {\mathcal F}_t \Big) = \mathbb P\Big(\tau^i_0 > s | {\mathcal F}_s \Big),\end{equation}

and for $l, k = 1,\ldots,n$ with $l \neq k$ , $\tau^l_0$ and $\tau^k_0$ are ${\mathbb F}$ -conditionally independent, i.e. if $t \in [0,\infty]$ and $r,s \in [0,t] \cap [0,\infty)$ , we have

(2.12) \begin{equation}\mathbb P\Big(\tau^l_0 > r, \tau^k_0 > s | {\mathcal F}_t \Big) = \mathbb P\Big(\tau^l_0 > r | {\mathcal F}_t \Big) P \Big(\tau^k_0 > s | {\mathcal F}_t\Big).\end{equation}

Remark 1. If we define ${\mathcal H}^{i,0}_t \,:\!=\, \sigma \big( \textbf{1}_{\big\{ \tau^i_0 \leqslant s \big\}}\,:\, 0 \leqslant s \leqslant t \big)$ , $i = 1,\ldots,n$ , then the condition (2.11) is equivalent to

\begin{equation*}\mathbb E[ X | {\mathcal F}_t] = \mathbb E[ X | {\mathcal F}_s],\end{equation*}

for each integrable ${\mathcal H}^{i,0}_s$ -measurable random variable X. The condition (2.12) is equivalent to the ${\mathcal F}_t$ -conditional independence between the $\sigma$ -algebras ${\mathcal H}^{l,0}_t$ and ${\mathcal H}^{k,0}_t$ .

Furthermore, if $F^i \,:\!=\, \big(F^i_t\big)_{t \geqslant 0}$ is the ${\mathbb F}$ -conditional cumulative process of $\tau^i_0$ ,

\begin{equation*}F^i_t \,:\!=\, \mathbb P\big( \tau^i_0 \leqslant t | \mathcal{F}_t\big), \ \ \ \ t \geqslant 0,\end{equation*}

we assume that there exists a locally integrable and ${\mathbb F}$ -progressively measurable process $\mu^i \,:\!=\, \big(\mu^i_t\big)_{t \geqslant 0}$ such that

(2.13) \begin{equation}e^{- \int_0^t \mu^i_s \textrm{d} s} = 1 - F^i_t \ \ \ \ \text{for all } t \geqslant 0.\end{equation}

We define $\Gamma^i \,:\!=\, \big(\Gamma^i_t\big)_{t \geqslant 0}$ as

(2.14) \begin{equation}\Gamma^i_t \,:\!=\, \int_0^t \mu^i_s \textrm{d} s, \ \ \ \ t \geqslant 0.\end{equation}

The process $\mu^i$ is called the intensity process of the random jump time $\tau^i_0$ , and the process $\Gamma^i$ is called the hazard process of $\tau^i_0$ . An explicit construction in Example 9.1.5 of [Reference Bielecki and Rutkowski11] shows that for a given family of locally integrable ${\mathbb F}$ -progressively measurable process $\mu^i$ , $i = 1,\ldots,n$ , it is always possible to construct random times $\tau^i_0$ , $i = 1,\ldots,n$ , such that $\Gamma^i$ is the hazard process of $\tau^i_0$ for every $i = 0,\ldots,n$ , and all the assumptions above are satisfied. For the sake of simplicity, we assume that the insurance portfolio is homogeneous.

Under this homogeneity condition, we denote the common ${\mathbb F}$ -conditional cumulative function and hazard process respectively by F and $\Gamma$ . The above assumption reflects the fact that, while the policy developments may not have a direct link to the information flow ${\mathbb F}$ , the accident occurrences $\tau^i_0$ , $i = 1,\ldots,n$ , are influenced by some common external systematic risk factors described by the occurrence intensity $\mu$ and observable from filtration ${\mathbb F}$ .

We now show how the general framework described above comprehends in a synthetic way both life and non-life insurance modeling, and compare our setting with the existing literature.

2.1. Life insurance

Life insurance policies typically do not have reporting delays and depend only on $\tau^i_0$ , $i = 1,\ldots,n$ , which actually represent the decease times. This can be included in our framework by setting $\theta^i \equiv 0$ , $\tau^i_j \equiv \infty$ for all $j > 1$ , and $X^i_j \equiv 1$ for all $j \in {\mathbb N}_0$ , and interpreting $\tau^i_0$ as the decease time of person i, where $i = 1,\ldots,n$ . The filtration ${\mathbb G}$ is hence reduced to

\begin{equation*}{\mathbb G} = {\mathbb F} \vee {\mathbb H}^1 \vee \ldots \vee {\mathbb H}^n,\end{equation*}

where

\begin{equation*}{\mathcal H}^i_t = \sigma \bigg( \textbf{1}_{\big\{ \tau^i_0 \leqslant s \big\}}, 0 \leqslant s \leqslant t \bigg), \ \ \ \ t \geqslant 0, \ i = 1,\ldots,n.\end{equation*}

In particular, the ${\mathbb F}$ -progressively measurable process $\mu$ is interpreted as mortality intensity in this context. The financial market is typically assumed to include mortality- or longevity-linked derivatives, such as longevity bonds, which pay off the longevity index value $e^{- \int_0^T \mu_s \textrm{d} s}$ at maturity T.

Life insurance within a hybrid market in this setting has been intensively studied in the literature; see e.g. [Reference Barbarin2], [Reference Biagini and Schreiber9], [Reference Biagini, Rheinländer and Schreiber7], and [Reference Biagini and Zhang10].

3. Valuation formulas

In this section, we state several results within the framework presented in Section 2, i.e. under the structure of Assumptions 1, 2, and3. We follow [Reference Bielecki and Rutkowski11, Section 5.1] for the presentation. These preliminary calculations are fundamental for providing the pricing formulas for non-life insurance claims in Section 5.1.

We start with an extension of the relation (2.11) and the ${\mathbb F}$ -independence (2.12) of $\tau^i_0$ , $i=1,\ldots,n$ . In particular, if these relations hold for the filtrations ${\mathbb H}^{i,0}$ , $i=1,\ldots,n$ , then they also hold for the filtrations ${\mathbb H}^i$ , $i = 1,\ldots,n$ .

Proof. The proof is straightforward in view of Lemma 1, Remark 1, and the independence conditions in Assumption 1. We outline the main steps for the first part of the lemma for completeness; the second part can be shown in a similar manner. See also the proofs of Lemma 3.2.1 and Lemma 3.2.2 in [Reference Zhang38].

In view of Lemma 1, (2.2), and (2.3), for any $t \in [0, \infty]$ and $l,k = 1,\ldots,n$ with $l \neq k$ , the $\sigma$ -algebras ${\mathcal H}^l_t$ and ${\mathcal H}^k_t$ are ${\mathcal F}_t$ -independent if and only if

\begin{align*}&\mathbb{E}\bigg[ \textbf{1}_{\big\{ \tau^l_0 + \theta^l + \tilde{\tau}^l_p \leqslant s \big\}} \textbf{1}_{\big\{ \tilde{X}^l_p \in B^l \big\}} \textbf{1}_{\big\{ \tau^k_0 + \theta^k + \tilde{\tau}^k_q \leqslant r \big\}} \textbf{1}_{\big\{ \tilde{X}^k_p \in B^k \big\}} \big| {\mathcal F}_t \bigg]\\[3pt]=& \mathbb{E}\bigg[ \textbf{1}_{\big\{ \tau^l_0 + \theta^l + \tilde{\tau}^l_p \leqslant s \big\}} \textbf{1}_{\big\{ \tilde{X}^l_p \in B^l \big\}} \big| {\mathcal F}_t \bigg] \mathbb{E}\bigg[ \textbf{1}_{\big\{ \tau^k_0 + \theta^k + \tilde{\tau}^k_q \leqslant r \big\}} \textbf{1}_{\big\{ \tilde{X}^k_p \in B^k \big\}} \big| {\mathcal F}_t \bigg],\end{align*}

where $s,r \in [0,t] \cap [0, \infty)$ (note that t may assume the value $\infty$ ), $B^l, B^k \in \mathcal{B}({\mathbb R}_+)$ , and $p, q \in {\mathbb N}_0$ . Without loss of generality, we take $p \neq 1$ and $q \neq 1$ . We consider the following deterministic functions:

\begin{equation*}f^l (x) \,:\!=\, \mathbb{E}\bigg[ \textbf{1}_{\big\{ \theta^l + \tilde{\tau}^l_p \leqslant s - x \big\}} \textbf{1}_{\big\{ \tilde{X}^l_p \in B^l \big\}} \bigg], \ \ \ \ f^l (x) = f^l (x)\textbf{1}_{\left\{ x \leqslant s \right\}},\end{equation*}

\begin{equation*}f^k (x) \,:\!=\, \mathbb{E}\bigg[ \textbf{1}_{\big\{ \theta^k + \tilde{\tau}^k_q \leqslant r - x \big\}} \textbf{1}_{\big\{ \tilde{X}^k_q \in B^k \big\}} \bigg], \ \ \ \ f^k (x) = f^k (x)\textbf{1}_{\left\{ x \leqslant r \right\}}.\end{equation*}

Hence, $f^l \big(\tau^l_0\big)$ and $f^k \big(\tau^k_0\big)$ are respectively ${\mathcal H}^{l,0}_t$ - and ${\mathcal H}^{k,0}_t$ -measurable. This together with Remark 1 and the independence conditions in Assumption 1 leads to the ${\mathcal F}_t$ -independence of ${\mathcal H}^l_t$ and ${\mathcal H}^k_t$ ,

\begin{align*}& \ \ \mathbb{E}\bigg[ \textbf{1}_{\big\{ \tau^l_0 + \theta^l + \tilde{\tau}^l_p \leqslant s \big\}} \textbf{1}_{\big\{ \tilde{X}^l_p \in B^l \big\}} \textbf{1}_{\big\{ \tau^k_0 + \theta^k + \tilde{\tau}^k_q \leqslant r \big\}} \textbf{1}_{\big\{ \tilde{X}^k_p \in B^k \big\}} \big| {\mathcal F}_t \bigg]\\&= \mathbb{E}\bigg[ \mathbb{E}\Big[ \textbf{1}_{\big\{ \tau^l_0 + \theta^l + \tilde{\tau}^l_p \leqslant s \big\}} \textbf{1}_{\big\{ \tilde{X}^l_p \in B^l \big\}} \textbf{1}_{\big\{ \tau^k_0 + \theta^k + \tilde{\tau}^k_q \leqslant r \big\}} \textbf{1}_{\big\{ \tilde{X}^k_p \in B^k \big\}} \big| {\mathcal F}_t \vee \sigma\big(\tau_0^l\big) \vee \sigma\big(\tau_0^k\big) \Big] \big| {\mathcal F}_t \bigg]\\&= \mathbb{E}\Bigg[ \mathbb E\big[ \textbf{1}_{\big\{ x + \theta^l + \tilde{\tau}^l_p \leqslant s \big\}} \textbf{1}_{\big\{ \tilde{X}^l_p \in B^l \big\}} \textbf{1}_{\big\{ y + \theta^k + \tilde{\tau}^k_q \leqslant r \big\}} \textbf{1}_{\big\{ \tilde{X}^k_p \in B^k \big\}} \big]\big|_{\substack{x = \tau^l_0\\ y = \tau^k_0}} \big| {\mathcal F}_t \Bigg]\\ &= \mathbb{E}\Big[ f^l \big(\tau^l_0\big) f^k \big(\tau^k_0\big) \big| {\mathcal F}_t \Big]\\ &= \mathbb{E}\Big[ f^l \big(\tau^l_0\big)\big| {\mathcal F}_t \Big] \mathbb E\Big[ f^k \big(\tau^k_0\big) \big| {\mathcal F}_t \Big]\\ &= \mathbb{E}\bigg[ \textbf{1}_{\big\{ \tau^l_0 + \theta^l + \tilde{\tau}^l_p \leqslant s \big\}} \textbf{1}_{\big\{ \tilde{X}^l_p \in B_l \big\}} \big| {\mathcal F}_t \bigg] \mathbb E\bigg[ \textbf{1}_{\big\{ \tau^k_0 + \theta^k + \tilde{\tau}^k_q \leqslant r \big\}} \textbf{1}_{\big\{ \tilde{X}^k_p \in B_k \big\}} \big| {\mathcal F}_t \bigg].\end{align*}

As a consequence of Lemma 3, the ${\mathbb G}$ -conditional expectation can be reduced to $({\mathbb F} \vee {\mathbb H}^i)$ -conditional expectation in most cases.

