Extended Reduced-Form Framework for Non-Life Insurance
aa r X i v : . [ q -f i n . M F ] O c t Extended Reduced-Form Framework for Non-LifeInsurance
Francesca Biagini ∗† Yinglin Zhang ‡ October 29, 2019
Abstract
In this paper we propose a general framework for modeling an insuranceliability cash flow in continuous time, by generalizing the reduced-form frame-work for credit risk and life insurance. In particular, we assume a nontrivialdependence structure between the reference filtration and the insurance inter-nal filtration. We apply these results for pricing non-life insurance liabilitiesin hybrid financial and insurance markets, while taking into account the roleof inflation under the benchmark approach. This framework offers at thesame time a general and flexible structure, and explicit and treatable pricingformula.
JEL Classification:
C02, G10, G19
Key words: non-life insurance, reduced-form framework, marked point pro-cess, benchmark approach, filtration dependence.
In this paper we propose a general framework for modeling an insurance liabilitycash flow in continuous time, by extending the classic reduced-form setting forcredit risk and life insurance. In particular, we consider a nontrivial dependencestructure between the reference filtration F and insurance internal filtration H .The global information flow available to the insurance company is represented by G = F ∨ H . In this way, we obtain for the first time a framework in continuoustime for non-life insurance, where filtration dependence is taken into account. Inview of the development of insurance-linked derivatives, which offer the possibility ∗ Main affiliation: Department of Mathematics, LMU Munich, Theresienstraße, 39, 80333Munich, Germany, Email: [email protected] † Secondary affiliation: Department of Mathematics, University of Oslo, Box 1053, Blindern,0316, Oslo, Norway. ‡ Department of Mathematics, LMU Munich, Theresienstraße, 39, 80333 Munich, Germany,Email: [email protected]
1f transferring insurance risks to the financial market, this bottom-up modelingapproach can be used for pricing and hedging both life and non-life insurance lia-bilities in hybrid financial and insurance markets. As an application of the generalframework structure, we derive pricing formulas for non-life insurance claims bytaking into account the role of inflation under the benchmark approach, which wasintroduced in [34].Historically, the mathematical modeling of life and non-life insurance liabilitiesin continuous time is quite asymmetric and risk mitigation of non-life portfoliosvia asset allocation is scarcely practiced. While there are a lot of recent worksconcerning life insurance, see e.g [28], [13], [14], [2], [8], [9], [7], [10], non-lifeinsurance is often studied in discrete time and/or state space, see e.g. [23], [24],[18]. We refer to e.g. [36] for a unified framework for life and non-life insurancein discrete time. Mathematical frameworks for non-life insurance in continuoustime can be found in e.g. [27], [16], [3], [32], [31] and [35]. However, these settingsdo not consider a nontrivial dependence structure between reference filtration andinsurance internal filtration. In particular, in e.g. [32] and [31], the insuranceinternal filtration is not distinguished from the reference filtration, and in e.g.[27], [16] and [3], reference and insurance internal filtrations are assumed to beindependent. The importance of considering a nontrivial dependence structurebetween filtrations, which represent different information flows in a hybrid market,is discussed in [4] in view of the recent introduction of insurance-linked derivatives.Derivatives based on occurrence intensity index, such as mortality derivatives,weather derivatives etc., play an important role in mitigating risks of insurancecompanies in the case of life and non-catastrophe non-life business. This last one,which includes car insurance, theft insurance, home insurance, etc., as opposedto catastrophe non-life insurance , covers high-probability low-cost events, and isoften neglected by the literature. This paper aims to fill this gap, as well as toprovide analytical results which can be used for the non-life insurance reservingproblem and the valuation of non-life non-catastrophe linked financial products,which are currently still not common but can be potentially attractive in the future.Recent non-life insurance literature in continuous time, see e.g. [3], [32], [31]and [35], commonly assumes the insurance internal information flow as given bythe natural filtration of a marked point process, which describes the insuranceclaim movement. Pricing and hedging formulas are then obtained by using thecompensator of this marked point process. However, as we discuss in Section4, this approach can not be always followed in the case of multiple filtrationswith nontrivial dependence. Indeed, with respect to a generic filtration, it is notalways true that there exists a marked point process with a given compensator,and the compensator does not always determine uniquely the law of the process.To overcome these difficulties, we propose a new framewok, which uses a directapproach as in Section 5.1 and 9.1.2 of [11] and allows an explicit bottom-upconstruction to treat more general filtrations. We note that, when our general See e.g. [12] for the distinction between catastrophe and non-catastrophe insurance. n claims. We assume that the reference filtration F includes in-formation related to the financial market and environmental, social and economicindicators. Following the classic non-life insurance modeling approach as in e.g.