Time-dynamic evaluations under non-monotone information generated by marked point processes

Consider life insurance contracts that are evaluated by using big data. Data from activity trackers, social media, etc., can improve individual forecasts of the mortality and morbidity of insured persons. By exercising the ‘right to erasure’ according to the General Data Protection Regulation of the European Union, the policyholder may ask the insurer to delete parts of the health-related data at discretion. Moreover, data providers might implement self-imposed information restrictions for data privacy reasons. For example, users of Google products can opt for an auto-delete of location history and activity data after a fixed time limit. As a result, the evaluation of an insurance liability \(\xi \) according to (1.1) will be restricted to sub-sigma-algebras \((\mathcal{G}_{t})_{t \geq 0}\) that are non-monotone in \(t\) due to data deletions.

We consider a credit rating process. In the Jarrow–Lando–Turnbull model, the filtration \((\mathcal{F}_{t})_{t \geq 0}\) is generated by a finite-state-space Markov chain \((R_{t})_{t \geq 0}\) that represents credit ratings; cf. Jarrow et al. [15]. The Markov property makes it possible to equivalently replace \(\mathcal{F}_{t}\) in (1.1) by the sub-sigma-algebra \(\mathcal{G}_{t}:= \sigma (R_{t})\). The Markov assumption can be motivated by the theoretical idea that a credit rating should fully describe the current risk profile of a prospective debtor so that historical ratings can be ignored. However, empirical data does not always support the Markov property, so that \(E_{Q}[ \xi / B(T) | \mathcal{G}_{t} ]\) may in fact differ from \(E_{Q}[ \xi / B(T) | \mathcal{F}_{t} ]\); cf. Lando and Skodeberg [17]. The information dynamics of \(\mathcal{G}_{t}=\sigma (R_{t})\) is non-monotone in \(t\).

We call a process \(X\) incrementally adapted to \(\mathbb{G}\) if the increment \(X_{t}-X_{s}\) is \(\sigma ( \mathcal{G}_{u}, u \in (s,t])\)-measurable for any interval \((s,t] \subseteq [0,\infty )\).

Let \(X\) be incrementally adapted to \(\mathbb{G}\). We say that \(X\) is an infinitesimal forward/backward martingale (IF/IB-martingale) with respect to \(\mathbb{G}\) if for each \(t \geq 0\) and any increasing sequence \((\mathcal{T}_{n}^{{{t}}})_{n\in \mathbb{N}}\) of partitions of \([0, {t]} \) with \(\lim _{n \rightarrow \infty } |\mathcal{T}_{n}^{{{t}}}|=0\), we have or respectively, assuming that the expectations and limits exist.

We call \(X\) infinitesimally forward/backward predictable (IF/IB-predictable) with respect to \(\mathbb{G}\) if for each \(t \geq 0\) and any increasing sequence \((\mathcal{T}_{n}^{{{t}}})_{n\in \mathbb{N}}\) of partitions of \([0, {t]}\) with \(\lim _{n \rightarrow \infty } |\mathcal{T}_{n}^{{{t}}}|=0\), we almost surely have or respectively, assuming that the expectations and limits exist.

Let \(X\) be incrementally adapted to \(\mathbb{G}\). We say that a process \(C\) is an infinitesimal forward/backward compensator of \(X\) (IF/IB-compensator) with respect to \(\mathbb{G}\) if \(C\) is incrementally adapted to \(\mathbb{G}\) and IF/IB-predictable and \(X-C\) is an IF/IB-martingale with respect to \(\mathbb{G}\), respectively.

We say that \(E[ \xi | \mathcal{G}_{t}]-E[ \xi | \mathcal{G}_{0}] = F_{t} - B_{t}\), \(t \geq 0\), is an infinitesimal martingale representation if \(F\) is an IF-martingale and \(B\) is an IB-martingale with respect to \(\mathbb{G}\).

Let \(X\) be an integrable càdlàg process. If there exists a unique càdlàg process \(X^{\mathbb{G}}\) such that almost surely for each \(t \geq 0\), we call \(X^{\mathbb{G}}\) the optional projection of \(X\) with respect to \(\mathbb{G}\).

We call \(E[ X_{t} | \mathcal{G}_{t}]-E[ \xi | \mathcal{G}_{0}] = F_{t} - B_{t} + C_{t}\), \(t \geq 0\), an infinitesimal representation if \(F\) is an IF-martingale, \(B\) is an IB-martingale and \(C\) is either an IB-compensator or an IF-compensator with respect to \(\mathbb{G}\).

Recall that \(T_{2i-1} \leq T_{2i}\), \(i \in \mathbb{N}\), is the only kind of order that we assume to hold between the random times \(T_{i}\), resulting from the natural assumption \(\tau _{i} \leq \sigma _{i} \), \(i \in \mathbb{N}\). This fact is relevant when an ordering unintentionally reveals additional information. For example, if we have a model where the innovation times \(\tau _{i}\) are ordered, i.e., \(T_{1} < T_{3} < T_{5} < \cdots \), then \(\mathcal{G}_{t}\) reveals among other things the exact number of deletions that have happened until \(t\). This can be an unwanted feature if the number of past deletions is itself a non-admissible piece of information. In many situations, we can avoid such an implied information effect by ordering the pairs \((T_{2i-1},T_{2i})\) in a non-informative way.