Corollary 2. If $0 \leqslant t \leqslant T < \infty$ , and Y is an integrable $({\mathcal F}_T \vee {\mathcal H}^{i}_T)$ -measurable random variable, then

\begin{equation*}\left. \mathbb E\left[ Y \right| {\mathcal G}_t \right] = \left. \mathbb E\big[ Y \right| {\mathcal F}_t \vee {\mathcal H}^{i}_t \big].\end{equation*}

Proof. It is sufficient to prove the statement for the indicator functions of the form $Y = \textbf{1}_A \textbf{1}_B$ where $A \in {\mathcal F}_T$ and $B \in {\mathcal H}^i_T$ . Given $C \in {\mathcal F}_t$ , $D^j \in {\mathcal H}^j_t$ , $j = 1,\ldots,n$ , it follows from Lemma 2 that

\begin{equation*}\int_{C \cap D^1 \cap \ldots \cap D^n} \textbf{1}_A \textbf{1}_B \textrm{d} P = \int_{C \cap D^1 \cap \ldots \cap D^n} \left. \mathbb E\big[ \textbf{1}_A \textbf{1}_B \right| {\mathcal F}_t \vee {\mathcal H}^{i}_t \big] \textrm{d} P.\end{equation*}

Details can also be found in the proof of Corollary 3.2.3 in [Reference Zhang38].

Another important consequence of Lemma 2 is the so-called H-hypothesis between the filtrations ${\mathbb F}$ and ${\mathbb G}$ , i.e. the property that every ${\mathbb F}$ -martingale is also a ${\mathbb G}$ -martingale.

Proof. By Lemma 6.1.1 of [Reference Bielecki and Rutkowski11], it is equivalent to the H-hypothesis between two filtrations ${\mathbb F} \subseteq {\mathbb G}$ that for any $t \geqslant 0$ and any bounded, ${\mathcal G}_t$ -measurable random variable $\eta$ , it holds that

(3.1) \begin{equation}\mathbb{E}\big[\eta \big| {\mathcal F}_\infty \big] = {E}\big[\eta \big| {\mathcal F}_t \big].\end{equation}

It is sufficient to prove (3.1) for indicator functions of the form $\textbf{1}_A \textbf{1}_{B^1} \ldots \textbf{1}_{B^n}$ , where $A \in {\mathcal F}_t$ , $B^i \in {\mathcal H}^i_t$ , $i = 1,\ldots,n$ . This can be achieved by applying Lemma 2 multiple times to get

\begin{align*}\mathbb E\left. \left[\textbf{1}_A \textbf{1}_{B^1} \ldots \textbf{1}_{B^n} \right| {\mathcal F}_\infty \right]&= \mathbb E\left. \left[\textbf{1}_A \textbf{1}_{B^1} \ldots \textbf{1}_{B^n} \right| {\mathcal F}_t \right].\end{align*}

Now we would like to derive some more explicit representations. We note that for every integrable random variable Y, $t \geqslant 0$ , $i = 1,\ldots,n$ , and $j \in {\mathbb N}_0$ , we have the decomposition

(3.2) \begin{equation} \mathbb E\big[ Y \big| {{\mathcal H}}^{i}_t \vee {\mathcal F}_t \big] = \mathbb E\bigg[ \textbf{1}_{\big\{ \tau^i_j > t \big\}} Y \big| {{\mathcal H}}^{i}_t \vee {\mathcal F}_t \bigg] + \mathbb E\bigg[ \textbf{1}_{\big\{ \tau^i_j \leqslant t \big\}} Y \big| {{\mathcal H}}^{i}_t \vee {\mathcal F}_t \bigg].\end{equation}

In the following we will evaluate separately the two components on the right-hand side of (3.2). The following lemma is important for deriving a representation of the first component.

Lemma 3. For any $t \geqslant 0$ , $i = 1,\ldots,n$ , and $j \in {\mathbb N}_0$ , we have

\begin{equation*}{\mathcal H}^i_t \vee {\mathcal F}_t \subseteq {\mathcal G}^{i,j}_t,\end{equation*}

where

(3.3) \begin{equation}{\mathcal G}^{i,j}_t \,:\!=\, \Big\{ A \in {\mathcal G} \,:\, \exists C \in {\mathcal H}^{i, <j}_t \vee {\mathcal F}_t, A \cap \Big\{\tau^i_j >t \Big\} = C \cap \Big\{\tau^i_j > t \Big\} \Big\}.\end{equation}

Proof. By Corollary 1, it holds that

\begin{equation*}{\mathcal H}^i_t = {\mathcal H}^{i, < j}_t \vee {\mathcal H}^{i, \geqslant j}_t.\end{equation*}

Hence, it is sufficient to show that both ${{\mathcal H}}^{i, \geqslant j}_t$ and ${{\mathcal H}}^{i, < j}_t \vee {\mathcal F}_t$ belong to ${\mathcal G}^{i,j}_t$ . In the first case, if $i > 1$ and $A = \big\{\tau^i_k \leqslant s\big\} \cap \big\{ X^i_k \in B\big\}$ for some $k \geqslant j$ , $0 \leqslant s \leqslant t$ , and $B \in \mathcal{B}({\mathbb R})$ , we take $C = \emptyset$ ; and similarly for $i=1$ and $A = \big\{\tau^1_k \leqslant s\big\} \cap \big\{\big(\theta_k, X^1_k\big) \in B\big\}$ for $k \geqslant j$ , $0 \leqslant s \leqslant t$ , and $B \in \mathcal{B}\big({\mathbb R}^2_+\big)$ . In the second case, if $A \in {{\mathcal H}}^{i, < j}_t \vee {\mathcal F}_t$ , we take $C = A$ .

The following proposition gives two representations of the first component on the right-hand side of (3.2). The representation (3.4) is analogous to Lemma 5.1.2 in [Reference Bielecki and Rutkowski11]; the representation (3.5) is new and will be used for our further discussion.

Proposition 1. For any $t \geqslant 0$ , $i = 1,\ldots,n$ , $j \in {\mathbb N}_0$ and any integrable ${\mathcal G}$ -measurable random variable Y, we have

(3.4) \begin{align} \mathbb E\Big[ \textbf{1}_{\big\{ \tau^i_j > t \big\}} Y \big| {{\mathcal H}}^{i}_t \vee {\mathcal F}_t \Big] &= \textbf{1}_{\big\{ \tau^i_j > t \big\}} \frac{ \mathbb E\Big[ \textbf{1}_{\big\{ \tau^i_j > t \big\}} Y \big| {\mathcal H}^{i, <j}_t \vee {\mathcal F}_t \Big]}{P \Big( \tau^i_j >t \big| {\mathcal H}^{i, <j}_t \vee {\mathcal F}_t \Big)} \end{align}

(3.5) \begin{align} \qquad\qquad\qquad &= \textbf{1}_{\big\{ \tau^i_j > t \big\}} \mathbb E\Big[ Y \big| {\mathcal H}^{i, \leqslant j}_t \vee {\mathcal F}_t \Big]. \end{align}

Proof. The equality (3.4) is equivalent to

\begin{align*} \mathbb E\bigg[ \textbf{1}_{\big\{ \tau^i_j > t \big\}} Y\ \ P \Big( \tau^i > t \big| {\mathcal H}^{i, <j}_t \vee {\mathcal F}_t\Big) \big| {{\mathcal H}}^{i}_t \vee {\mathcal F}_t \bigg] = \textbf{1}_{\big\{ \tau^i_j >t \big\}} \mathbb E\bigg[ \textbf{1}_{\big\{ \tau^i_j > t \big\}} Y \big| {\mathcal H}^{i, <j}_t \vee {\mathcal F}_t \bigg].\end{align*}

We note that the right-hand side is $\big({{\mathcal H}}^{i}_t \vee {\mathcal F}_t\big)$ -measurable. Hence, it suffices to show that for any $A \in {{\mathcal H}}^{i}_t \vee {\mathcal F}_t$ ,

\begin{align*}\int_A \textbf{1}_{\big\{ \tau^i_j > t \big\}} Y\ \ P \Big( \tau^i_j > t \big|{\mathcal H}^{i, <j}_t \vee {\mathcal F}_t \Big) \textrm{d} P= \int_A \textbf{1}_{\big\{ \tau^i_j >t \big\}} \mathbb E\bigg[ \textbf{1}_{\big\{ \tau^i_j > t \big\}} Y \big|{\mathcal H}^{i, <j}_t \vee {\mathcal F}_t \bigg] \textrm{d} P.\end{align*}

By Lemma 3, there is an event $C \in {\mathcal H}^{i, <j}_t \vee {\mathcal F}_t$ such that

\begin{equation*}A \cap \big\{ \tau^i_j > t \big\} = C \cap \big\{ \tau^i_j > t \big\};\end{equation*}

hence

\begin{align*}&\int_A \textbf{1}_{\left\{ \tau^i_j > t \right\}} Y \left.P \left( \tau^i_j > t \right| {\mathcal H}^{i, <j}_t \vee {\mathcal F}_t \right) \textrm{d} P\\[3pt]=&\int_C \textbf{1}_{\left\{ \tau^i_j >t \right\}} Y \left.P \left( \tau^i_j > t \right|{\mathcal H}^{i, <j}_t \vee {\mathcal F}_t \right) \textrm{d} P \\[3pt]=&\int_C \left.\mathbb E\left[ \textbf{1}_{\left\{ \tau^i_j >t \right\}} Y \right|{\mathcal H}^{i, <j}_t \vee {\mathcal F}_t \right] \left.\mathbb E\left[ \textbf{1}_{\left\{ \tau^i_j >t \right\}} \right|{\mathcal H}^{i, <j}_t \vee {\mathcal F}_t \right] \textrm{d} P \\[3pt]=&\int_C \left.\mathbb E\left[ \textbf{1}_{\left\{ \tau^i_j >t \right\}} \left.\mathbb E\left[ \textbf{1}_{\left\{ \tau^i_j > t \right\}} Y \right|{\mathcal H}^{i, <j}_t \vee {\mathcal F}_t \right] \right| {\mathcal H}^{i, <j}_t \vee {\mathcal F}_t \right] \textrm{d} P \\[3pt]=&\int_C \textbf{1}_{\left\{ \tau^i_j >t \right\}} \left.\mathbb E\left[ \textbf{1}_{\left\{ \tau^i_j > t \right\}} Y \right|{\mathcal H}^{i, <j}_t \vee {\mathcal F}_t \right] \textrm{d} P \\[3pt]=& \int_A \textbf{1}_{\left\{ \tau^i_j >t \right\}} \left. \mathbb E\left[ \textbf{1}_{\left\{ \tau^i_j > t \right\}} Y \right| {\mathcal H}^{i, <j}_t \vee {\mathcal F}_t \right] \textrm{d} P.\end{align*}

The equality (3.5) can be proved in the same way. We only need to observe that

\begin{equation*}{\mathcal G}^{i,j}_t \subseteq \Big\{ A \in {\mathcal G} \,:\, \exists C \in {\mathcal H}^{i, \leqslant j}_t \vee {\mathcal F}_t, A \cap \big\{\tau^i_j > t \big\} = C \cap \big\{\tau^i_j > t \big\} \Big\}.\end{equation*}

Hence, the $\sigma$ -algebra ${\mathcal H}^{i,<j}_t$ in (3.4) can be replaced by ${\mathcal H}^{i,\leqslant\, j}_t$ . This concludes the proof.