[1] and [26], we assume the insurance internal filtration H , which represents in-ternal information of an insurance company given by the claim movements, to begenerated by a family of marked point processes, describing sequences of reportingtimes and associated losses. As typically in the case of non-life insurance, accidenttimes and their related damages are unknown until the moment of reporting. Weare able to capture these features and at the same time to introduce a dependencestructure between filtrations F and H by providing a nontrivial extension of theclassical reduced-form framework. In particular, we model the accident times of therelated insurance securities as F -conditionally independent random variables witha common F -adapted intensity process µ . Random delay between accident timeand the first reporting is modelled in the first mark and subsequent developmentof the claim is modelled by a time shift of an independent marked point processwith respect to the first reporting. This structure includes the life insurance caseand allows to obtain analytical valuation formulas, which can be expressed in termof the accident intensity µ , the delay distribution and the updating distribution, asillustrated in the preliminary calculations in Section 3. We then apply these resultsfor pricing insurance liabilities in a hybrid market under the benchmark approachof [34]. The hybrid nature of the combined market is given by the presence ofderivatives related to the intensity process µ on the financial market and by theinfluence of inflation and benchmark portfolio in the valuation of insurance liabili-ties. Here we focus only on pricing non-life insurance claims and obtain analyticalpricing formulas, which can also be useful for future design of insurance-linkedderivatives, especially non-catastrophe non-life derivatives.This paper is organised as follows. In Section 2 we construct a general frame-work for insurance liability cash flow in continuous time under a nontrivial depen-dence between the reference and the insurance internal filtrations, applicable bothto life and non-life insurance, and give a brief comparison with the existing insur-ance frameworks in the literature. In Section 3 we give some useful preliminaryvaluation results in this setting. In Section 4 we discuss the compensator approach.In Section 5 we describe the hybrid nature of the combined market and derive thereal world pricing formula for non-life insurance reserving under the benchmarkapproach. 3 General framework
In this section we construct a general framework for modeling an insurance liabilitycash flow. We consider a filtered probability space (Ω , G , G , P ) , where G := ( G t ) t > , G = G ∞ , and G is trivial.W assume that G = F ∨ H , where F := ( F t ) t > and H := ( H t ) t > are filtrationsrepresenting respectively a reference information flow and the internal informa-tion flow only available to the insurance company. Hence G describes the globalinformation flow available to the insurance company. The reference filtration F typically includes information related to the financial market, and to environmen-tal, political and social indicators. While we do not specify the structure of thereference filtration F , we assume that the insurance internal filtration H is gener-ated by a family of marked point processes, representing the times and amountsof losses of the insurance portfolio, as in e.g. [1], [22], [29] and [30]. Filtrations F and H are not supposed be independent. Without loss of generality, we assumethat all filtrations satisfy the conditions of completeness and right-continuity. Ifnot otherwise specified, all relations in this paper hold in the P -a.s. sense. For adetailed background of marked point processes we refer to e.g. [25], [15] and [20].In the following we use the classic terminology of non-life insurance, see e.g. [37]and [33], and specify the filtration H as follows.We consider an insurance portfolio with n policies. For i -th policy with i =1 , ..., n , the insurance company is typically informed about the accident occurredat a random time τ i only after a random delay θ i , which can be very long especiallyin the case of non-life insurance. Once the accident is reported at time τ i , where τ i := τ i + θ i , (2.1)both the accident time τ i , the reporting delay θ i and the impact size of the accident,described by a nonnegative random variable X i , become available information. Inparticular, we assume that for all i = 1 , ..., n , τ i > P -a.s.Let N be the set of natural numbers without zero. We describe the i -th insur-ance policy movement by a marked point process ( τ ij , Θ ij ) j ∈ N with 2-dimensionalnonnegative marks. That is, the sequence ( τ ij ) j ∈ N is a point process, where τ ij : (Ω , G , P ) → ( R + , B ( R + )) , j ∈ N , and (Θ ij ) j ∈ N is a sequence of 2-dimensional nonnegative random variables, with Θ ij : (Ω , G , P ) → ( R , B ( R )) , j ∈ N . For every j ∈ N , the random time τ ij describes the reporting time of j -th eventrelated to i -th policy. The mark components Θ ij describe the reporting delay andthe impact size of the corresponding event, respectively, which are known only ifthe event is reported. More precisely, we set τ i with mark Θ i = ( θ i , X i ) , (2.2)4nd τ ij +1 = τ i + ˜ τ ij with mark Θ ij +1 = (0 , X ij +1 ) := (0 , ˜ X ij ) , (2.3)for j > , where (˜ τ ij , ˜ X ij ) j ∈ N is an auxiliary marked point process, which describesupdating and development after the first reporting at τ i . For the sake of simplicity,we here assume that only the first reporting delay is different from zero, since in thispaper we focus on modeling the first accident times τ i and their relation with thereference filtration. However our setting can be easily generalized by consideringnon zero random delays in (2.3). We set furthermore that the marked point process (˜ τ ij , ˜ X ij ) j ∈ N is simple, i.e. lim j →∞ ˜ τ ij = ∞ , and ˜ τ ij < ˜ τ ij +1 , if ˜ τ ij < ∞ , and satisfies the following integrability condition E ∞ X j =1 { ˜ τ ij t } ˜ X ij < ∞ for all t > , (2.4)for i = 1 , ..., n . In particular, the random times ( τ ij ) j ∈ N are strictly ordered: τ < τ < · · · < τ j < τ j +1 < · · · ,τ < τ < · · · < τ j < τ j +1 < · · · , ... τ n < τ n < · · · < τ nj < τ nj +1 < · · · . (2.5)Note that we may have ∞ = τ ij = τ ij +1 = ... , in such a case infinite value standsfor an event which never happens. For the sake of simplicity we assume also thefollowing. Assumption 2.1. Homogeneous delay : the random delays θ i , i = 1 , ..., n , have the same distri-bution.2. Homogeneous development : the marked point processes (˜ τ ij , ˜ X ij ) j ∈ N , i =1 , ..., n , have the same distribution.3. Independent first mark : the first marks X i , i = 1 , ..., n , are mutually inde-pendent and independent from F ∞ ∨ σ ( τ ) ∨ ... ∨ σ ( τ n ) .4. Independent delay : the random delays θ i , i = 1 , ..., n , are mutually indepen-dent and independent from F ∞ ∨ σ (( τ , X )) ∨ ... ∨ σ (( τ n , X n )) .5. Independent development : the marked point processes (˜ τ ij , ˜ X ij ) j ∈ N + , i =1 , .., n are mutually independent and independent from F ∞ ∨ σ (( τ , θ , X )) ∨ ... ∨ σ (( τ n , θ n , X n )) .