Without loss of generality, suppose here that \(0 \not \in E \). We define an infinite-dimensional process \((\Gamma _{t})_{t \geq 0} \) by Then, using the fact that the paths of \((\Gamma _{t})_{t \geq 0} \) are componentwise càdlàg, the information \(\mathcal{G}_{t} \) and \(\mathcal{G}^{-}_{t}\) can be alternatively represented as where the left limit \(\Gamma _{t-}\) is defined componentwise. However, \(\mathcal{G}^{-}_{t}\) is usually different from the left set-limit \(\mathcal{G}_{t-}\), and the latter set-limit might not even exist. For example, consider a model with only two jumps \(T_{1}\), \(T_{2}\) in finite time and a trivial mark \(Z_{2}=\mathrm{{const}}\). It is not difficult to choose \(T_{1}, T_{2}\) in such a way that the events \(\{T_{1} \leq s < T_{2}\}\) and \(\{T_{1} \leq u < T_{2}\}\) are different for each \(s < u \leq t\). In this case, \(\liminf _{s \uparrow t} \mathcal{G}_{s}= \bigvee _{ s < t} \bigcap _{ s < u < t} \mathcal{G}_{u} \) equals the completed trivial sigma-algebra, whereas \(\limsup _{s \uparrow t} \mathcal{G}_{s}= \bigcap _{ s < t} \bigvee _{ s < u < t} \mathcal{G}_{u} \) equals \(\mathcal{G}^{-}_{t}\).

Suppose that \(X=(X_{t})_{t\geq 0}\) is a càdlàg process that satisfies Then the optional projection \(X^{\mathbb{G}}\) according to Definition 2.6exists, and we have \(X^{\mathbb{G}}_{t-}=E[ X_{t-} | \mathcal{G}^{-}_{t}]\) almost surely for each \(t > 0\). If \(X\) has integrable variation on compacts, then \(X^{\mathbb{G}}\) has paths of finite variation on compacts.

For any integrable random variable \(\xi \) and any sets \(M \in \mathcal{M}\) and \(I \in \mathcal{N}\), we almost surely have with the convention that \(0/0:=0\).

The left-hand sides of (4.3) almost surely equal the conditional expectations that one obtains when the families \(\mathbb{G}\) and \(\mathbb{G}^{-}\) of sigma-algebras are replaced by their non-completed versions. Therefore, in the remaining proof, we ignore the extension by \(\mathcal{Z}\) in the definitions of \(\mathbb{G}\) and \(\mathbb{G}^{-}\).

For each \(H \in \sigma (Z_{M}) \), there exists a \(G \in \mathcal{G}_{t} \) such that \(H \cap A_{t}^{M} = G \cap A_{t}^{M} \), and for each \(G \in \mathcal{G}_{t} \), there exists an \(H \in \sigma (Z_{M}) \) such that \(G \cap A_{t}^{M} = H \cap A_{t}^{M}\). Thus This implies that the random variable \(\mathbb{I}^{M}_{t} \frac{E_{M,R_{I}} [ \xi \mathbb{I}^{M}_{t} ]}{E_{M,R_{I}}[ \mathbb{I}^{M}_{t}]}\) is \((\mathcal{G}_{t}\vee \sigma (R_{I}))\)-measurable, and for each \(G \in \mathcal{G}_{t} \vee \sigma (R_{I})\), we obtain i.e., the first equation in (4.3) holds. By replacing (4.4) by we can analogously show that the second equation in (4.3) holds. □

For each \(M \in \mathcal{M}, I \in \mathcal{N}, r \in [0,\infty) \times E_{I} \) and each càdlàg process \(X\) that satisfies condition (4.1), the stochastic processes have càdlàg paths. Moreover, their left limits can be obtained by replacing \(X_{t} \mathbb{I}_{t}^{M}\) by \(X_{t-} \mathbb{I}_{t-}^{M}\).

Apply the dominated convergence theorem. □

With the conventions \(0/0:=0\) and \(1/0:=\infty \), we have for each \(M\in \mathcal{M} \) almost surely that

Let \(\tau \) and \(\sigma \) be any two nonnegative random times such that \(\tau \leq \sigma \). At first we are going to show that almost surely. For each \((t,s) \in [0,\infty )^{2}\), we define the unbounded rectangles and the countably generated set Let \(\partial B\) and \(B^{\circ }\) be the boundary and the interior of \(B\). Any line of the form intersects \(\partial B\) at most at one point, since for any two points \(y,y' \in L_{x}\) with \(y\neq y'\), we either have \(y \in A_{y'}^{\circ }\) or \(y' \in A_{y}^{\circ }\). Therefore the set is countable, and is countably generated. The sets \(N_{B}=\{(\tau ,\sigma ) \in B\}\) and \(N_{C}=\{(\tau ,\sigma ) \in C\}\) are both nullsets since they equal countable unions of nullsets.