Now we focus on the second component on the right-hand side of (3.2). The following lemma gives a slightly more general result.

Lemma 4. For any $t \geqslant 0$ , $i = 1,\ldots,n$ , $j \in {\mathbb N}_0$ , any $\sigma$ -algebra $\mathcal{A} \subseteq {\mathcal G}$ , and any integrable ${\mathcal G}$ -measurable random variable Y, we have

\begin{equation*} \mathbb E\bigg[ \textbf{1}_{\big\{ \tau^i_j \leqslant t \big\}} Y \big| {\mathcal H}^{i,\leqslant\, j}_t \vee \mathcal{A} \bigg] = \mathbb E\bigg[ \textbf{1}_{\big\{ \tau^i_j \leqslant t \big\}} Y \big| {\mathcal H}^{i,\leqslant\, j}_\infty \vee \mathcal{A} \bigg].\end{equation*}

Proof. We note that the left-hand side is $\Big({\mathcal H}^{i,\leqslant\, j}_\infty \vee \mathcal{A}\Big)$ -measurable. Since the marked point process $\big(\tau^i_j, \Theta^i_j\big)_{j \in {\mathbb N}_0}$ is simple, i.e. the strict monotonicity (2.5) holds, if $A \in {\mathcal H}^{i,\leqslant\, j}_\infty \vee \mathcal{A}$ , then $A \cap \big\{\tau^i_j \leqslant t\big\} \in {\mathcal H}^{i,\leqslant\, j}_t \vee \mathcal{A}$ , and

\begin{align*}\int_{A} \textbf{1}_{\big\{ \tau^i_j \leqslant t \big\}} Y\ \textrm{d} P &= \int_{A \cap \{\tau^i_j \leqslant t\}} Y\ \textrm{d} P = \int_{A \cap \{\tau^i_j \leqslant t\}} \mathbb E\Big[ Y | {\mathcal H}^{i,\leqslant\, j}_t \vee \mathcal{A} \Big] \textrm{d} P\\&= \int_A \left. \mathbb E\left[ \textbf{1}_{\big\{ \tau^i_j \leqslant t \big\}} Y \right| {\mathcal H}^{i,\leqslant\, j}_t \vee \mathcal{A} \right] \textrm{d} P.\end{align*}

This concludes the proof.

Remark 2. Since we have

\begin{equation*}{\mathcal H}^{i,\leqslant\, j}_\infty = \sigma\!\left(\tau^i_h, h=1,\ldots,j \right),\end{equation*}

Lemma 4 shows that, if $\tau^i_j$ has occurred before time t, then partial information about $\tau^i_j$ up to t is equivalent to full information about all the random times $\tau^i_h$ , $h=1,\ldots,j$ . In particular, if Y is a function of $\tau^i_1, \ldots, \tau^i_j$ , i.e. $Y = f (\tau^i_1, \ldots, \tau^i_j)$ , then the conditional expectation is simply

\begin{equation*} \mathbb E\Big[ \textbf{1}_{\big\{ \tau^i_j \leqslant t \big\}} Y \big| {\mathcal H}^{i,\leqslant\, j}_t \vee \mathcal{A} \Big] = \textbf{1}_{\big\{ \tau^i_j \leqslant t \big\}} Y.\end{equation*}

We summarize the above results in the following representation theorem.

Theorem 4. For any $t \geqslant 0$ , $i = 1,\ldots,n$ , $j \in {\mathbb N}_0$ and any integrable ${\mathcal G}$ -measurable random variable Y, we have

\begin{equation*} \mathbb E\big[ Y \big| {{\mathcal H}}^{i}_t \vee {\mathcal F}_t \big] = \textbf{1}_{\big\{ \tau^i_j \leqslant t \big\}} \mathbb E\Big[ Y \big| {\mathcal H}^{i,\leqslant\, j}_\infty \vee {{\mathcal H}}^{i, > j}_t \vee {\mathcal F}_t \Big] + \textbf{1}_{\big\{ \tau^i_j > t \big\}} \mathbb E\Big[ Y \big| {\mathcal H}^{i, \leqslant j}_t \vee {\mathcal F}_t \Big].\end{equation*}

If furthermore Y is $\big({{\mathcal H}}^{i}_T \vee {\mathcal F}_T\big)$ -measurable, then

\begin{equation*} \mathbb E\big[ Y \big| {\mathcal G}_t \big] = \textbf{1}_{\big\{ \tau^i_j \leqslant t \big\}} \mathbb E\Big[ Y \big| {\mathcal H}^{i,\leqslant\, j}_\infty \vee {{\mathcal H}}^{i, > j}_t \vee {\mathcal F}_t \Big] + \textbf{1}_{\big\{ \tau^i_j > t \big\}} \mathbb E\Big[ Y \big| {\mathcal H}^{i, \leqslant j}_t \vee {\mathcal F}_t \Big].\end{equation*}

Proof. Since

\begin{equation*} \mathbb E\big[ Y \big| {{\mathcal H}}^{i}_t \vee {\mathcal F}_t \big] = \mathbb E\bigg[ \textbf{1}_{\big\{ \tau^i_j \leqslant t \big\}} Y \big| {{\mathcal H}}^{i}_t \vee {\mathcal F}_t \bigg] + \mathbb E\bigg[ \textbf{1}_{\big\{ \tau^i_j > t \big\}} Y \big| {{\mathcal H}}^{i}_t \vee {\mathcal F}_t \bigg],\end{equation*}

the first part is a straightforward consequence of Proposition 1 and Lemma 4. For the second part, it suffices to apply Corollary 2.

We now prove some results which we will use to solve the reserve estimation problem in Section 5.1. For this purpose, our approach allows us to obtain analytical formulas in a general setting in continuous time, where filtration dependence is taken into account. As we illustrate in Section 4, this is not possible in such generality when using more classical approaches. Let $0 \leqslant t \leqslant T < \infty$ , and let $Z \,:\!=\, (Z_t)_{t \in[0,T]}$ be a continuous, bounded, and ${\mathbb F}$ -adapted process. For $i = 1,\ldots,n$ , we now consider

(3.6) \begin{equation}Y = \sum^{\textbf{N}^i_T}_{j = \textbf{N}^i_t} X^i_j Z_{\tau^i_j} = \sum^{\infty}_{j = 1} \textbf{1}_{\big\{ t <\tau^i_j \leqslant T \big\}} X^i_j Z_{\tau^i_j},\end{equation}

and compute

(3.7) \begin{equation}\mathbb E\left. \left[ Y \right| {\mathcal G}_t \right] = \mathbb E\left. \left[ \sum^{\textbf{N}^i_T}_{j = \textbf{N}^i_t} X^i_j Z_{\tau^i_j} \right| {\mathcal G}_t \right].\end{equation}

In particular, as before, we study separately the two components of the decomposition of (3.7) with respect to the first reporting time $\tau^i_1$ , i.e.

(3.8) \begin{equation}\mathbb E\left. \left[ \sum^{\textbf{N}^i_T}_{j = \textbf{N}^i_t} X^i_j Z_{\tau^i_j} \right| {\mathcal G}_t \right] = \mathbb E\left. \left[ \textbf{1}_{\left\{ \tau^i_1 > t \right\}} \sum^{\textbf{N}^i_T}_{j = \textbf{N}^i_t} X^i_j Z_{\tau^i_j} \right| {\mathcal G}_t \right] + \mathbb E\left. \left[ \textbf{1}_{\left\{ \tau^i_1 \leqslant t \right\}} \sum^{\textbf{N}^i_T}_{j = \textbf{N}^i_t} X^i_j Z_{\tau^i_j} \right| {\mathcal G}_t \right],\end{equation}

and derive more explicit formulas in terms of the intensity process $\mu$ , the distribution of delay $\theta^i$ , and the distribution of development $N^i$ after the first reporting. We start with the ${\mathbb F}$ -conditional expectation of $\tau^i_1$ .

Lemma 5. For any $i = 1,\ldots,n$ and $t \geqslant 0$ , we have

(3.9) \begin{equation}\mathbb P\big( \tau^i_1 \leqslant t \big| {\mathcal F}_t \big) = \int_0^t G(t - s) e^{-\int^s_0 \mu_v \textrm{d} v} \mu_s \textrm{d} s,\end{equation}

where G is the cumulative distribution function of $\theta^i$ given in (2.6).

Proof. Note that by Assumption 1, $\theta^i$ is independent of ${\mathcal F}_t \vee \sigma\big(\tau^i_0\big)$ . Furthermore, both $\theta^i$ and $\tau^i_0$ are P-a.s. nonnegative. Therefore, we have

\begin{align*}\mathbb P \big( \tau^i_1 \leqslant t \big| {\mathcal F}_t \big) &= \mathbb E \Big[ \textbf{1}_{\big\{ \tau^i_0 + \theta^i \leqslant t \big\}} \Big| {\mathcal F}_t \Big] \\&= \mathbb E \bigg[ \mathbb E \bigg[ \textbf{1}_{\big\{ \tau^i_0 \leqslant t \big\}}\textbf{1}_{\big\{ \tau^i_0 + \theta^i \leqslant t \big\}} \bigg| {\mathcal F}_t \vee \sigma\big(\tau^i_0\big) \bigg] \bigg| {\mathcal F}_t \bigg]\\&= \mathbb E \bigg[ \textbf{1}_{\big\{ \tau^i_0 \leqslant t \big\}} \mathbb E\bigg[ \textbf{1}_{\big\{ \theta^i \leqslant t -x \big\}} \bigg]\bigg|_{x = \tau^i_0} \bigg| {\mathcal F}_t \bigg] \\&= \mathbb E \bigg[ \textbf{1}_{\big\{ \tau^i_0 \leqslant t \big\}} {G}\big(t - \tau^i_0\big) \big| {\mathcal F}_t \bigg].\end{align*}

To conclude we only need to show

(3.10) \begin{equation}\mathbb E \bigg[ \textbf{1}_{\big\{ \tau^i_0 \leqslant t \big\}} {G}\big(t - \tau^i_0\big) \big| {\mathcal F}_t \bigg] = \int_0^t {G}(t - s) e^{-\int^s_0 \mu_v \textrm{d} v} \mu_s \textrm{d} s.\end{equation}

This can be done in the same way as for Proposition 5.1.1 of [Reference Bielecki and Rutkowski11], in view of the relation (2.11) and the fact that G is continuous according to Assumption 2. First the relation (3.10) can be shown for a piecewise constant function G; it is then obtained as a limit for continuous G.

In the expression (3.9) of Lemma 5, the parameter t also appears in the integrand. The following corollary improves the relation (3.9) and shows that the process of conditional expectation $(\mathbb P\left. \left( \tau^i_1 \leqslant t \right| {\mathcal F}_t \right))_{t \geqslant 0}$ is absolutely continuous with respect to the Lebesgue measure.

Corollary 4. For any $i=1,\ldots,n$ , we have

(3.11) \begin{equation}\mathbb P\left. \left( \tau^i_1 \leqslant t \right| {\mathcal F}_t \right) = \int_0^t \left( \alpha_0 e^{-\int^s_0 \mu_v \textrm{d} v} \mu_s + \int_0^s g(s - u) e^{-\int^u_0 \mu_v \textrm{d} v} \mu_u \textrm{d} u \right)\textrm{d} s,\end{equation}

where $\alpha_0$ and g are defined in (2.6).