5e emphasize that the above assumptions are general enough. The homogeneityassumptions can be satisfied by subdividing opportunely the insurance portfolio.The independence assumptions reflect the fact that reporting delays θ i , occurrencesand size of the losses after the first reporting time, described by (˜ τ ij , ˜ X ij ) j ∈ N + , aretypically idiosyncratic factors which are independent to each other and independentfrom the reference information. However, we introduce a dependence structure bymodeling F -progressively measurable occurrence intensities of the accidents, as wewill present in (2.13) and (2.14). This will reflect the assumption that the occur-rence intensity of accidents can be deduced from the reference information flowrepresented by F , while further updates of accident events (˜ τ ij , ˜ X ij ) j ∈ N + are typi-cally insurance portfolio specific and are not available as third-party or referenceinformation. We assume furthermore that the distribution of delay variables θ i , i = 1 , ..., n has the following structure. Assumption 2.2.
The common cumulative distribution function G : [0 , + ∞ ) → [0 , of θ i , i = 1 , ..., n , assigns probability α at 0 and has a density function g for x > , i.e, G ( x ) = α + Z x g ( y )d y, x ∈ R + . (2.6)According to the above assumption, the delays are nonnegative and may have amixed distribution. In this way, we cover both the case of life insurance with θ i ≡ , i.e. g = 0 , and the case of non-life insurance with non-null delays.For every i = 1 , ..., n , we define the marked cumulative process N i by N i ( t, B )( ω ) := ∞ X j =1 { τ ij ( ω ) t } { Θ ij ( ω ) ∈ B } , for every t > , B ∈ B ( R ) , ω ∈ Ω . The process ( N it ) t > defined by N it := N i ( t, R ) = ∞ X j =1 { τ ij t } , t > , is called ground process associated to the marked point process. At any time t > ,the random variable N it counts the number of occurrence of τ ij up to time t . Inthe literature, the name marked point process refers sometimes to the process N i .Indeed, there is a unique correspondence between the marked point process andits marked cumulative process. More precisely, { τ ij t } = { N it > j } , (2.7)for all t > and { Θ ij ∈ B } = ∞ [ K ′ =1 ∞ \ K = K ′ ∞ [ k =1 { N i ( k − / K = n − , N i ( k/ K , B ) − N i (( k − / K , B ) = 1 } , (2.8)6or all B ∈ B ( R ) . See equations (2.8), (2.9) of [20] and Lemma 2.2.2 of [25].We consider the filtrations H i, := ( H i, t ) t > with H i, t := σ (cid:16) { τ i s } { ( θ i ,X i ) ∈ B } , s t, for all B ∈ B ( R ) (cid:17) , for all t > , and H i,j := ( H i,jt ) t > , j > , with H i,jt := σ (cid:16) { τ ij s } { X ij ∈ B } , s t, for all B ∈ B ( R + ) (cid:17) , for all t > . It holds that H i,j ∞ = σ ( τ ij ) ∨ σ ( X ij ) for j > . In particular, in view of (2.1) we have H i, ∞ = σ ( τ i ) ∨ σ (( θ i , X i )) = σ ( τ i ) ∨ σ (( θ i , X i )) . (2.9)Let H i := ( H it ) t > be the natural filtration of the marked cumulative process N i ,that is for all t > , H it = σ ( N i ( s, B ) , s t, for all B ∈ B ( R )) . The internal information flow of the insurance company is described by the filtra-tion H := ( H t ) t > , where H t := H t ∨ ... ∨ H nt , t > . (2.10)Similarly, for i = 1 , .., n , we call ˜ N i the corresponding marked cumulativeprocesses associated to the marked point processes (˜ τ ij , ˜ X ij ) j ∈ N + and ˜ H i the corre-sponding filtration, respectively. Similarly, all other notations associated to theselast processes will be denoted by the symbol " ∼ ". Lemma 2.3.