Suppose now that \(Z(\omega ) = \infty \) for an arbitrary but fixed \(\omega \in \Omega \). We necessarily have \(\tau (\omega ) < \sigma (\omega )\). Since \(t \mapsto E[ \mathbf{1}_{\{\tau \leq t < \sigma \}} ]\) is a càdlàg function, at least one of the following statements is true:

In case (1), we have \(P[(\tau ,\sigma )\in A_{(u,u)}] =E[ \mathbf{1}_{\{\tau \leq u < \sigma \}} ] =0\) and \((\tau (\omega ),\sigma (\omega ))\) is in \(A_{(u,u)}^{\circ }\), so that we can conclude that \(\omega \in N_{B}\).

In case (2), we can argue analogously to case (1), but need to replace the definition of \(A_{(t,s)}\) by \(\{ (t',s') : t'< t, s\leq s'\}\) and define a corresponding nullset \(N_{B}'\). We obtain that \(\omega \in N_{B}'\).

In case (3), we have \(P[(\tau ,\sigma )\in A_{(\tau (\omega ),\tau (\omega ))}]=E[ \mathbf{1}_{\{\tau \leq \tau (\omega ) < \sigma \}} ] =0\) as well as \((\tau (\omega ),\sigma (\omega )) \in A_{(\tau (\omega ),\tau ( \omega ))} \subseteq B \cup \partial B\). If \((\tau (\omega ),\sigma (\omega )) \in B\), then \(\omega \in N_{B}\). If \((\tau (\omega ),\sigma (\omega )) \in \partial B \), then the whole line segment is in \(\partial B\) because of \((\tau (\omega ), \tau (\omega )) \in \partial B\) and the rectangular shape of the sets \(A_{(t,s)}\). On this line, there is at least one intersection with \(C\), so that we can conclude that \(\omega \in N_{C}\).

In case (4), we can argue similarly to case (3), but need to replace the definition of \(A_{(t,s)}\) by \(\{ (t',s') : t'< t, s\leq s'\}\) and define corresponding nullsets \(N_{B}'\) and \(N_{C}'\).

All in all, we have \(P[Y=\infty ] \leq P[N_{B} \cup N_{C}\cup N_{B}' \cup N_{C}']=0\), i.e., (4.6) holds.

Now let \(M \in \mathcal{M}\) be arbitrary but fixed and choose \(\tau \) and \(\sigma \) as the random times where \(\mathbb{I}_{t}^{M}\) jumps from zero to one and jumps back to zero, respectively. Suppose that \(P_{Z_{M}=z}\) is a regular version of \(P[\, \cdot \,| Z_{M} = z]\) with corresponding expectation \(E_{Z_{M}=z}[\,\cdot \,]\). Then from (4.6), we can conclude that for each choice of \(z\). Replacing both \(z\) by \(Z_{M}\), where we use the insertion rule for conditional expectations for the inner \(z\), and taking the unconditional expectation on both sides of the equation, we end up with  □

Motivated by Proposition 4.2, we set since this process almost surely equals \(X^{\mathbb{G}}_{t}\) for each \(t \geq 0\). Note that there are at most a countable number of conditional expectations involved; so the corresponding regular versions are simultaneously unique up to evanescence. For each compact interval \([0,t]\) and almost each \(\omega \in \Omega \), the set is finite due to the assumption (3.1). If \(E_{M} [ \mathbb{I}^{M}_{t} ](\omega )\neq 0\), Lemma 4.3 yields that If \(E_{M} [ \mathbb{I}^{M}_{t} ](\omega )=0\), Proposition 4.4 implies that \(\mathbb{I}^{M}_{t}=0\) for almost all \(\omega \in \Omega \), where the exception nullset does not depend on the choice of \(t\). So (4.8) is almost surely true on \([0,\infty )\), since \(\mathbb{I}^{M}_{t}(\omega )=0\) implies that there is a whole interval \([t,t+\epsilon _{\omega })\) where the right-continuous jump path \(s \mapsto \mathbb{I}^{M}_{s}(\omega )\) is constantly zero. Similarly, we can show that the process \(Y\) almost surely has left limits, which are of the form According to Proposition 4.2, \(Y_{t-}\) almost surely equals \(E[ X_{t-} | \mathcal{G}^{-}_{t}]\). As càdlàg processes are uniquely defined by their values on countable dense subsets of the time line, our choice for \(X^{\mathbb{G}}\) is almost surely the only possible modification of \((E[X_{t} | \mathcal{G}_{t}])_{t\geq 0}\).

The variation of \(Y\) on \([0,t]\) is bounded by where \(\mathcal{T}^{t}\) is any partition of \([0,t]\). As \(C_{M}(\omega ):= \sup _{t} \mathbb{I}^{M}_{t}(\omega )/E_{M} [ \mathbb{I}^{M}_{t} ](\omega )\) is finite for almost each \(\omega \in \Omega \) (see Proposition 4.4) and the variation of \(L_{M}(s):= E_{M}[\mathbb{I}^{M}_{s}] \) is bounded by 2, the latter bound is dominated by which is finite for almost each \(\omega \in \Omega \) since \(X\) has integrable variation on compacts and \(\mathcal{M}_{t}(\omega )\) is finite. □

For each \(I \in \mathbb{N}\), the mappings \(\nu _{I}\) and \(\rho _{I}\) can be uniquely extended to random measures on \(([0, \infty ) \times E_{I}, \mathcal{B}([0, \infty ) \times E_{I}))\).