Note that in the following lemma we do not assume that Z is ${\mathbb F}$ -adapted and the boundedness condition can be generalized.

Lemma 6. If the process $Z \,:\!=\, (Z_u)_{u \in [t, T]}$ is left-continuous and bounded and $Z_t$ is ${\mathcal F}_T$ -measurable for all $t \geqslant 0$ , then we have

\begin{align*}\mathbb E \bigg[ \textbf{1}_{\big\{ t < \tau^i_1 \leqslant T \big\}} Z_{\tau^i_1} \big| {\mathcal F}_t \bigg] = \mathbb E \bigg[ \int_t^T Z_u \textrm{d} \mathbb P \big( \tau^i_1 \leqslant u \big| {\mathcal F}_u \big) \big| {\mathcal F}_t \bigg],\end{align*}

for $i = 1,\ldots,n$ and $t \in [0,T]$ .

Proof. The argument is similar to Proposition 5.1.1 of [Reference Bielecki and Rutkowski11], which we outline for completeness. We assume first that both Z and $\tilde{Z}$ are stepwise constant, i.e. without loss of generality,

\begin{equation*}Z_u = \sum_{j = 0}^n Z_{t_j} \textbf{1}_{\left\{ t_j < u \leqslant t_{j+1} \right\}}, \ \ \ \ \tilde{Z}_u = \sum_{j = 0}^n \tilde{Z}_{t_j} \textbf{1}_{\left\{ t_j < u \leqslant t_{j+1} \right\}},\end{equation*}

for $t < u \leqslant T$ , where $t_0 = t < \ldots < t_{j + 1} = T$ , $Z_{t_j}$ is ${\mathcal F}_{T}$ -measurable, and $\tilde{Z}_{t_j}$ is independent from ${\mathcal F}_T \vee \sigma\big(\tau^i_1\big)$ for all $j = 0,\ldots,n$ . By Lemma 2, it holds that

(3.12) \begin{align} &\mathbb E \bigg[ \textbf{1}_{\big\{ t < \tau^i_1 \leqslant T \big\}} \tilde{Z}_{\tau^i_1} Z_{\tau^i_1} \bigg| {\mathcal F}_t \bigg] \nonumber \\&= \mathbb E \Bigg[ \sum_{j = 0}^n \mathbb E \bigg[ \tilde{Z}_{t_j} Z_{t_j} \textbf{1}_{\big\{ t_j < \tau^i_1 \leqslant t_{j+1} \big\}} \bigg| {\mathcal F}_{T } \bigg] \bigg| {\mathcal F}_t \Bigg] \nonumber\\&= \mathbb E \Bigg[ \sum_{j = 0}^n E\big[\tilde{Z}_{t_j}\big] Z_{t_j} \bigg( \mathbb E \Big[ \textbf{1}_{\big\{ \tau^i_1 \leqslant t_{j+1} \big\}} \bigg| {\mathcal F}_{t_{j + 1}} \Big] - \mathbb E \bigg[ \textbf{1}_{\big\{ \tau^i_1 \leqslant t_{j} \big\}} \bigg| {\mathcal F}_{t_{j}} \bigg] \bigg) \bigg| {\mathcal F}_t \Bigg] \end{align}

(3.13) \begin{align} = \mathbb E \bigg[ \int_t^T E\big[\tilde{Z}_u\big] Z_u \textrm{d} \mathbb P\big( {\tau^i_1 \leqslant u} \big| {\mathcal F}_{u} \big) \big| {\mathcal F}_t \bigg]. \qquad\qquad\qquad\qquad\ \ \qquad\end{align}

In the general case, it is sufficient to find stepwise constant approximations for Z and $\tilde{Z}$ . Since Z is bounded and $E[\tilde{Z}]$ is continuous and bounded on [t,T], the Riemann sum under the sign of conditional expectation in (3.12) converges to the Lebesgue–Stieltjes integral in the expression (3.13); hence the convergence of the conditional expectations follows as well by the dominated convergence theorem.

Now we are able to calculate the first component on the right-hand side of (3.8). We define

(3.14) \begin{equation}\begin{array}{*3{>{\displaystyle}l}}\tilde m(t) \,:\!=\, E \left[ \sum^{\tilde{\textbf{N}}_t}_{j = 1} \tilde{X}^i_j \right] \ \ \ \ &\text{if } t \geqslant 0,\\\tilde m(t) \,:\!=\, 0 \ \ \ \ &\text{if }t < 0,\end{array}\end{equation}

where $\tilde{\textbf{N}}$ denotes the ground process of $\Big(\tilde{\tau}^i_j, \tilde{X}^i_j\Big)_{j \in {\mathbb N}_0}$ , i.e.

(3.15) \begin{equation}\tilde{\textbf{N}}_t \,:\!=\, \sum^\infty_{j=1} \textbf{1}_{\left\{ \tilde{\tau}^i_j \leqslant t \right\}}, \ \ \ \ t \geqslant 0.\end{equation}

Note that $\tilde m$ does not depend on i because of Assumption 1(2).

The following result also holds under different integrability and measurability conditions.

Proposition 2. Let $Z \,:\!=\, (Z_t)_{t \in [0,T]}$ be a continuous, bounded, and ${\mathbb F}$ -adapted process and let Y be as in (3.6); then for any $t \in [0,T]$ ,

\begin{align*}&\mathbb E \bigg[ \textbf{1}_{\big\{ \tau^i_1 > t \big\}} Y \big| {\mathcal H}^{i}_t \vee {\mathcal F}_t \bigg]\\[3pt]&= \textbf{1}_{\big\{ \tau^i_1 > t \big\}} \frac{\mathbb E \Big[ \int_t^T \big(E\big[X^i_1\big] Z_u + \int_u^T Z_v \textrm{d} \tilde{m}(v - u)\big) \textrm{d} \mathbb P \big( {\tau^i_1 \leqslant u} \big| {\mathcal F}_{u} \big) \big| {\mathcal F}_t \Big]}{\mathbb P \big( \tau^i_1 > t \big| {\mathcal F}_t \big)},\end{align*}

where $\tilde m$ is defined in (3.14).

Proof. By applying (3.5) in Proposition 1 to Y as defined in (3.6), we get

(3.16) \begin{align}&\mathbb E \bigg[ \textbf{1}_{\big\{ \tau^i_1 > t \big\}} Y \bigg| {\mathcal H}^{i}_t \vee {\mathcal F}_t \bigg] =\textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb E \bigg[ Y \bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \bigg] \nonumber \\&= \textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb E \bigg[ \sum^{\infty}_{j = 1} \textbf{1}_{\big\{ \tau^i_j \leqslant T \big\}}X^i_j Z_{\tau^i_j} \bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \bigg]\nonumber \\&= \textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb E \bigg[ \textbf{1}_{\big\{ \tau^i_1 \leqslant T \big\}}X^i_1 Z_{\tau^i_1} \bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \bigg] + \textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb{E} \Bigg[ \sum^{\infty}_{j = 2} \textbf{1}_{\big\{ \tau^i_j \leqslant T \big\}}X^i_j Z_{\tau^i_j} \bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg]. \end{align}

For the first component of (3.16), it is sufficient to use (3.4) in Proposition 1 and an argument similar to Proposition 5.1.1 of [Reference Bielecki and Rutkowski11], taking into account the independence condition in Assumption 1(3) and Lemma 2. We have hence

\begin{align*} \textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb E \bigg[ \textbf{1}_{\big\{ \tau^i_1 \leqslant T \big\}}X^i_1 Z_{\tau^i_1} \Big| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \bigg]&= \mathbb E \bigg[ \textbf{1}_{\big\{ t < \tau^i_1 \leqslant T \big\}}X^i_1 Z_{\tau^i_1} \Big| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \bigg]\\[3pt] &= \textbf{1}_{\big\{ \tau^i_1 > t \big\}} \frac{\mathbb E \bigg[ \textbf{1}_{\big\{ t < \tau^i_1 \leqslant T \big\}}X^i_1 Z_{\tau^i_1} \Big| {\mathcal F}_t \bigg]}{\mathbb P \big( \tau^i_1 > t \big| {\mathcal F}_t \big)}\\[3pt]&= \textbf{1}_{\big\{ \tau^i_1 > t \big\}} \frac{\mathbb E \Big[ \int_t^T E\big[X^i_1\big] Z_u \textrm{d} \mathbb P \big( {\tau^i_1 \leqslant u} \big| {\mathcal F}_{u} \big) \Big| {\mathcal F}_t \Big]}{\mathbb P \big( \tau^i_1 > t \big| {\mathcal F}_t \big)}. \end{align*}

Now we focus on the second component of (3.16). We assume first that restricted to the interval [t,T], Z is a bounded, stepwise, ${\mathbb F}$ -predictable process, i.e.

(3.17) \begin{equation}Z_u = \sum_{k = 0}^n Z_{t_k} \textbf{1}_{\left\{ t_k < u \leqslant t_{k+1} \right\}},\end{equation}

for $t < u \leqslant T$ , where $t_0 = t < \ldots < t_{n + 1} = T$ and $Z_{t_k}$ is ${\mathcal F}_{t_k}$ -measurable for all $k = 0,\ldots,n$ . In this case, we have

(3.18) \begin{align}&\textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb{E} \Bigg[ \sum^{\infty}_{j = 2} \textbf{1}_{\big\{ \tau^i_j \leqslant T \big\}}X^i_j Z_{\tau^i_j} \Bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg] \nonumber\\&= \textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb{E} \Bigg[ \sum^{\infty}_{j = 1} \textbf{1}_{\big\{ t < \tau^i_1 + \tilde{\tau}^i_j \leqslant T \big\}} \tilde{X}^i_j Z_{\tau^i_j} \Bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg] \nonumber \\&= \textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb{E} \Bigg[ \sum_{k=0}^n \sum^{\infty}_{j = 1} \textbf{1}_{\big\{ t_k < \tau^i_1 + \tilde{\tau}^i_j \leqslant t_{k + 1} \big\}} \tilde{X}^i_j Z_{t_k} \Bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg] \nonumber\\&= \textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb{E} \Bigg[ \sum_{k=0}^n Z_{t_k} \mathbb E\Bigg[ \sum^{\infty}_{j = 1} \textbf{1}_{\big\{ t_k < \tau^i_1 + \tilde{\tau}^i_j \leqslant t_{k + 1} \big\}} \tilde{X}^i_j \Bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_{t_k} \vee \sigma\big(\tau^i_1\big) \Bigg] \Bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg] \nonumber \\&= \textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb{E} \Bigg[ \sum_{k=0}^n Z_{t_k} \mathbb E\Bigg[ \sum^{\infty}_{j = 1} \textbf{1}_{\big\{ t_k < x + \tilde{\tau}^i_j \leqslant t_{k + 1} \big\}} \tilde{X}^i_j \Bigg]\Bigg|_{x = \tau^i_1} \Bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg] \nonumber \\&= \textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb{E} \Bigg[ \sum_{k=0}^n Z_{t_k} \big( \tilde{m}\big(t_{k + 1} - \tau^i_1\big) - \tilde{m}\big(t_{k} - \tau^i_1\big) \big) \Bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg], \end{align}

where in the second-to-last equality we use the independence between the marked point process $\Big(\tilde{\tau}^i_j, \tilde{X}^i_j\Big)_{j \in {\mathbb N}_0}$ and the $\sigma$ -algebra ${\mathcal H}^{i,1}_\infty \vee {\mathcal F}_\infty$ in Assumption 1. This shows that for any bounded, stepwise, ${\mathbb F}$ -predictable process Z, we have

\begin{equation*}\textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb{E} \Bigg[ \sum^{\infty}_{j = 2} \textbf{1}_{\big\{ \tau^i_j \leqslant T \big\}}X^i_j Z_{\tau^i_j} \Bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg] = \textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb{E}\Bigg[ \int_t^T Z_u \textrm{d} \tilde{m}\big(u - \tau^i_1\big) \Big| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg].\end{equation*}