For every i = 1 , ..., n , we have H i = W j ∈ N + H i,j .Proof. Clearly, we have H it ⊆ _ j ∈ N + H i,jt . For the other inclusion, it is sufficient to show that for all s t and B ∈ B ( R ) , { τ ij s } ∩ { Θ ij ∈ B } ∈ H it . This follows directly from (2.7) and (2.8).We now introduce the following notation, which is useful in the sequel. For i =1 , ..., n , j ∈ N + , we define H i, j := _ k j H i,k , H i, > j := _ k > j H i,k , similarly for H i,>j and H i,
If we define H i, t := σ (cid:16) { τ i s } : 0 s t (cid:17) , i = 1 , ..., n , then con-dition (2.11) is equivalent to E [ X |F t ] = E [ X |F s ] , for each integrable H i, s -measurable random variable X . Condition (2.12) is equiv-alent to the F t -conditional independence between the σ -algebras H l, t and H k, t . Furthermore, if F i := ( F it ) t > is the F -conditional cumulative function of τ i , F it := P (cid:0) τ i t (cid:12)(cid:12) F t (cid:1) , t > , we assume that there exists a locally integrable and F -progressively measurableprocess µ i := ( µ it ) t > , such that e − R t µ is d s = 1 − F it for all t > . (2.13)We define Γ i := (Γ it ) t > as Γ it := Z t µ is d s, t > . (2.14)The process µ i is called intensity process of the random jump time τ i and theprocess Γ i is called hazard process of τ i . An explicit construction in Example9.1.5 of [11] shows that for a given family of locally integrable F -progressivelymeasurable process µ i , i = 1 , ..., n , it is always possible to construct random times τ i , i = 1 , ..., n , such that Γ i is the hazard process of τ i for every i = 0 , ..., n , andall the assumptions above are satisfied. For the sake of simplicity, we assume thatthe insurance portfolio is homogeneous. 8 ssumption 2.6. The accident times τ i , i = 1 , ..., n , have the same intensityprocess. Under the homogeneity condition, we denote the common F -conditional cumula-tive function, hazard process and intensity process respectively by F , Γ and µ . Theabove assumption reflects the fact that, while the policy developments may nothave direct link to the information flow F , the accident occurrences τ i , i = 1 , ..., n ,are influenced by some common systematic risk-factors, and the common condi-tional intensity µ is deducible from the reference information flow.We now show how the general framework described above comprehends in asynthetic way both life and non-life insurance modeling, and compare our settingwith the existing literature. Life insurance policies typically do not have reporting delay and depend only on τ i , i = 1 , ..., n , which actually represent the decease times. This can be includedin our framework by setting θ i ≡ , τ ij ≡ ∞ for all j > and X ij ≡ for all j ∈ N + , and interpreting τ i as the decease time of person i , where i = 1 , ..., n . Thefiltration G is hence reduced to G = F ∨ H ∨ ... ∨ H n , where H it = σ (cid:16) { τ i s } , s t (cid:17) , t > , i = 1 , ..., n. In particular, the F -progressively measurable process µ is interpreted as mortalityintensity in this context. The financial market is typically assumed to includemortality or longevity linked derivatives, such as longevity bond, which pays offthe longevity index value e − R T µ s d s at maturity T .Life insurance within hybrid market under this setting has been intensivelystudied in the literature, see e.g. [2], [9], [7] and [10]. The framework in Section 2 in its full generality describes the case of non-life insur-ance. Indeed, non-life insurance policies typically have reporting delay, i.e. θ i = 0 ,which can also count to several years. For i -th policy, we interpret X ij as paymentamount at the j -th random times τ ij ; the exact accident time τ i and first paymentamount X i is known only after reporting at time τ i . Further developments mayoccur after the first reporting and before the settlement of claim. The total num-ber of developments ( τ ij ) j ∈ N + is unknown as well as the amount of correspondingpayments ( X ij ) j ∈ N + . The accident time τ i admits an F -progressively measurableintensity process µ related to the underlying risk. If liquidly traded derivativesrelated to the µ process are available on the financial market, they could be usedfor hedging systematic risks related to non-life portfolio.9he above described setting gives a nontrivial extention of the underlyingframeworks in e.g. [16], [3], [32] and [31]. In e.g. [16] and [3], the referencefiltration F is assumed to be independent from the insurance internal filtration H generated by the non-life portfolio movement. The interaction between thefinancial and the insurance markets is thus captured only by means of interestrate and/or inflation risk. On the contrary, in e.g. [32] and [31], it is assumedthat G = H = F . Financial products used for hedging purpose are in these casesliquidly traded catastrophe derivatives and/or reinsurance contracts, which sharesimilar risk structure of the target non-life insurance portfolio. Considering a moregeneral setting, where F and H are not necessarily independent or equal, is techni-cally challenging, as we discuss in Section 4. However, the extended reduce-formframework proposed in this paper allows to consider a nontrivial dependence struc-ture between filtrations F and H and still to derive analytical pricing formulas fornon-life insurance liabilities. Furthermore, beside the financial instruments usedin e.g. [16], [3], [32] and [31], it is possible to use intensity µ related derivativesas hedging instrument, see discussion in Section 5. This last type of derivatives isstill not common but is potentially attractive for covering systematic risks arisingfrom non-catastrophe non-life insurance. In this Section, we state several results under the above structure assumptions, byfollowing Section 5.1 of [11] for the presentation. These preliminary calculations arefundamental for providing pricing formulas of non-life insurance claims in Section5.1.We start with extension of relation (2.11) and the F -independence (2.12) of τ i , i = 1 , ..., n . In particular, if these relations hold for the filtrations H i, , i = 1 , ..., n ,then they also hold for the filtrations H i , i = 1 , ..., n . Lemma 3.1.