Suppose that the mappings \((t,e,\omega ) \mapsto F_{I}(t,e)(\omega )\), \(I \in \mathcal{N}\), are jointly measurable and satisfy If \(F_{I}(t,e)\) is \(\mathcal{G}_{t}^{-}\)-measurable for each \((t,e)\), then for each \(B \in \mathcal{E}_{I}\), the jump process has the IF-compensator If \(F_{I}(t,e)\) is \(\mathcal{G}_{t}\)-measurable for each \((t,e)\), then for each \(B \in \mathcal{E}_{I}\), the jump process has the IB-compensator

For each \(M \in \mathcal{M}\) and \(t \geq 0\), we almost surely have

For each \(t \geq 0\) and \(M \in \mathcal{M}\), (3.1) implies that almost surely. Therefore, applying the monotone convergence theorem yields almost surely for each \(M \in \mathcal{M}\) and \(t \geq 0\). □

The processes \((\mathbb{I}^{M}_{u-})\) and \((P_{M}[A^{M}_{u-}])\) are jointly measurable with respect to \((u,\omega )\) since they are left-continuous in \(u\); see Lemma 4.3. The mapping \((u,e,\omega ) \mapsto P_{M,R_{I}=(u,e)}[A_{u-}^{M}]\) is jointly measurable with respect to \((u,e,\omega )\) since \(P_{M,R_{I}=(u,e)}[A_{s-}^{M}]\) is left-continuous in \(s\) and jointly measurable with respect to \((u,e,\omega )\in [0,\infty )^{|I|}\times E_{I}\times \Omega \); see Lemma 4.3. Thus for any fixed \(A \in \mathcal{B}([0, \infty ) \times E_{I})\), the mapping \(\omega \mapsto \nu _{I} ( A)(\omega )\) is measurable. Moreover, for almost each \(\omega \in \Omega \), the mapping \(A \mapsto \nu _{I} (A)(\omega )\) is a locally finite measure on \(([0, \infty ) \times E_{I} , \mathcal{B}([0, \infty ) \times E_{I} ))\). This can be seen by combining Proposition 4.4 and (5.2) and using the fact that \(P_{M, R_{I}=(u,e)}[A_{u-}^{M}]\) is bounded by 1. Hence \(\nu _{I}\) has a unique extension to a random measure on \(([0, \infty ) \times E_{I} , \mathcal{B}([0, \infty ) \times E_{I}))\). Similar conclusions hold for the mappings \(\rho _{I}\). □

Suppose that the mappings \((t,e,\omega ) \mapsto F_{I}(t,e)(\omega )\), \(I \in \mathcal{N}\), are jointly measurable and satisfy (5.1). For each \(t >0\) and \(B \in \mathcal{E}_{I} \), we almost surely have for any increasing sequence \((\mathcal{T}_{n}^{{{t}}})_{n\in \mathbb{N}}\) of partitions of \([0, {t]}\) with \(\lim _{n \rightarrow \infty } |\mathcal{T}_{n}^{{{t}}}|=0\) and for \(G_{I}\) and \(H_{I}\) defined by

By decomposing \(F\) into a positive part \(F^{+}\) and a negative part \(F^{-}\), it suffices to prove the first equation for the nonnegative mappings \(F^{+}\) and \(F^{-}\) only. Therefore, without loss of generality, we suppose from now on that \(F\) is nonnegative.

Let \(\mathcal{M}_{t}=\mathcal{M}_{t}(\omega )\) be defined as in (4.7). In the following, we use the notation \(J_{k}:=(t_{k},t_{k+1}]\). Since \(\sum _{M \in \mathcal{M}_{t}} \mathbb{I}^{M}_{t_{k}}=1\) for any \(t_{k}\), applying (4.3), the monotone convergence theorem and the law of total probability gives for almost each \(\omega \in \Omega \). For \(u \in (0,t]\), let \(J^{u}\) be the unique interval \((t_{k},t_{k+1}]\) from \(\mathcal{T}_{n}^{{{t}}}\) such that \(t_{k}< u \leq t_{k+1}\), and let \(t(u)\) be the left end point of \(J^{u}\). Then we can write Taking the limit for \(n\rightarrow \infty \), we obtain for almost each \(\omega \in \Omega \) that using that \(\mathcal{M}_{t}\) is finite for almost each \(\omega \) and applying the monotone convergence and the dominated convergence theorem. Note that Proposition 4.4, the assumption (5.1) and \(0 \leq \mathbb{I}_{t(u)}^{M} F_{I}\bullet \mu _{I}(J^{u} \times B ) \leq F_{I}\bullet \mu _{I}((0,t] \times B )\) ensure the existence of an integrable majorant. For \(n \rightarrow \infty \), we have \(t(u) \uparrow u\) and \(J^{u} \downarrow \{u\}\); so the dominated convergence theorem implies that In summary, the right-hand side of (5.3) equals the integral \(G_{I}\bullet \nu _{I}((0,t ]\times B)\), and we can conclude that the first equation in Proposition 5.4 holds. The proof of the second is similar. □