A continuous bounded process Z can be approximated by a sequence of bounded, stepwise, and ${\mathbb F}$ -predictable processes; i.e. there is a sequence $Z^n$ of the form (3.17) such that

\begin{equation*}Z^n \longrightarrow Z \ \ \ \ \text{and} \ \ \ \ |Z^n| \leqslant M,\end{equation*}

with $M > 0$ . Since $\tilde m$ is right-continuous and monotone, the Lebesgue–Stieltjes integral

(3.19) \begin{equation}\int_t^T Z_u \textrm{d} \tilde{m}\big(u - \tau^i_1\big)\end{equation}

is well defined. It holds by the Lebesgue theorem that

\begin{equation*}\int_t^T Z^n_u \textrm{d} \tilde{m}\big(u - \tau^i_1\big) \longrightarrow \int_t^T Z_u \textrm{d} \tilde{m}\big(u - \tau^i_1\big).\end{equation*}

Furthermore,

(3.20) \begin{equation}\bigg|\int_t^T Z^n_u \textrm{d} \tilde{m}\big(u - \tau^i_1\big) \bigg| \leqslant M \left| \int_t^T \textrm{d} \tilde{m}\big(u - \tau^i_1\big)\right| = M \bigg|\tilde{m}\big(T - \tau^i_1\big) - \tilde{m}\big(t - \tau^i_1\big)\bigg|.\end{equation}

The right-hand side of (3.20) is uniformly bounded by (3.14) and (2.4). By again applying the Lebesgue theorem, we obtain also the convergence of the conditional expectations

\begin{equation*}\textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb{E}\Bigg[ \int_t^T Z^n_u \textrm{d} \tilde{m}\big(u - \tau^i_1\big) \bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg] \longrightarrow \mathbb{E}\Bigg[ \int_t^T Z_u \textrm{d} \tilde{m}\big(u - \tau^i_1\big) \bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg].\end{equation*}

We note that $\tilde m(u) = 0 $ for $u < 0$ ; hence,

\begin{align*}\textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb{E} \Bigg[ \sum^{\infty}_{j = 2} \textbf{1}_{\big\{ \tau^i_j \leqslant T \big\}}X^i_j Z_{\tau^i_j} \big| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg]&= \textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb{E} \Bigg[ \int_t^T Z_u \textrm{d} \tilde{m}\big(u - \tau^i_1\big) \big| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg]\\&= \mathbb{E} \Bigg[ \textbf{1}_{\big\{ t < \tau^i_1 \leqslant T \big\}} \int_t^T Z_u \textrm{d} \tilde{m}\big(u - \tau^i_1\big) \big| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg].\end{align*}

By again applying (3.4) in Proposition 1 to the above expression, we get

\begin{align*}\textbf{1}_{\big\{ \tau^i_1 > t \big\}} \mathbb{E} \Bigg[ \sum^{\infty}_{j = 2} \textbf{1}_{\big\{ \tau^i_j \leqslant T \big\}}X^i_j Z_{\tau^i_j} \Bigg| {\mathcal H}^{i,1}_t \vee {\mathcal F}_t \Bigg]&= \textbf{1}_{\big\{ \tau^i_1 > t \big\}}\frac{ \mathbb{E} \bigg[ \textbf{1}_{\big\{ t < \tau^i_1 \leqslant T \big\}} \int_t^T Z_u \textrm{d} \tilde{m}\big(u - \tau^i_1\big) \big| {\mathcal F}_t \bigg]}{\mathbb P \big( \tau^i_1 > t \big| {\mathcal F}_t \big)}.\end{align*}

Let $\tilde Z_s \,:\!=\, \int_t^T Z_u \textrm{d} \tilde{m}(u - s)$ , $s \in [0,T]$ . We note that $\tilde m$ is right-continuous and monotone. On one hand, for fixed $s \in [0,T]$ , the function $d_s(u) \,:\!=\, \tilde m (u - s)$ , $u \in [0,T]$ , is also right-continuous and monotone and defines the cumulative distribution function of a finite positive measure, in view of (2.4). On the other hand, for fixed $u \in [0,T]$ , the function $\tilde m (u - s)$ , $s \in [0,T]$ , is left-continuous in s; i.e. for every series $s_n \nearrow s$ , we have the pointwise convergence

\begin{equation*}\lim_{s_n \nearrow s} d_{s_n}(u) = d_s(u) \ \ \ \ \text{for all } u \in [0,T]\end{equation*}

of the cumulative distribution functions, equivalent to convergence in distribution or weak convergence in measure. Note that a series of positive finite measures $(\nu_n)_{n \in {\mathbb N}}$ converges weakly to a positive finite measure $\nu$ if for all bounded continuous functions f the following holds:

\begin{equation*} \int f \textrm{d} \nu_n \longrightarrow \int f \textrm{d} \nu. \end{equation*}

This yields the convergence

\begin{equation*}\tilde Z_{s_n} \longrightarrow \tilde Z_s, \ \ \ \ P\text{-a.s.};\end{equation*}

that is, $\tilde Z_s \,:\!=\, \int_t^T Z_u \textrm{d} \tilde{m}(u - s)$ , $s \in [0,T]$ , is left-continuous. Furthermore, it is also bounded. Now we apply Lemma 6 and obtain

\begin{align*}&\textbf{1}_{\left\{ \tau^i_1 > t \right\}}\frac{ \mathbb{E} \bigg[ \textbf{1}_{\left\{ t < \tau^i_1 \leqslant T \right\}} \int_t^T Z_u \textrm{d} \tilde{m}\big(u - \tau^i_1\big) \Big| {\mathcal F}_t \bigg]}{\mathbb P \big( \tau^i_1 > t \big| {\mathcal F}_t \big)}\\&\quad =\textbf{1}_{\left\{ \tau^i_1 > t \right\}}\frac{ \mathbb{E} \bigg[ \textbf{1}_{\left\{ t < \tau^i_1 \leqslant T \right\}} \tilde Z_{\tau^i_1} \Big| {\mathcal F}_t \bigg]}{\mathbb P \big( \tau^i_1 > t \big| {\mathcal F}_t \big)}\\ &\quad = \textbf{1}_{\left\{ \tau^i_1 > t \right\}} \frac{\mathbb E \bigg[ \int_t^T \tilde Z_u \textrm{d} \mathbb P \Big( {\tau^i_1 \leqslant u} \Big| {\mathcal F}_{u} \Big) \Big| {\mathcal F}_t \bigg]}{\mathbb P \big( \tau^i_1 > t \big| {\mathcal F}_t \big)}\\ &\quad = \textbf{1}_{\left\{ \tau^i_1 > t \right\}} \frac{\mathbb E \bigg[ \int_t^T \left(\int_t^T Z_v \textrm{d} \tilde{m}(v - u) \right) \textrm{d} \mathbb P \Big( {\tau^i_1 \leqslant u} \Big| {\mathcal F}_{u} \Big) \Big| {\mathcal F}_t \bigg]}{\mathbb P \big( \tau^i_1 > t \big| {\mathcal F}_t \big)}.\end{align*}

As the last step, we note that for $u < s$ , $\int_t^T Z_u \textrm{d} \tilde{m}(u - s) = \int_s^T Z_u \textrm{d} \tilde{m}(u - s) $ since $\tilde{m}(u - s) = 0$ . This concludes the proof.

Remark 4. The proof of Proposition 2 relies on Assumption 1. Another sufficient condition would be the continuity of $\tilde m$ , as in the case of a compound Poisson process or a Cox process with continuous intensity process and integrable marks. Indeed, since $\tilde m (u) = 0$ for $u < 0$ ,

\begin{align*}\textbf{1}_{\left\{ \tau^i_1 > t \right\}} \left|\int_t^T Z^n_u \textrm{d} \tilde{m}\big(u - \tau^i_1\big) \right| &\leqslant \textbf{1}_{\left\{ \tau^i_1 > t \right\}} M \left| \int_t^T \textrm{d} \tilde{m}\big(u - \tau^i_1\big)\right|\\&= \textbf{1}_{\left\{ t < \tau^i_1 \leqslant T \right\}} M |\tilde{m}\big(T - \tau^i_1\big) - \tilde{m}\big(t - \tau^i_1\big)|,\end{align*}

and the right-hand side is uniformly bounded if $\tilde m$ is continuous.

The following proposition gives a representation of the second component on the right-hand side of (3.8).

Proposition 3. Under the same assumptions as those of Proposition 2, if for each $i= 1,\ldots,n$ , the process

\begin{equation*}\left(\sum^{\tilde{\textbf{N}}_t}_{j = 1} \tilde{X}^i_j\right)_{t \in[0,T]},\end{equation*}

where $\tilde{\textbf{N}}$ is defined in (3.15), is of independent increments with respect to its natural filtration $\tilde{{\mathbb H}}^i$ , then for $t \in [0,T]$ and Y as in (3.6), it holds that

\begin{equation*}\left. \mathbb E\left[ \textbf{1}_{\left\{ \tau^i_1 \leqslant t \right\}} Y \right| {\mathcal H}^{i}_t \vee {\mathcal F}_t \right] =\textbf{1}_{\left\{ \tau^i_1 \leqslant t \right\}} \left. \left. \mathbb E\left[ \int_t^T Z_{u} \textrm{d} \tilde{m}({u - x}) \right| {\mathcal H}^{i,1}_\infty \vee \tilde{{\mathcal H}}^{i}_{t - x} \vee {\mathcal F}_t \right] \right|_{x = \tau^i_1},\end{equation*}

for $i=1,\ldots,n$ .

Proof. It follows from Lemma 4 that

\begin{align*} \left. \mathbb E\left[ \textbf{1}_{\left\{ \tau^i \leqslant t \right\}} Y \right| {\mathcal H}^{i}_t \vee {\mathcal F}_t \right]= \left. \mathbb E\left[ \textbf{1}_{\left\{ \tau^i_1 \leqslant t \right\}} Y \right| {\mathcal H}^{i,1}_\infty \vee {\mathcal H}^{i, >1}_t \vee {\mathcal F}_t \right].\end{align*}