For any t ∈ [0 , ∞ ] and l, k = 1 , ..., n with l = k , the σ -algebras H lt and H kt are F t -independent. Furthermore, for any s t ∞ and i = 1 , ..., n ,if X is H is -measurable, then E [ X | F t ] = E [ X | F s ] .Proof. The proof is straightforward in view of Lemma 2.3, Remark 2.5 and theindependence conditions in Assumption 2.1. For details, see proof of Lemma 3.2.1and Lemma 3.2.2 in [38].As a consequence of Lemma 3.1, the G -conditional expectation can be reduced to F ∨ H i -conditional expectation in most cases. Corollary 3.2. If t T < ∞ , and Y is an integrable ( F T ∨ H iT ) -measurablerandom variable, then E [ Y | G t ] = E [ Y | F t ∨ H it (cid:3) . roof. It is sufficient to prove the statement for the indicator functions of the form Y = A B where A ∈ F T and B ∈ H iT , by using Lemma 3.1. Further details areshown in the proof of Corollary 3.2.3 in [38].An other important consequence of Lemma 3.1 is the so called H -hypothesis be-tween filtrations F and G , i.e. the property that every F -martingale is also a G -martingale. Corollary 3.3.
The H -hypothesis holds between filtrations F and G .Proof. It follows by Lemma 6.1.1 of [11] and Lemma 3.1. See also Corollary 3.2.4in [38].Now we would like to derive some more explicit representations. We note thatfor every integrable random variable Y , t > , i = 1 , ..., n and j ∈ N + , we have thedecomposition E [ Y | H it ∨ F t (cid:3) = E h { τ ij >t } Y (cid:12)(cid:12)(cid:12) H it ∨ F t i + E h { τ ij t } Y (cid:12)(cid:12)(cid:12) H it ∨ F t i . (3.1)In the following we will evaluate separately the two components on the right-handside of (3.1). The following lemma is important for deriving a representation ofthe first component. Lemma 3.4.
For any t > , i = 1 , ..., n and j ∈ N + , we have H it ∨ F t ⊆ G i,jt , where G i,jt := n A ∈ G : ∃ C ∈ H i,
By Corollary 2.4, it holds that H it = H i,
Equality (3.3) is equivalent to E h { τ ij >t } Y P (cid:0) τ i > t (cid:12)(cid:12) H i,
We note that the left-hand side is ( H i, j ∞ ∨A ) -measurable. Since the markedpoint process ( τ ij , Θ ij ) j ∈ N is simple, i.e. the strict monotonicy (2.5) holds, if A ∈H i, j ∞ ∨ A , then A ∩ { τ ij t } ∈ H i, jt ∨ A , and Z A { τ ij t } Y d P = Z A ∩{ τ ij t } Y d P = Z A ∩{ τ ij t } E [ Y | H i, jt ∨ A i d P = Z A E h { τ ij t } Y (cid:12)(cid:12)(cid:12) H i, jt ∨ A i d P. This concludes the proof.
Remark 3.7.
Since we have H i, j ∞ = σ (cid:0) τ ih , h = 1 , ..., j (cid:1) , Lemma 3.6 shows that, if τ ij has occurred before time t , then partial informationabout τ ij up to t is equivalent to full information about all the random times τ ih , h = 1 , ..., j . In particular, if Y is a function of τ i , ..., τ ij , i.e. Y = f ( τ i , ..., τ ij ) ,then the conditional expectation is simply E h { τ ij t } Y (cid:12)(cid:12)(cid:12) H i, jt ∨ A i = { τ ij t } Y. We summarize the above results in the following representation theorem.
Theorem 3.8.
For any t > , i = 1 , ..., n , j ∈ N + and any integrable G -measurablerandom variable Y , we have E [ Y | H it ∨ F t (cid:3) = { τ ij t } E [ Y | H i, j ∞ ∨ H i,>jt ∨ F t i + { τ ij >t } E [ Y | H i, jt ∨ F t i . If furthermore Y is ( H iT ∨ F T ) -measurable, then E [ Y | G t ] = { τ ij t } E [ Y | H i, j ∞ ∨ H i,>jt ∨ F t i + { τ ij >t } E [ Y | H i, jt ∨ F t i . Proof.
Since E [ Y | H it ∨ F t (cid:3) = E h { τ ij t } Y (cid:12)(cid:12)(cid:12) H it ∨ F t i + E h { τ ij >t } Y (cid:12)(cid:12)(cid:12) H it ∨ F t i , the first part is a straightforward consequence of Proposition 3.5 and Lemma 3.6.For the second part, it suffices to apply Corollary 3.2.13e now show some results which we will use to solve the reserve estimationproblem in Section 5.1. For this purpose, our approach allows to for obtain analyt-ical formulas in a general setting in continuous time, where filtration dependence istaken into account. As we illustrate in Section 4, this is not possible in such a gen-erality by using more classical approaches. Let t T < ∞ and Z := ( Z t ) t ∈ [0 ,T ] be a continuous, bounded and F -adapted process. For i = 1 , ..., n , we now consider Y = N iT X j = N it X ij Z τ ij = ∞ X j =1 { t<τ ij T } X ij Z τ ij , (3.5)and compute E [ Y | G t ] = E N iT X j = N it X ij Z τ ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t . (3.6)In particular, similarly to before, we study separately the two components of thedecomposition of (3.6) with respect to the first reporting time τ i , i.e. E N iT X j = N it X ij Z τ ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t = E { τ it >t } N iT X j = N it X ij Z τ ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t + E { τ it t } N iT X j = N it X ij Z τ ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t , (3.7)and derive more explicit formulas in terms of the intensity process µ , the distribu-tion of delay θ i , and the distribution of development N i after the first reporting.We start with the F -conditional expectation of τ i . Lemma 3.9.