Under the assumptions of Proposition 5.4, for each \(t \geq 0\) and \(B \in \mathcal{E}_{I} \), we almost surely have for any increasing sequence \((\mathcal{T}_{n}^{{{t}}})_{n\in \mathbb{N}}\) of partitions of \([0, {t]}\) with \(\lim _{n \rightarrow \infty } | \mathcal{T}_{n}^{{{t}}}|=0\).

By decomposing \(G\) into a positive part \(G^{+}\) and a negative part \(G^{-}\), it suffices to prove the first equation for the nonnegative mappings \(G^{+}\) and \(G^{-}\) only. Therefore, without loss of generality, we suppose from now on that \(G\) is nonnegative.

From the definition of \(\nu _{I}\) and the monotone convergence theorem, we get From Proposition 4.2, we know that \(G_{I}(u,e)\) is \(\mathcal{G}^{-}_{u}\)-measurable for each \((u,e)\). This fact and (4.5) imply that Applying the Fubini–Tonelli theorem and the monotone convergence theorem gives The latter expectation is finite according to (5.1). Hence for each \(M \in \mathcal{M}\), we almost surely have Let \(J_{k}:=(t_{k},t_{k+1}]\). From the dominated convergence theorem, we obtain since \(\mathbb{I}^{M}\) is bounded by 1 and because of the second line in (5.4). By using the first line in (5.4), the dominated convergence theorem moreover yields By applying the Fubini–Tonelli theorem, we can show that the last term equals Using Proposition 4.4 and the dominated convergence theorem, we therefore obtain where the second equality uses that (4.5) and the \(\mathcal{G}^{-}_{u}\)-measurability of \(G_{I}(u,e)\) allows us to pull \(G_{I}(u,e)\) out of the conditional expectation \(E_{M}[\mathbb{I}^{M}_{u-} G_{I}(u,e)] \). Summing the latter equation over \(M \in \mathcal{M}_{t}\) for \(\mathcal{M}_{t}\) as in (4.7) and applying Proposition 4.2, we obtain almost surely. Thus we can conclude that the first equation in Proposition 5.5 holds. The proof of the second is similar. □

Let \(G_{I}\) and \(H_{I}\) be defined as in Proposition 5.4. If \(F_{I}(t,e)\) is \(\mathcal{G}_{t}^{-}\)-measurable for each \((t,e)\), then Proposition 4.2 implies that \(G_{I}(t,e)=F_{I}(t,e)\) almost surely. Similarly, if \(F_{I}(t,e)\) is \(\mathcal{G}_{t}\)-measurable for each \((t,e)\), we have almost surely that \(H_{I}(t,e)=F_{I}(t,e)\). With this fact and by subtracting the limit equations in Propositions 5.4 and 5.5, we obtain that satisfy the defining limit equations for IF/IB-martingales. IF/IB-predictability of the compensators follows from Proposition 5.5. Note that all involved processes are incrementally adapted to \(\mathbb{G}\) because of (4.4) and (4.5). □

Let \(\xi \) be an integrable random variable. Then for each \(t \geq 0\), (1.2) holds almost surely for For each \(I \in \mathcal{N}\) and \(e \in E_{I}\), the process \(u \mapsto G_{I}(u-,u,e)\) is \(\mathbb{G}^{-}\)-adapted and the process \(u \mapsto G_{I}(u,u,e)\) is \(\mathbb{G}\)-adapted.

Let \(\xi \) be an integrable random variable. Then for each \(t \geq 0\), we have

As (6.3) is additive in \(\xi \), it suffices to show the equation for nonnegative and bounded random variables \(\xi \) only. The general case then follows from monotone convergence applied to both parts of the sequence \(\xi _{n} := (\xi _{n} \wedge n)^{+} - (-\xi _{n} \wedge n)^{+} \), \(n \in \mathbb{N}\). Therefore, in the remaining proof, we suppose that \(0 \leq \xi \leq C \) for a finite real number \(C\).

Let \(U_{t_{k}}(\omega ):= \sup \{ s \in (t_{k} ,\infty ) : T_{j}(\omega ) \not \in (t_{k},s), j \in \mathbb{N}\}\), i.e., \(U_{t_{k}}\) is the time of the first occurrence of a random time strictly after \(t_{k}\). Since \(1=\sum _{I \in \mathcal{N}} \mathbf{1}_{\{U_{t_{k}}=Q_{I}\}} \), we can conclude that for \(B_{I,k}=(t_{k},t_{k+1}] \times E_{I}\), where we use the fact that unless \(t_{k}< Q_{I} \leq t_{k+1}\). Because of (5.2) and we can apply the dominated convergence theorem on the last line in (6.4), which leads to (6.3). Note here that for \(t_{k+1} \downarrow u\) and \(t_{k} \uparrow u\) implies that  □