As in the proof of Proposition 2, we assume first Z of the form (3.17). Similar calculations lead to

\begin{align*} &\left. \mathbb E\left[ \textbf{1}_{\big\{ \tau^i_1 \leqslant t \big\}} Y \right| {\mathcal H}^{i,1}_\infty \vee {\mathcal H}^{i, >1}_t \vee {\mathcal F}_t \right] \nonumber \\ &= \textbf{1}_{\big\{ \tau^i_1 \leqslant t \big\}} \left. \mathbb E\left[ \sum^{\infty}_{j = 1} \textbf{1}_{\big\{ t <\tau^i_1 + \tilde{\tau}^i_{j} \leqslant T \big\}}\tilde{X}^i_j Z_{\tau^i_j} \right| {\mathcal H}^{i,1}_\infty \vee {\mathcal H}^{i, >1}_t \vee {\mathcal F}_t \right] \nonumber \\ &= \textbf{1}_{\big\{ \tau^i_1 \leqslant t \big\}} \left. \left. \mathbb E\left[ \sum_{i = 0}^n \sum^{\infty}_{j = 1} \textbf{1}_{\big\{ t_i < x + \tilde{\tau}^i_{j} \leqslant t_{i+1} \big\}}\tilde{X}^i_j Z_{t_i} \right| {\mathcal H}^{i,1}_\infty \vee {\mathcal H}^{i, >1}_t \vee {\mathcal F}_t \right] \right|_{x = \tau^i_1} \nonumber \\&= \textbf{1}_{\big\{ \tau^i_1 \leqslant t \big\}} \left. \left. \mathbb E\left[ \sum_{i = 0}^n \sum^{\infty}_{j = 1} \textbf{1}_{\big\{ t_i < x + \tilde{\tau}^i_{j} \leqslant t_{i+1} \big\}}\tilde{X}^i_j Z_{t_i} \right| {\mathcal H}^{i,1}_\infty \vee \tilde{{\mathcal H}}^{i}_{t - x} \vee {\mathcal F}_t \right] \right|_{x = \tau^i_1},\end{align*}

where the last step follows from the definitions of the filtrations. Using the tower property, the independence between the marked point process $\Big(\tilde{\tau}^i_j, \tilde{X}^i_{j}\Big)_{j \in {\mathbb N}_0}$ and ${\mathcal F}_\infty \vee {\mathcal H}^{i,1}_\infty$ (see Assumption 1), and the independence of increments of the process

\begin{equation*}\left(\sum^{\tilde{\textbf{N}}_t}_{j = 1} \tilde{X}^i_j\right)_{t \in[0,T]},\end{equation*}

we get furthermore

(3.21) \begin{align}&\left. \mathbb E\!\left[ \textbf{1}_{\left\{ \tau^i_1 \leqslant t \right\}} Y \right| {\mathcal H}^{i,1}_\infty \vee {\mathcal H}^{i, >1}_t \vee {\mathcal F}_t \right] \nonumber \\=& \textbf{1}_{\left\{ \tau^i_1 \leqslant t \right\}} \left. \left. \mathbb E\!\left[ \sum_{i = 0}^n Z_{t_i} \left( \left. \mathbb E\!\left[ \sum^{\infty}_{j = 1} \textbf{1}_{\left\{ \tilde{\tau}^i_{j} \leqslant t_{i+1} - x \right\}}\tilde{X}^i_j \right| {\mathcal H}^{i,1}_\infty \vee \tilde{{\mathcal H}}^{i}_{t - x} \vee {\mathcal F}_{t_{i}} \right] \right. \right. \right. \right. \nonumber \\&- \left.\left. \left. \left. \left. \mathbb E\!\left[ \sum^{\infty}_{j = 1} \textbf{1}_{\left\{ \tilde{\tau}^i_{j} \leqslant t_i -x \right\}}\tilde{X}^i_j \right| {\mathcal H}^{i,1}_\infty \vee \tilde{{\mathcal H}}^{i}_{t - x} \vee {\mathcal F}_{t_i} \right] \right) \right| {\mathcal H}^{i,1}_\infty \vee \tilde{{\mathcal H}}^{i}_{t - x} \vee {\mathcal F}_t \right] \right|_{x = \tau^i_1} \nonumber \\=& \textbf{1}_{\left\{ \tau^i_1 \leqslant t \right\}} \left. \left. \mathbb E\!\left[ \sum_{i = 0}^n Z_{t_i} \left( \tilde{m}({t_{i+1} - x}) - \sum^{\tilde{\textbf{N}}_{t-x}}_{j = 1}\tilde{X}^i_j - \tilde{m}({t_{i} - x}) + \sum^{\tilde{\textbf{N}}_{t-x}}_{j = 1}\tilde{X}^i_j \right) \right| {\mathcal H}^{i,1}_\infty \vee \tilde{{\mathcal H}}^{i}_{t - x} \vee {\mathcal F}_t \right] \right|_{x = \tau^i_1}\nonumber \\=& \textbf{1}_{\left\{ \tau^i_1 \leqslant t \right\}} \left. \left. \mathbb E\!\left[ \sum_{i = 0}^n Z_{t_i} \left( \tilde{m}({t_{i+1} - x}) - \tilde{m}({t_{i} - x}) \right) \right| {\mathcal H}^{i,1}_\infty \vee {\mathcal F}_t \right] \right|_{x = \tau^i_1}. \end{align}

This yields that for any bounded, stepwise, ${\mathbb F}$ -predictable process Z, we have

\begin{align*}\left. \mathbb E\left[ \textbf{1}_{\left\{ \tau^i_1 \leqslant t \right\}} Y \right| {\mathcal H}^{i}_t \vee {\mathcal F}_t \right] =\textbf{1}_{\left\{ \tau^i_1 \leqslant t \right\}} \left. \left. \mathbb E\left[ \int_t^T Z_{u} \textrm{d} \tilde{m}({u - x}) \right| {\mathcal H}^{i,1}_\infty \vee {\mathcal F}_t \right] \right|_{x = \tau^i_1}.\end{align*}

If Z is continuous, bounded, and ${\mathbb F}$ -adapted, then Z can be approximated by a sequence of bounded, stepwise, and ${\mathbb F}$ -predictable processes. This together with the fact that $\tilde m$ is right-continuous and monotone guarantees that the Riemann sum in (3.21) under the sign of conditional expectation converges to the Lebesgue–Stieltjes integral, using the same arguments as Proposition 2.

We summarize the results in the following theorem, which gives an explicit representation of ${\mathbb G}$ -conditional expectation with respect to the first reporting time $\tau^i_1$ . Note that, like most of our results, the conclusion also holds under alternative integrability and measurability conditions.

Theorem 5. Let $Z \,:\!=\, (Z_t)_{t \in [0, T]}$ be a continuous, bounded, and ${\mathbb F}$ -adapted process, and let Y be of the form (3.6). If the process

\begin{equation*}\left(\sum^{\tilde{\textbf{N}}_t}_{j = 1} \tilde{X}^i_j\right)_{t \in[0,T]} \end{equation*}

has independent increments and $\tilde m$ is as defined in (3.14), then

\begin{align*}\mathbb E\!\left. \left[ Y \right| {\mathcal G}_t \right] =& \textbf{1}_{\left\{ \tau^i_1 \leqslant t \right\}} \left. \left. \mathbb E\!\left[ \int_t^T Z_{u} \textrm{d} \tilde{m}({u - x}) \right| {\mathcal H}^{i,1}_\infty \vee \tilde{{\mathcal H}}^{i}_{t - x} \vee {\mathcal F}_t \right] \right|_{x = \tau^i_1}\\& + \textbf{1}_{\left\{ \tau^i_1 > t \right\}} \frac{\mathbb E\!\left. \left[ \int_t^T \left(E\big[X^i_1\big] Z_u + \int_u^T Z_v \textrm{d} \tilde{m}(v - u)\right) \textrm{d} \mathbb P\left. \left( {\tau^i_1 \leqslant u} \right| {\mathcal F}_{u} \right) \right| {\mathcal F}_t \right]}{\mathbb P\left. \left( {\tau^i_1 > t} \right| {\mathcal F}_{t} \right)},\end{align*}

for $i=1,\ldots,n$ , where

\begin{equation*}\mathbb P\left. \left( {\tau^i_1 \leqslant t} \right| {\mathcal F}_{t} \right) = \int_0^t \left( \alpha_0 e^{-\int^u_0 \mu_v \textrm{d} v} \mu_u + \int_0^u g(u - v) e^{-\int^v_0 \mu_s \textrm{d} s} \mu_v \textrm{d} v \right) \textrm{d} u,\end{equation*}

with $\alpha_0$ and g defined in (2.6).

Compared to Theorem 4, Theorem 5 is more explicit and has the advantage that the representation is expressed as function of $\mu$ , the distribution of $\theta^i$ , and the distribution of $\Big(\tilde{\tau}^i_j, \tilde{X}^i_j\Big)_{j \in {\mathbb N}_0}$ . This result will be useful for the concrete reserving problem in a hybrid market in Section 5.

4. Comparison with the compensator approach

In this section, we compare our framework with the compensator approach for non-life insurance in the existing literature. Within this section, the filtration ${\mathbb H}$ denotes the natural filtration of a marked point process $(\tau_n, X_n)_{n \in {\mathbb N}_0}$ , with marked cumulative process N, and ${\mathbb G}$ is a generic enlargement of ${\mathbb H}$ . We set ${\mathcal H} \,:\!=\, {\mathcal H}_\infty$ and ${\mathcal G} \,:\!=\, {\mathcal G}_\infty$ .

In most of the current literature, e.g. [Reference Barbarin, De Launois and Devolder3], [Reference Norberg and Savina32], [Reference Norberg31], and [Reference Schmidt35], the study of non-life insurance contracts is based on modeling the ${\mathbb G}$ -compensator of N, since the ${\mathbb G}$ -compensator is involved in the pricing formula and in the calculation of the hedging strategy. In the reduced-form framework for life insurance, the direct modeling approach and the compensator approach coincide; see e.g. [Reference Bielecki and Rutkowski11]. However, the compensator approach presents several difficulties in a non-life insurance setting with nontrivial dependence between the filtrations.

Theorem 14.2.IV(a) of [Reference Daley and Vere-Jones15] shows that given a marked point process $(\tau_n, X_n)_{n \in {\mathbb N}_0}$ with finite first moment measure, its ${\mathbb G}$ -compensator $\Lambda$ always exists and is $(l \otimes P)$ -almost everywhere unique, where l denotes the Lebesgue measure on ${\mathbb R}_+$ . In particular, for all $(t, B, \omega) \in {\mathbb R}_+ \times \mathcal{B}({\mathbb R}_+) \times \Omega$ , the following relation holds:

(4.1) \begin{equation}\Lambda(t, B, \omega) = \int_0^t \kappa(B| s, \omega) {\Lambda}(\textrm{d} s, \omega),\end{equation}

where $\kappa(B| s, \omega)$ , $B \in \mathcal{B}({\mathbb R}_+)$ , $s \geqslant 0$ , $\omega \in \Omega$ , is the unique predictable kernel such that for all $A \in {\mathcal G}_s$ , $0 < s < t$ , $B \in \mathcal{B}({\mathbb R}_+)$ ,

\begin{equation*}\int_A \int_s^t N(u, B) (\omega) \textrm{d} u P(\textrm{d} \omega) = \int_A \int_s^t \kappa(B| u, \omega) {\textbf{N}}_u( \omega) \textrm{d} u P(\textrm{d} \omega).\end{equation*}

However, under general conditions it is not always true that given a ${\mathbb G}$ -mark-predictable and cumulative process $\Lambda$ , there exists a marked point process $(\tau_n, X_n)_{n \in {\mathbb N}_0}$ with ${\mathbb G}$ -compensator $\Lambda$ . The problem is first mentioned in [Reference Jacod21], where the case with ${\mathbb G} = {\mathbb H}$ is solved. An extension of the existence theorem to the case of ${\mathbb G} = {\mathbb F} \otimes {\mathbb H}$ , i.e. when the filtrations ${\mathbb F}$ and ${\mathbb H}$ are independent, is provided in [Reference Dettweiler17]. Furthermore while the law of N is uniquely determined by the ${\mathbb H}$ -compensator, this is not true for the ${\mathbb G}$ -compensator. (See the discussion in [Reference Jacod21] and in [Reference Jacobsen20, Section 4.8].) Consequently, the literature with the compensator approach is mostly limited to the cases of ${\mathbb G} \equiv {\mathbb H}$ (see e.g. [Reference Norberg and Savina32], [Reference Norberg31]) or ${\mathbb G} = {\mathbb F} \otimes {\mathbb H}$ (see e.g. [Reference Barbarin, De Launois and Devolder3]).