For any i = 1 , ..., n and t > , we have P (cid:0) τ i t (cid:12)(cid:12) F t (cid:1) = Z t G ( t − s ) e − R s µ v d v µ s d s, (3.8) where G is the cumulative distribution function of θ i given in (2.6).Proof. Note that by Assumption 2.1, θ i is independent from F t ∨ σ ( τ i ) . Further-more, both θ i and τ i are P -a.s. nonnegative. Therefore, we have P (cid:0) τ i t (cid:12)(cid:12) F t (cid:1) = E h { τ i + θ i t } (cid:12)(cid:12)(cid:12) F t i = E h E h { τ i t } { τ i + θ i t } (cid:12)(cid:12)(cid:12) F t ∨ σ ( τ i ) i(cid:12)(cid:12)(cid:12) F t i = E h { τ i t } E (cid:2) { θ i t − x } (cid:3)(cid:12)(cid:12) x = τ i (cid:12)(cid:12)(cid:12) F t i = E h { τ i t } G ( t − τ i ) (cid:12)(cid:12)(cid:12) F t i . To conclude we only need to show E h { τ i t } G ( t − τ i ) (cid:12)(cid:12)(cid:12) F t i = Z t G ( t − s ) e − R s µ v d v µ s d s. (3.9)14his can be done in the same way as for Proposition 5.1.1 of [11], in view of relation(2.11) and the fact that G is continuous according to Assumption 2.2. Remark 3.10.
Note that from (3.9) we can derive the conditional probability thatthe accident has incurred, but not yet reported (IBNR events in the terminologyused in the insurance sector).
In expression (3.8) of Lemma 3.9, the parameter t appears also in the integrand.The following corollary improves relation (3.8) and shows that the process of con-ditional expectation ( P (cid:0) τ i t (cid:12)(cid:12) F t (cid:1) ) t > is absolutely continuous with respect tothe Lebesgue measure. Corollary 3.11.
For any i = 1 , ..., n , we have P (cid:0) τ i t (cid:12)(cid:12) F t (cid:1) = Z t (cid:18) α e − R s µ v d v µ s + Z s g ( s − u ) e − R u µ v d v µ u d u (cid:19) d s, (3.10) where α and g are defined in (2.6).Proof. This follows immediately from Assumption 2.2 and relation (3.8).
Lemma 3.12.
If the process Z := ( Z u ) u ∈ [ t,T ] is left-continuous and bounded and Z t is F T -measurable for all t > , then we have E h { t<τ i T } Z τ i (cid:12)(cid:12)(cid:12) F t i = E (cid:20)Z Tt Z u d P (cid:0) τ i u (cid:12)(cid:12) F u (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , for i = 1 , ..., n and t ∈ [0 , T ] .Proof. The proof is similar to the one in Proposition 5.1.1 of [11] by assuming first Z stepwise constant and by using Lemma 3.1. The convergence of the conditionalexpectation follows by the boundedness of Z and by the convergence of Riemannsum under the sign of conditional expectation to Lebesgue–Stieltjes integral, whichcoincides with Lebesque integral in view of Corollary 3.11. Note that we do notnecessarily assume the F -adaptedness of Z . For details, we refer to the proof ofLemma 3.2.13 in [38].Now we are able to calculate the first component on the right-hand side of(3.7). We define ˜ m ( t ) := E ˜ N t X j =1 ˜ X ij , if t > , ˜ m ( t ) := 0 , if t < , (3.11) We do not assume that Z is F -adapted. We emphasize that the boundedness condition can be generalized. ˜ N denotes the ground process of (˜ τ ij , ˜ X ij ) j ∈ N + , i.e. ˜ N t := ∞ X j =1 { ˜ τ ij t } , t > . (3.12)Note that ˜ m does not depend on i because of Assumption 2.1 (2). Proposition 3.13.