Fix \(M \in \mathcal{M}\) and define \(M+1:= \{i+1 : i \in M\}\). If \(\mathbb{I}^{M}_{u-}=1\), then only random times from the index set can occur at time \(u\). If \(\mathbb{I}^{M}_{u}=1\), then only random times from the index set can occur at time \(u\). Therefore (6.3) can be represented as where Furthermore, using \(M'\cap M'' = \emptyset \), we can show that for \(t>0\), which implies that \(E_{M}[ \mathbb{I}_{t-}^{M} \mathbb{I}_{t}^{M} \xi ]= K_{t}+L_{t-}\). Analogously we obtain In the specific case \(\xi =1\), we write \(k_{t}\) and \(\ell _{t}\) instead of \(K_{t}\) and \(L_{t}\). By applying integration by parts pathwise for each \(\omega \in \Omega \), we get that The equation formed by the first and last line can be rewritten as because of With the convention \(0/0:=0\) and by using the Radon–Nikodým theorem, we may multiply by \((k_{t}+\ell _{t-})^{-1}\) on both sides, which leads to Because of (6.3) and \(\mathrm{d}\mathbb{I}_{t}^{M} = \sum _{I \in \mathcal{N}} ( \mathbb{I}_{t}^{M} -\mathbb{I}_{t-}^{M} ) \mu _{I} (\mathrm{d}t \times E_{I} )\), the last equation can be rewritten to Let \(I \in \mathcal{N}\) be arbitrary but fixed. Then for each \(M \in \mathcal{M}\), there exists an \(\tilde{M} \in \mathcal{M}\), and for each \(\tilde{M} \in \mathcal{M}\), there exists an \(M \in \mathcal{M}\) such that As a consequence, for almost each \(\omega \in \Omega \), we have Because of summing (6.5) over \(M \in \mathcal{M}\) and adding (6.6) yields for almost each \(\omega \in \Omega \) that we have (1.2) and (6.2) after rearranging the addends. By applying Proposition 4.2, we can see that \(G_{I}(u-,u,e)\) is \(\mathcal{G}_{u-}\)-measurable and \(G_{I}(u,u,e)\) is \(\mathcal{G}_{u}\)-measurable for each \((I,e)\). □

Let \(X\) be a càdlàg process that satisfies (4.1) and has an IB-compensator with respect to \(\mathbb{G}\), denoted as \(X^{IB}\). Then almost surely with If \(X\) has an IF-compensator with respect to \(\mathbb{G}\), denoted as \(X^{IF}\), then (7.2) still holds but with \(X_{t}^{IB}\) replaced by \(X_{t}^{IF}\) and \(X_{u-}\) replaced by \(X_{u}\) in (7.3).

Let \(h(M,t)(\omega ): \mathcal{M} \times [0,\infty ) \times \Omega \rightarrow \mathbb{R}\) be measurable and suppose that \(|h(M,t)| \leq Z\) for an integrable majorant \(Z\). Let \(\gamma \) be the sum of the Lebesgue measure and a countable number of Dirac measures, for deterministic time points \(0\leq t_{1} < t_{2} < \cdots \) that increase to infinity. Then the càdlàg process \(X\) defined by has the IB-compensator In order to see this, apply Proposition 4.2, the dominated convergence theorem, Proposition 4.4 and Lemma 4.3 in order to obtain that almost surely, where \(\mathcal{M}_{t}\) is defined as in (4.7). If \(s\mapsto h(M,s)\) is \(\mathbb{G}\)-adapted for each \(M\), we have \(X=X^{IB}\). Likewise we can show that the càdlàg process has the IF-compensator If \(s\mapsto h(M,s)\) is \(\mathbb{G}^{-}\)-adapted for each \(M\), we have \(Y=Y^{IF}\).

The theorem follows from the additive decomposition and from applying Theorem 6.1 for each summand \(E[ X_{t_{k}} | \mathcal{G}_{t_{k+1}}] - E[ X_{t_{k}} | \mathcal{G}_{t_{k}}]\). The sum \(\sum _{\mathcal{T}_{n}^{{{t}}}} (E[ X_{t_{k}} | \mathcal{G}_{t_{k+1}}] - E[ X_{t_{k}} | \mathcal{G}_{t_{k}}])\) has a representation of the form (1.2) in case of \(t_{k} < s \leq t_{k+1}\) for \(G_{I}(s, u,e)\) defined by (7.3). Because of the càdlàg property of \(X\), by applying the dominated convergence theorem pathwise for almost each \(\omega \in \Omega \), we end up with (7.2) and (7.3). The alternative decomposition leads to the second variant with \(X^{IB}\) replaced by \(X^{IF}\) and \(X_{u-}\) by \(X_{u}\) in (7.3). □

Without loss of generality, suppose here that \(0 \not \in E\). Motivated by Remark 3.2, for any \(t > 0\) and any integrable random variable \(\xi \), define for \(e \in E_{I}\), where \(J_{t}:= \sum _{I \in \mathcal{N}} \mu _{I}(\{t\}\times E_{I})\) indicates whether there is a stopping event at time \(t\). One can then show that the integrands in (7.2) are almost surely equal to for each \(t>0\), \(I \in \mathcal{N}\) and \(e \in E_{I}\). The differences on the right-hand side have intuitive interpretations. The first line describes the difference in expectation between a change scenario and a remain scenario if we are currently at time \(t-\) and are looking forward in time. Similarly, the second line describes the difference in expectation between a change scenario and a remain scenario if we are currently at time \(t\) and are looking backward in time. In (7.2), these differences in expectation are integrated with respect to the compensated forward and backward scenario dynamics.