In the following we provide a sufficient condition in the general case of ${\mathbb G} = {\mathbb F} \vee {\mathbb H}$ , such that the law of N is uniquely determined by $\Lambda$ . Similarly to e.g. [Reference Norberg and Savina32] and [Reference Norberg31], we assume that the ${\mathbb G}$ -compensator of $(\tau_n, X_n)_{n \in {\mathbb N}_0}$ has the following form:

(4.2) \begin{equation}\Lambda(t,B) = \int_0^t \int_B \lambda_s \eta_s (\textrm{d} x) \textrm{d} s \ \ \ \ \text{for all } t \geqslant 0, \ B \in \mathcal{B}\big({\mathbb R}_+\big),\end{equation}

where $\lambda \,:\!=\, (\lambda_t)_{t \geqslant 0}$ is a ${\mathbb G}$ -progressively measurable process and the mapping $\eta$ given by

\begin{align*}\eta\,:\, {\mathbb R}_+ \times \mathcal{B}\big({\mathbb R}_+\big) \times \Omega &\longrightarrow \big({\mathbb R}_+, \mathcal{B}\big({\mathbb R}_+\big)\big)\\(t, B, \omega) &\mapsto \eta_t(B)(\omega)\end{align*}

is such that for every $t \geqslant 0$ , $\omega \in \Omega$ , $\eta(t,\cdot,\omega)$ is a probability measure on $\big({\mathbb R}_+, \mathcal{B}\big({\mathbb R}_+\big)\big)$ , and for every $B \in \mathcal{B}({\mathbb R}_+)$ , $(\eta_t(B))_{t \geqslant 0}$ is a ${\mathbb G}$ -progressively measurable process. Clearly, we have

\begin{equation*}{\Lambda}_t = \int_0^t \lambda_s \textrm{d} s \ \ \ \ \text{for all } t \geqslant 0.\end{equation*}

In particular, we can choose a predictable version of both $\lambda$ and $\eta$ ; see Section 14.3 of [Reference Daley and Vere-Jones15] for details. The processes $\lambda$ and $\eta$ can be interpreted respectively as jump intensity and jump size intensity. We recall that a marked point process $(\tau_n, X_n)_{n \in {\mathbb N}_0}$ has independent marks if the marks $(X_n)_{n \in {\mathbb N}_0}$ are mutually independent given $\textbf{N}$ .

Proof. By Proposition 6.4.IV(a) of [Reference Daley and Vere-Jones15], the law of a marked point process with independent marks is uniquely determined by the kernel $\kappa$ and the distribution of N. According to the relations (4.1) and (4.2), the kernel $\kappa$ is given by

\begin{equation*}\kappa(B|t, \omega) = \eta_t(B)(\omega), \ \ \ \ (t, B, \omega) \in {\mathbb R}_+ \times \mathcal{B}({\mathbb R}_+) \times \Omega.\end{equation*}

Corollary 4.8.5 of [Reference Jacobsen20] and Theorem 14.2.IV(c) of [Reference Daley and Vere-Jones15] show that, if N is simple and of the form (4.2), the process $(E[\lambda_t|{\mathcal H}_t])_{t \geqslant 0}$ uniquely determines the distribution of N on $(\Omega, {\mathcal H})$ . If in addition $\lambda$ is ${\mathbb H}$ -adapted, then by Theorem 4.8.1 of [Reference Jacobsen20], the distribution of N on $(\Omega, {\mathcal G})$ is also uniquely determined.

Nevertheless, Proposition 4 requires the jump intensity process $\lambda$ to be ${\mathbb H}$ -adapted in order to have N uniquely defined in law, which is an unnatural condition in our context.

By contrast, the approach proposed in Section 2 allows us to take into account a dependence structure between the filtrations ${\mathbb H}$ and ${\mathbb G}$ by directly modeling the ${\mathbb F}$ -adapted intensity process $\mu$ . Furthermore, this allows us to obtain analytical results for valuation formulas as shown in Section 3.

5. Pricing and hedging of non-life insurance liability cash flows in a hybrid market

In this section, we address the issue of pricing and hedging non-life insurance liability cash flows by applying the results of Section 3 under the benchmarked risk-minimization approach. We assume that P is the real-world measure and consider a general structure for a hybrid insurance and financial market. We fix a time horizon T with $0 < T < \infty$ , and denote the inflation index process by $I\,:\!=\,(I_t)_{t \in [0,T]}$ , which represents the percentage increments of the Consumer Price Index and follows a nonnegative $(P, {\mathbb F})$ -semimartingale. We distinguish real price value, i.e. inflation-adjusted, from nominal price value, which can be converted to the real value at any time $t \in [0,T]$ if divided by the inflation index $I_t$ . If not otherwise specified, all prices are expressed as nominal values.

We consider d liquidly traded primary assets on the financial market described by the price process vector $S \,:\!=\, \big(S^1_t, \ldots, S^d_t\big)_{t \in [0, T]}$ , which follows a real-valued $(P, {\mathbb F})$ -semimartingale. We assume that there is a publicly accessible index, based on the intensity process $\mu$ and modeled by the process $L\,:\!=\,(L_t)_{t \in [0,T]}$ with

\begin{equation*}L_t \,:\!=\, e^{-\Gamma_t}, \ \ \ \ t \in [0,T];\end{equation*}

see e.g. [Reference Cairns, Blake and Dowd13]. This index reflects the underlying systematic risk factors related to the insurance portfolio, such as mortality risk, weather risk, car accident risk, etc. We distinguish three kinds of primary assets as elements of the vector S:

1. 1. Financial assets, such as the zero-coupon bond, call and put options, futures, etc.

2. 2. Inflation-linked derivatives, such as inflation-linked zero-coupon bonds (also called zero-coupon Treasury Inflation-Protected Securities (TIPS)), which pay off $I_T$ (equivalent to 1 real unit) at time T; inflation-linked call and put options; etc.

3. 3. Macro-risk-factor-linked derivatives based on the index L, such as longevity bonds which pay off $L_T$ at time T, weather-index-based derivatives, etc. Note that apart from longevity bonds, other macro-index-based derivatives are still not common in the market.

We denote by $L(S,P,{\mathbb G})$ the space of ${\mathbb R}^d$ -valued ${\mathbb G}$ -predictable S-integrable processes. Following the definitions in [Reference Biagini and Pratelli6], we define the portfolio or value process $S^\delta\,:\!=\,\big(S^\delta_t\big)_{t \in [0,T]}$ associated to a trading strategy $\delta\,:\!=\, (\delta_t)_{t \in [0,T]}$ in $L(S,P,{\mathbb G})$ as the following càdlàg optional process:

\begin{equation*}S^\delta_{t-} = \delta_t^\top S_t = \sum_{i=1}^d \delta^i_t S^i_t, \ \ \ \ t \in [0,T].\end{equation*}

It is called self-financing if

\begin{equation*}S^\delta_t = S_0^\delta + \int_0^t \delta_{u-}^\top \textrm{d} S_u = S_0^\delta + \sum_{i=1}^d \int_0^t \delta^i_{u-} \textrm{d} S^i_u, \ \ \ \ t \in [0,T].\end{equation*}

We introduce the following set:

\begin{equation*}\mathcal{V}^{+}_x = \big\{ S^\delta \text{ self-financing }\,:\, \delta \in L(S, {\mathbb P}, {\mathbb G}), \ S_0^\delta = x > 0, \ S^\delta > 0\big\}.\end{equation*}

Definition 3. A benchmark or numéraire portfolio $S^{*}\,:\!=\,(S^{*}_t)_{t \in [0,T]}$ is an element of $\mathcal{V}^{+}_1$ such that

\begin{equation*}\frac{S^\delta_s}{S^{*}_s} \geqslant \mathbb E\left[ \left. \frac{S^\delta_t}{S^{*}_t} \right| {\mathcal G}_s \right], \ \ \ \ s,t \in [0,T], \ \ t \geqslant s.\end{equation*}

We follow the approach of [Reference Platen and Heath34] and work under the following assumption.

In [Reference Hulley and Schweizer19], it is shown that Assumption 6 is weaker than assuming the existence of an equivalent martingale measure. As discussed in [Reference Biagini4], this weak no-arbitrage assumption is more suitable for modeling a hybrid market as in our case and allows us to work directly under the real-world measure P. Extensive background on the benchmark approach and its application can be found in [Reference Platen and Heath34]. Note that, as discussed in [Reference Platen and Heath34], the benchmark portfolio $S^*$ can be identified with a sufficiently diversified portfolio such as the MSCI World Index.

Given a generic random variable or process X, we denote by $\hat X\,:\!=\, X/S^*$ the benchmarked value of X. The following lemma is proved in [Reference Biagini, Cretarola and Platen5].

For the sake of simplicity, we assume the following conditions, which are similar to the ones in [Reference Biagini and Zhang10].

The payment stream in real units of the insurance company towards policyholders is modeled by a nonnegative $(P, {\mathbb G})$ -semimartingale $D \,:\!=\, ( D_t)_{t \in [0, T]}$ . We denote by $A\,:\!=\,(A_t)_{t \in [0,T]}$ the nominal benchmarked cumulative payment, namely

(5.1) \begin{equation}A_t \,:\!=\, \int^t_0 \frac{I_u}{S^*_u} \ \textrm{d} D_u, \ \ \ \ t \in [0,T].\end{equation}

Definition 4. We call the following formula the real-world pricing formula associated to A:

(5.2) \begin{equation}V_t \,:\!=\, \frac{S^*_t}{I_t} \mathbb E \left[{A_T - A_t}|{\mathcal G}_t\right] = \frac{S^*_t}{I_t} \mathbb E \left[\int_{]t, T]} \frac{I_u}{S^*_u} \textrm{d} D_u |{\mathcal G}_t\right],\end{equation}

for $t \in [0, T]$ .

Here $V_t$ in (5.2) is expressed as a real value, i.e. an inflation-adjusted value. According to the benchmark approach of [Reference Platen and Heath34], a portfolio’s process is fair if its benchmarked value process is a P-martingale. The real-world pricing formula (5.2) then provides the fair portfolio of minimal price among all replicating self-financing portfolios for a given benchmarked claim $\hat H$ , if $\hat H$ is hedgeable. In the case of incomplete market models, it corresponds to the benchmarked risk-minimizing price for the payment process A at time t, if A is square-integrable, i.e.

\begin{equation*}\sup_{t \in [0,T]} \mathbb E\left[{A_t^2}\right] < \infty.\end{equation*}

For hedging purposes, we now use the relation between the benchmark approach and risk minimization, illustrated in [Reference Platen and Heath34] and [Reference Biagini, Cretarola and Platen5] for a single payoff and in the appendix of [Reference Biagini and Zhang10] for the case of dividend payments. In particular, Theorem A.7 of [Reference Biagini and Zhang10] shows that if

(5.3) \begin{equation}A_T = \mathbb{E}[A_T] + \int_0^T \left( \delta^A_u \right)^\top \textrm{d} \hat S_u + L^A_T, \ \ \ \ P\text{-a.s.},\end{equation}

is the Galtchouk–Kunita–Watanabe decomposition of $A_T$ , where $\int_0^{\cdot} \delta^A_u \textrm{d} \hat S_u$ is P-strongly orthogonal to $L^A$ , then $\delta^A$ is the unique benchmarked risk-minimizing strategy for A, i.e. the trading strategy which minimizes the expected quadratic risk as in Definition A.4 of [Reference Biagini and Zhang10]. Furthermore, the associated benchmarked cumulative cost process $C_t^{\bar \delta}$ is given by

\begin{equation*}C_t^{\bar \delta} = \mathbb E\left[{A_T}\right] + L^A_t, \ \ \ \ t \in [0,T],\end{equation*}

and the benchmarked value process $\hat S_t^{\bar \delta}$ is given by the discounted value of the real-world pricing formula (5.2),

\begin{equation*}\hat S_t^{\bar \delta} = \mathbb E\left.\left[{A_T - A_t}\right|{\mathcal G}_t\right] = \frac{I_t}{S^*_t} V_t, \ \ \ \ t \in [0,T].\end{equation*}

Note that the decomposition (5.3) shows the orthogonality between the perfectly hedgeable part $\int_0^{T} \left( \delta^A_u \right)^\top \textrm{d} \hat S_u$ of $A_T$ and the totally unhedgeable part $\left( \mathbb E\left[{A_T}\right] + L^A_T \right)$ , covered by the benchmarked cumulative cost process C.