Let Z := ( Z t ) t ∈ [0 ,T ] be a continuous, bounded and F -adapted process and Y be as in (3.5), then for any t ∈ [0 , T ] , E h { τ i >t } Y (cid:12)(cid:12)(cid:12) H it ∨ F t i = { τ i >t } E hR Tt (cid:16) E [ X i ] Z u + R Tu Z v d ˜ m ( v − u ) (cid:17) d P (cid:0) τ i u (cid:12)(cid:12) F u (cid:1)(cid:12)(cid:12)(cid:12) F t i P (cid:0) τ i > t (cid:12)(cid:12) F t (cid:1) where ˜ m is defined in (3.11).Proof. By applying (3.4) in Proposition 3.5 to Y defined in (3.5), we get E h { τ i >t } Y (cid:12)(cid:12)(cid:12) H it ∨ F t i = { τ i >t } E [ Y | H i, t ∨ F t i = { τ i >t } E ∞ X j =1 { τ ij T } X ij Z τ ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H i, t ∨ F t = { τ i >t } E h { τ i T } X i Z τ i (cid:12)(cid:12)(cid:12) H i, t ∨ F t i + { τ i >t } E ∞ X j =2 { τ ij T } X ij Z τ ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H i, t ∨ F t . (3.13)For the first component of (3.13), it is sufficient to use (3.3) in Proposition 3.5 andan argument similar to Proposition 5.1.1 of [11], taking into account the indepen-dence condition in Assumption 2.1 (3) and Lemma 3.1. We have hence { τ i >t } E h { τ i T } X i Z τ i (cid:12)(cid:12)(cid:12) H i, t ∨ F t i = E h { t<τ i T } X i Z τ i (cid:12)(cid:12)(cid:12) H i, t ∨ F t i = { τ i >t } E h { t<τ i T } X i Z τ i (cid:12)(cid:12)(cid:12) F t i P (cid:0) τ i > t (cid:12)(cid:12) F t (cid:1) = { τ i >t } E hR Tt E [ X i ] Z u d P (cid:0) τ i u (cid:12)(cid:12) F u (cid:1)(cid:12)(cid:12)(cid:12) F t i P (cid:0) τ i > t (cid:12)(cid:12) F t (cid:1) . Note that the result also holds under different integrability and measurability conditions. [ t, T ] , Z is a bounded, stepwise, F -predictable process, i.e. Z u = n X k =0 Z t k { t k t } E ∞ X j =2 { τ ij T } X ij Z τ ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H i, t ∨ F t = { τ i >t } E ∞ X j =1 { t<τ i +˜ τ ij T } ˜ X ij Z τ ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H i, t ∨ F t = { τ i >t } E n X k =0 ∞ X j =1 { t k <τ i +˜ τ ij t k +1 } ˜ X ij Z t k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H i, t ∨ F t = { τ i >t } E n X k =0 Z t k E ∞ X j =1 { t k <τ i +˜ τ ij t k +1 } ˜ X ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H i, t ∨ F t k ∨ σ ( τ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H i, t ∨ F t = { τ i >t } E n X k =0 Z t k E ∞ X j =1 { t k
18f the cumulative distribution functions, equivalent to the convergence in distribu-tion or weak convergence in measure . This yields the convergence ˜ Z s n −→ ˜ Z s , P − a.s. , that is, ˜ Z s := R Tt Z u d ˜ m ( u − s ) , s ∈ [0 , T ] , is left-continuous. Futhermore, it is alsobounded. Now we apply Lemma 3.12 and obtain { τ i >t } E h { t<τ i T } R Tt Z u d ˜ m ( u − τ i ) (cid:12)(cid:12)(cid:12) F t i P (cid:0) τ i > t (cid:12)(cid:12) F t (cid:1) = { τ i >t } E h { t<τ i T } ˜ Z τ i (cid:12)(cid:12)(cid:12) F t i P (cid:0) τ i > t (cid:12)(cid:12) F t (cid:1) = { τ i >t } E hR Tt ˜ Z u d P (cid:0) τ i u (cid:12)(cid:12) F u (cid:1)(cid:12)(cid:12)(cid:12) F t i P (cid:0) τ i > t (cid:12)(cid:12) F t (cid:1) = { τ i >t } E hR Tt (cid:16)R Tt Z v d ˜ m ( v − u ) (cid:17) d P (cid:0) τ i u (cid:12)(cid:12) F u (cid:1)(cid:12)(cid:12)(cid:12) F t i P (cid:0) τ i > t (cid:12)(cid:12) F t (cid:1) . As the last step, we note that for u < s , R Tt Z u d ˜ m ( u − s ) = R Ts Z u d ˜ m ( u − s ) since ˜ m ( u − s ) = 0 . This concludes the proof. Remark 3.14.
The proof of Proposition 3.13 relies on assumption (3.11). Anothersufficient condition would be the continuity of ˜ m , such as in the case of a compoundPoisson process or a Cox process with continuous intensity process and integrablemarks. Indeed, since ˜ m ( u ) = 0 for u < , { τ i >t } (cid:12)(cid:12)(cid:12)(cid:12)Z Tt Z nu d ˜ m ( u − τ i ) (cid:12)(cid:12)(cid:12)(cid:12) { τ i >t } M (cid:12)(cid:12)(cid:12)(cid:12)Z Tt d ˜ m ( u − τ i ) (cid:12)(cid:12)(cid:12)(cid:12) = { t<τ i T } M | ˜ m ( T − τ i ) − ˜ m ( t − τ i ) | , and the right-hand side is uniformly bounded if ˜ m is continuous. The following proposition gives a representation of the second component onthe right-hand side of (3.7).
Proposition 3.15.