Consider a life insurance contract where the insurer collects health-related information about the insured with the aim to improve forecasts of the individual future insurance liabilities. For example, this can involve data from activity trackers or social media. Here, the marked point process includes the time of death \(\tau _{1}\), which is recorded as \(\zeta _{1}:=\tau _{1}\), and further health-related information \((\tau _{i}, \zeta _{i})_{i \geq 2}\). Upon request of the policyholder with reference to the ‘right to erasure’ according to the General Data Protection Regulation of the European Union, or as a self-imposed data privacy effort of the data provider, the insurer deletes parts of the health-related data at certain time points, i.e., we expand \((\tau _{i}, \zeta _{i})_{i \geq 2}\) by deletion times \((\sigma _{i})_{i \geq 2}\). For completeness, we define \(\sigma _{1}:= \infty \).

In the classical insurance modelling without data deletion, the time dynamics of the expected future insurance payments is commonly described by Thiele’s equation; see e.g. Møller [19] and Djehiche and Löfdahl [12]. Suppose that \(A_{t}\) gives the aggregated benefit cash flow of the life insurance contract on \([0,t]\), including survival benefits with rate \(a(t)\) and a death benefit of \(\alpha (t)\) upon death at time \(t\), i.e., We assume here that \(a:[0,\infty ) \rightarrow \mathbb{R}\) and \(\alpha :[0,\infty ) \rightarrow \mathbb{R}\) are bounded. For a given interest intensity \(\phi :[0,\infty ) \rightarrow [0,\infty )\) and a finite contract horizon \(T\), the process describes the discounted future liabilities of the insurer seen from time \(t\). As the càdlàg process \(X=(X_{t})_{t \geq 0}\) is neither adapted to \(\mathbb{F}\) nor to \(\mathbb{G}\), an insurer has to work with the optional projection instead (the so-called prospective reserve), i.e., the insurer aims to calculate in case that there is no data deletion and in case that information deletions may occur. The process \(X^{\mathbb{G}}\) is a well-defined càdlàg process according to Theorem 4.1.

By applying (7.1) and Itô’s lemma, we can derive the so-called stochastic Thiele equation with terminal condition \(X^{\mathbb{F}}_{T}=0\); cf. Møller [19, Eq. (2.17)]. The integrand \(F_{I}(t,e)\), which almost surely equals according to Remark 7.3, is a key quantity in life insurance risk management and is known as sum at risk. Equation (8.1) can be interpreted as a backward stochastic differential equation (BSDE) with solution \((X^{\mathbb{F}},(F_{I})_{I})\); see Djehiche and Löfdahl [12] for Markovian and Christiansen and Djehiche [4] for non-Markovian models. The BSDE (8.1) is in particular relevant if the life insurance payments \(a\) and \(\alpha \) depend on the current policy value so that the insurance cash flow \(A\) is only implicitly defined.

By applying Theorem 7.1 and Itô’s lemma and using the fact that the process \(A\) equals its own IB-compensator (since \(\sigma _{1} = \infty \)), we are able to derive an analogous equation for \(X^{\mathbb{G}}\), namely with terminal condition \(X^{\mathbb{G}}_{T}=0\). This equation can be interpreted as a new type of BSDE with solution \((X^{\mathbb{G}},(G_{I})_{I})\), featuring an IF-martingale and an IB-martingale instead of a classical martingale. The IF-martingale in the first line describes the impact of new information on the optional projection \(X^{\mathbb{G}}\). The IB-martingale in the second line quantifies the effect on \(X^{\mathbb{G}}\) of information deletions. The integrands \(G_{I}(t-,t,e)\) and \(G_{I}(t,t,e)\), which are almost surely equal to according to Remark 7.3, generalise the classical definition of the sum at risk. They are needed in life insurance risk management for sensitivity analyses, safe-side calculations, contract modifications and surplus decompositions.

If the policyholder may decide about data deletions at discretion, then the resulting value changes of the insurance contract can be systematically exploited by the policyholder, leading to a kind of data privacy arbitrage. Since it is the IB-martingale in (8.2) that measures the value changes due to data deletions at times \((\sigma _{i})_{i \geq 2}\), it represents the potential data privacy arbitrage. A simple solution for avoiding data privacy arbitrage could be to charge the IB-martingale as a fee upon a data deletion request. The fee can also be negative and represents then a bonus payment. However, more complex risk sharing schemes will be needed in insurance practice that moreover distinguish between different causes for data deletions. By following the concept of Schilling et al. [26] to interpret martingale representations as risk factor decompositions, we may interpret the infinitesimal martingale parts in (8.2) as an additive surplus decomposition that can distinguish between numerous kinds of jump events \(\mu _{I}\), \(I\subseteq \mathbb{N}\), \(\vert I \vert <\infty \). Such an additive decomposition of the insurer’s surplus is an important step for aligning insurance risk management to the digital age.