5.1. Pricing and hedging non-life insurance claims

In the setting outlined above, we now apply the results of Section 3 to compute the real-world pricing formula for non-life insurance claims, under the interpretation of Section 2.2. The cumulative payment at time t for ith policy, expressed as a real value, is given by

\begin{equation*}\sum_{j = 1}^{\infty} \textbf{1}_{\big\{ \tau^i_j \leqslant t \big\}} X^i_j = \sum_{j = 1}^{\textbf{N}^i_t} X^i_j.\end{equation*}

The nominal benchmarked cumulative payment process $A \,:\!=\, (A_t)_{t \in [0,T]}$ is hence

(5.4) \begin{equation}{A}_t \,:\!=\, \int_0^t \frac{I_s}{S^*_s} \textrm{d} D_s = \sum^n_{i=1} \sum_{j = 1}^{\textbf{N}^i_t} \frac{I_{\tau^i_j}}{S^*_{\tau^i_j}} X^i_j, \ \ \ \ t \in [0,T].\end{equation}

Following [Reference Arjas1], the claim reserving problem in the context of non-life insurance can be formulated as the estimation of A. In particular, we are interested in estimating not only the overall reserve $E[A_T]$ , but also the conditional reserve $ \mathbb E\left.\left[{A_T - A_t}\right|{\mathcal G}_t\right]$ at a given time t, i.e. a predictor of the remaining nominal payment $A_T - A_t$ , given the information up to time t. Unlike in the life insurance case, the risk related to non-life insurance policies is related not only to the accident itself, but also to the first reporting delay (this is the case of incurred but not reported (IBNR) claims), and to the time and the size of developments after the first reporting. We now focus on pricing and hedging the remaining nominal payment ${A}_T - A_t$ , for $t \in [0,T]$ . We assume that the process $I/S^*$ is ${\mathbb F}$ -conditionally independent of $\tau^i_1$ , for all $i=1,\ldots,n$ , and that the cumulative payments related to the marked point processes $\big(\tilde{\tau}^i, \tilde{X}^i_j\big)_{j \in {\mathbb N}_0}$ , $i = 1,\ldots,n$ ,

\begin{equation*}\sum_{j = 1}^{\tilde{\textbf{N}}^i_t} \tilde{X}^i_j, \ \ \ \ t \in [0,T], \ \ \ \ i = 1,\ldots,n,\end{equation*}

are independent and identically distributed compound Poisson processes, i.e. the $\tilde{\textbf{N}}^i$ are mutually independent Poisson processes with parameter $\lambda$ , and the $\tilde{X}^i_j$ are independent and identically distributed integrable nonnegative random variables independent of $\tilde{\textbf{N}}^i$ with expectation $E\big[\tilde{X}^i_j\big] = m$ . In this case, we have

\begin{equation*}\tilde{m}(t) = \lambda m t, \ \ \ \ t \in [0,T],\end{equation*}

where $\tilde m$ is defined in (3.14).

In view of the above assumptions, all conditions in Theorem 5 are satisfied in the case of $Y = A_T - A_t$ , for $t \in [0,T]$ . Let $R_t$ be the number of reported claims at time t, i.e.

\begin{equation*}R_t \,:\!=\, \sum_1^n \textbf{1}_{\left\{ \tau^i_1 \leqslant t \right\}}, \ \ \ \ t \in [0,T].\end{equation*}

The real-world pricing formula (5.2) together with Corollary 2, Theorem 5, and Assumption 7 yields

(5.5) \begin{align}V_t \frac{I_t}{S^*_t} &= \mathbb E\left.\left[{A_T - A_t}\right|{\mathcal G}_t\right] = \mathbb E\left.\left[{\sum^n_{i=1} \sum_{j = \textbf{N}^i_t}^{\textbf{N}^i_T} \frac{I_{\tau^i_j}}{S^*_{\tau^i_j}} X^i_j}\right|{\mathcal G}_t\right] \nonumber\\&= \sum^n_{i=1} \mathbb{E}\left. \left[\sum_{j = \textbf{N}^i_t}^{\textbf{N}^i_T} \frac{I_{\tau^i_j}}{S^*_{\tau^i_j}} X^i_j\right| {\mathcal F}_t \vee {\mathcal H}^i_t \right] \nonumber\\&= \lambda m R_t \left. \mathbb E\left[ \int_t^T \frac{I_u}{S^*_u} \textrm{d} u \right| {{\mathcal H}}^{i,1}_\infty \vee {\mathcal F}_t \right] \nonumber\\& \quad + (n - R_t) \frac{ \mathbb E\left. \left[ \int_t^T \left(E\big[X^i_1\big] \frac{I_u}{S^*_u} + \lambda m \int_u^T \frac{I_v}{S^*_v} \textrm{d} v \right) \textrm{d} \mathbb P\left. \left( {\tau^i_1 \leqslant u} \right| {\mathcal F}_{u} \right) \right| {\mathcal F}_t \right]}{e^{-\int^t_0 \mu_u \textrm{d} u} + \int_0^t \bar{G}(t - u) e^{-\int^u_0 \mu_v \textrm{d} v} \mu_u \textrm{d} u} \nonumber\\&= \lambda m R_t \int_t^T \left. \mathbb E\left[ \frac{I_u}{S^*_u} \right| {\mathcal F}_t \right] \textrm{d} u \nonumber\\& \quad + (n - R_t) \frac{ \mathbb E\left. \left[ \int_t^T \left(E\big[X^i_1\big] \frac{I_u}{S^*_u} + \lambda m \int_u^T \frac{I_v}{S^*_v} \textrm{d} v \right) \textrm{d} \mathbb P\left. \left( {\tau^i_1 \leqslant u} \right| {\mathcal F}_{u} \right) \right| {\mathcal F}_t \right]}{e^{-\int^t_0 \mu_u \textrm{d} u} + \int_0^t \bar{G}(t - u) e^{-\int^u_0 \mu_v \textrm{d} v} \mu_u \textrm{d} u} \nonumber\\&= \lambda m R_t (T-t) \frac{I_t}{S^*_t} \nonumber\\&\quad + (n - R_t) \frac{ \mathbb E\left. \left[ \int_t^T \left(E\big[X^i_1\big] \frac{I_u}{S^*_u} + \lambda m \int_u^T \frac{I_v}{S^*_v} \textrm{d} v \right) \textrm{d} \mathbb P\left. \left( {\tau^i_1 \leqslant u} \right| {\mathcal F}_{u} \right) \right| {\mathcal F}_t \right]}{e^{-\int^t_0 \mu_u \textrm{d} u} + \int_0^t \bar{G}(t - u) e^{-\int^u_0 \mu_v \textrm{d} v} \mu_u \textrm{d} u}, \end{align}

where the conditional probability function $\mathbb P\left. \left( {\tau^i_1 \leqslant t} \right| {\mathcal F}_{t} \right)$ is given in (3.11), i.e.

\begin{equation*}\mathbb P\left. \left( \tau^i_1 \leqslant t \right| {\mathcal F}_t \right) = \int_0^t \left( \alpha_0 e^{-\int^s_0 \mu_v \textrm{d} v} \mu_s + \int_0^s g(s - u) e^{-\int^u_0 \mu_v \textrm{d} v} \mu_u \textrm{d} u \right)\textrm{d} s.\end{equation*}

The first component on the left-hand side of (5.5),

(5.6) \begin{equation}\lambda m R_t (T-t) \frac{I_t}{S^*_t},\end{equation}

corresponds to already reported claims. We observe that the valuation of this part does not involve the updating information after the first reporting. The second component on the right-hand side of (5.5),

(5.7) \begin{equation}(n - R_t)\frac{ \mathbb E\left. \left[ \int_t^T \left(E\big[X^i_1\big] \frac{I_u}{S^*_u} + \lambda m \int_u^T \frac{I_v}{S^*_v} \textrm{d} v \right) \textrm{d} \mathbb P\left. \left( {\tau^i_1 \leqslant u} \right| {\mathcal F}_{u} \right) \right| {\mathcal F}_t \right]}{e^{-\int^t_0 \mu_u \textrm{d} u} + \int_0^t \bar{G}(t - u) e^{-\int^u_0 \mu_v \textrm{d} v} \mu_u \textrm{d} u},\end{equation}

which can be further explicitly computed, corresponds to unreported claims and includes both IBNR claims and claims not yet incurred. The standard literature on non-life insurance is mainly focused on IBNR claims. However, for the pricing problem it is more appropriate to consider the entire expression (5.5). As already mentioned at the beginning of this section, this price equals the benchmarked risk-minimizing price, if we assume square-integrability of the claim.

We now calculate the associated benchmarked risk-minimizing strategy. For additional details, we refer to [Reference Zhang38]. We mainly focus on the second component (5.7) related to not-yet-reported claims, since hedging strategy is additive with respect to claims, and the first component (5.6), related to already reported claims, is perfectly hedgeable by trading inflation-linked zero-coupon bonds. By the same arguments as in Proposition 4.11 in [Reference Barbarin2] and Section 4.1 of [Reference Biagini and Zhang10], the benchmarked risk-minimizing strategy $\bar{\delta}$ associated to not-yet-reported claims is given by

\begin{equation*}\bar{\delta}_t = (n - R_{t_{-}}) \left(e^{-\int^t_0 \mu_u \textrm{d} u} + \int_0^t \bar{G}(t - u) e^{-\int^u_0 \mu_v \textrm{d} v} \mu_u \textrm{d} u \right)^{-1}\phi_t, \ \ \ \ t \in [0,T],\end{equation*}

where $\phi_t$ is the benchmarked risk-minimizing strategy at t related to

(5.8) \begin{equation}U_t \,:\!=\, \mathbb E\left. \left[ \int_0^T \left(\mathbb E\big[X^i_1\big] \frac{I_u}{S^*_u} + \lambda m \int_u^T \frac{I_v}{S^*_v} \textrm{d} v \right) \textrm{d} \mathbb P\left. \left( {\tau^i_1 \leqslant u} \right| {\mathcal F}_{u} \right) \right| {\mathcal F}_t \right], \ \ \ \ t \in [0,T].\end{equation}

More precisely, the vector process $\phi \,:\!=\, (\phi_t)_{t\in[0,T]}$ follows from the Galtchouk–Kunita–Watanabe decomposition of $(U_t)_{t \in [0,T]}$ ,

\begin{align*}U_t =& \mathbb E\left[ \int_0^T \left(\mathbb E\big[X^i_1\big] \frac{I_u}{S^*_u} + \lambda m \int_u^T \frac{I_v}{S^*_v} \textrm{d} v \right) \textrm{d} \mathbb P\left. \left( {\tau^i_1 \leqslant u} \right| {\mathcal F}_{u} \right) \right] + \int_0^t \phi^{\top}_u \textrm{d} \hat{S}_u + L^U_t.\end{align*}

Note that the benchmarked risk-minimizing strategy covers only the perfectly hedgeable part of the liability cash flow A. It is, however, the best possible hedging strategy in the benchmarked risk-minimizing sense. Indeed, the totally unhedgeable part, which is orthogonal to the hedgeable part, represents the basis hedging risk.

The form of V in (5.5) suggests the design of derivatives which can be used to hedge risks in this market model. In particular, since U involves only the stochastic processes $S^*$ , I, and $\mu$ , it is sufficient to introduce three kinds of instruments, namely pure financial assets, inflation-linked derivatives, and macro-risk-factor-linked derivatives, to hedge risks derived from $S^*$ , I, and $\mu$ respectively.