A popular approximation concept in credit rating modelling is to pretend that the credit rating process is Markovian even if the empirical data does not fully support this assumption. Suppose that credit ratings are updated at integer times only. By setting \(\tau _{i}:=i-1\) and \(\sigma _{i}:=i\) for \(i \in \mathbb{N}\) and defining \(\zeta _{i}\) as the credit rating at time \(\tau _{i}\), the rating process \(R=(R_{t})_{t \geq 0}\) has the representation and satisfies The jumps of the process \(R\) correspond to the random counting measures \(\mu _{I}\). In the Jarrow–Lando–Turnbull model, the rating space \(E\) is finite, \((R_{i})_{i \in \mathbb{N}_{0}}\) is assumed to be a Markov chain, and for \(r_{i},r_{i+1} \in E\) and \(i \in \mathbb{N}_{0}\), where \(Q\) is the risk-neutral measure and \(\pi \) is a deterministic function on \(\mathbb{N}_{0} \times E\). The latter formula allows us to estimate \(Q\) from market data by a two step method. First, the transition probabilities \(P[R_{i+1}= r_{i+1}| R_{i}=r_{i} ]\) are estimated from observed credit rating time series. Then the function \(\pi \) is calibrated such that the risk-neutral values of credit rating derivatives conform with observed market prices. Once we have \(Q\), we can use the (classical) martingale representation (6.1) in order to explicitly construct hedges for financial claims \(\xi \); see e.g. Last and Penrose [18]. For example, by arguing analogously to (8.1), the claim \(\xi =h(R_{T})\) has the martingale representation The integral in the first line describes the investments in the risk-free asset \(B\). The second line corresponds to risky investments. It can be rewritten in terms of the tradable assets in a complete financial market; cf. Last and Penrose [18, Sect. 5], which yields a trading strategy that can be used to replicate the claim \(\xi \).

A standard estimator for the state occupation probabilities of the Markov chain \(R\) with respect to \(P\) is the Aalen–Johansen estimator, which directly corresponds to the Nelson–Aalen estimator for the compensators \(\lambda _{I}\) of the random counting measures \(\mu _{I}\). Under the assumption that \(R\) is Markovian, the Nelson–Aalen estimator consistently estimates \(\lambda _{I}=\nu _{I}\). If \(R\) is not Markovian, then the Nelson–Aalen estimator still consistently estimates \(\nu _{I}\), see Datta and Satten [9], but now \(\nu _{I} \neq \lambda _{I}\). In other words, if we ignore the information beyond \(\mathbb{G}\) in the estimation of \(\lambda _{I}\) due to an incorrect Markov assumption, then we actually estimate the infinitesimal forward compensator \(\nu _{I}\) instead of the classical compensator \(\lambda _{I}\). Similarly, ignoring the information beyond \(\mathbb{G}\) upon estimating \(F_{I}\) and (1.1) from market data means that we unintentionally end up with the integrands and \(B(t-) E_{Q}[ h(R_{T})/B(T) |\mathcal{G}_{t}] \) rather than the integrands and \(B(t-) E_{Q}[ h(R_{T})/B(T) |\mathcal{F}_{t}] \). For the correct interpretation of the latter conditional expectations with mixed conditions, see Remark 7.3. All in all, by ignoring the information beyond \(\mathbb{G}\) in the estimation and calculation of (8.3) due to an incorrect Markov assumption, we unintentionally end up with instead of the right-hand side of (8.3). This unintentional modification distorts the replicating trading strategy for the claim \(h(R_{T})\) which was included in (8.3). Do we still correctly replicate \(h(R_{T})\)? By applying Theorem 6.1 instead of (6.1) and using that \(\mathcal{F}_{0}=\mathcal{G}_{0}\) and \(\mathcal{G}_{T}= \sigma (R_{T}) \vee \mathcal{Z}\), we get analogously to (8.3) that Equation (8.5) implies that the distorted trading strategy we might derive from (8.4) is actually not a hedge for \(\xi =h(R_{T})\). The hedging error is given by the third line in (8.5). To sum up, by estimating and calculating (8.3) under an incorrect Markov assumption for \(R\), we unintentionally replace the (classical) \(\mathbb{F}\)-martingale in (8.3) by the \(\mathbb{G}\)-IF-martingale in (8.4) (the risk-free investment is also affected), and the corresponding \(\mathbb{G}\)-IB-martingale is just the hedging error.

Schilling et al. [26] interpret martingale representations as additive risk factor decompositions. Likewise we can read the (infinitesimal) martingale parts in (8.3) and (8.5) as linear risk factor decompositions. The relevance of such decompositions in credit risk modelling is explained in Rosen and Saunders [25].