Under the same assumptions of Proposition 3.13, if for each i = 1 , ..., n , the process (cid:16)P ˜ N t j =1 ˜ X ij (cid:17) t ∈ [0 ,T ] , where ˜ N is defined in (3.12), is of A series of positive finite measures ( ν n ) n ∈ N converges weakly to a positive finite measure ν ,if for all bounded continuous functions f , it holds Z f d ν n −→ Z f d ν ndependent increments with respect to its natural filtration ˜ H i , then for t ∈ [0 , T ] and Y as in (3.5), it holds E h { τ i t } Y (cid:12)(cid:12)(cid:12) H it ∨ F t i = { τ i t } E (cid:20)Z Tt Z u d ˜ m ( u − x ) (cid:12)(cid:12)(cid:12)(cid:12) H i, ∞ ∨ ˜ H it − x ∨ F t (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) x = τ i , for i = 1 , ..., n .Proof. It follows from Lemma 3.6 that E (cid:2) { τ i t } Y (cid:12)(cid:12) H it ∨ F t (cid:3) = E h { τ i t } Y (cid:12)(cid:12)(cid:12) H i, ∞ ∨ H i,> t ∨ F t i . As in the proof of Proposition 3.13, we assume first Z of the form (3.14). Similarcalculations lead to E h { τ i t } Y (cid:12)(cid:12)(cid:12) H i, ∞ ∨ H i,> t ∨ F t i = { τ i t } E ∞ X j =1 { t<τ i +˜ τ ij T } ˜ X ij Z τ ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H i, ∞ ∨ H i,> t ∨ F t = { τ i t } E n X i =0 ∞ X j =1 { t i
Let Z := ( Z t ) t ∈ [0 ,T ] be a continuous, bounded and F -adaptedprocess , Y be of the form (3.5). If the process (cid:16)P ˜ N t j =1 ˜ X ij (cid:17) t ∈ [0 ,T ] , has independentincrements and ˜ m is defined in (3.11), then E [ Y | G t ] = { τ i t } E (cid:20)Z Tt Z u d ˜ m ( u − x ) (cid:12)(cid:12)(cid:12)(cid:12) H i, ∞ ∨ ˜ H it − x ∨ F t (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) x = τ i + { τ i >t } E hR Tt (cid:16) E [ X i ] Z u + R Tu Z v d ˜ m ( v − u ) (cid:17) d P (cid:0) τ i u (cid:12)(cid:12) F u (cid:1)(cid:12)(cid:12)(cid:12) F t i P (cid:0) τ i > t (cid:12)(cid:12) F t (cid:1) , for i = 1 , ..., n , where P (cid:0) τ i t (cid:12)(cid:12) F t (cid:1) = Z t (cid:18) α e − R u µ v d v µ u + Z u g ( u − v ) e − R v µ s d s µ v d v (cid:19) d u, with α and g defined in (2.6).Proof. It is enough to combine Corollary 3.2, Lemma 3.9, Corollary 3.11, Propo-sition 3.13 and Proposition 3.15.Compared to Theorem 3.8, Theorem 3.16 is more explicit and has the advantagethat the representation is expressed as function of µ , the distribution of θ i and thedistribution of (˜ τ ij , ˜ X ij ) j ∈ N + . This result will be useful for the concrete reservingproblem in hybrid market in Section 5. Note that the result of Theorem 3.16 also holds under different integrability and measurabilityconditions. Comparison with the compensator approach
In this section, we compare our framework with the compensator approach fornon-life insurance in the existing literature. Within this section, the filtration H denotes the natural filtration of a marked point process ( τ n , X n ) n ∈ N , with markedcumulative process N , and G is a generic enlargement of H . We set H := H ∞ and G := G ∞ .In most of the current literature, e.g. [3], [32], [31] and [35], the study of non-lifeinsurance contracts is based on modeling the G -compensator of N , since the G -compensator is involved in the pricing formula and in the calculation of the hedgingstrategy. In the reduced-form framework for life insurance, the direct modelingapproach and the compensator approach coincide, see e.g. [11]. However, thecompensator approach presents several difficulties in a non-life insurance settingwith nontrivial filtrations’ dependence. Definition 4.1.
The G -mark-predictable σ -algebra on the product space R + ×B ( R + ) × Ω is the σ -algebra generated by sets of the form ( s, t ] × B × A where < s < t , B ∈ B ( R + ) and A ∈ G s . Definition 4.2.
The G -compensator of a marked point process ( τ n , X n ) n ∈ N isany G -mark-predictable, cumulative process Λ( t, B, ω ) such that, (Λ( t, B )) t > with Λ( t, B )( · ) := Λ( t, B, · ) is the G -compensator of the point process ( N ( t, B )) t > .We use the notation (Λ t ) t > , Λ t := Λ( t, R + ) , to denote the G -compensator of theground process ( N t ) t > . Theorem 14.2.IV(a) of [15] shows that given a marked point process ( τ n , X n ) n ∈ N with finite first moment measure, its G -compensator Λ always exists and is ( l ⊗ P ) -a.e. unique, where l denotes the Lebesgue measure on R + . In particular, for all ( t, B, ω ) ∈ R + × B ( R + ) × Ω , the following relation holds Λ( t, B, ω ) = Z t κ ( B | s, ω )Λ(d s, ω ) , (4.1)where κ ( B | s, ω ) , B ∈ B ( R + ) , s > , ω ∈ Ω , is the unique predictable kernel suchthat for all A ∈ G s , < s < t, B ∈ B ( R + ) , Z A Z ts N ( u, B )( ω )d uP (d ω ) = Z A Z ts κ ( B | u, ω ) N u ( ω )d uP (d ω ) . However, under general conditons it is not always true that given a G -mark-predictable and cumulative process Λ , there exists a marked point process ( τ n , X n ) n ∈ N with G -compensator Λ . The problem is first mentioned in [21], where the case with G = H is solved. An extention of the existence theorem to the case of G = F ⊗ H ,i.e. when the filtrations F and H are independent, is provided in [17]. Further-more while the law of N is uniquely determined by the H -compensator, this isnot true for the G -compensator. See discussion in [21] and Section 4.8 of [20].22onsequently, the literature with the compensator approach is mostly limited tothe cases of G ≡ H , see e.g. [32], [31], or G = F ⊗ H , see e.g. [3].In the following we provide a sufficient condition in the general case of G = F ∨ H , such that the law of N is uniquely determined by Λ . Similarly to e.g. [32]and [31], we assume that the G -compensator of ( τ n , X n ) n ∈ N has the following form Λ( t, B ) = Z t Z B λ s η s (d x )d s for all t > , B ∈ B ( R + ) , (4.2)where λ := ( λ t ) t > is a G -progressively measurable process and the mapping ηη : R + × B ( R + ) × Ω −→ ( R + , B ( R + ))( t, B, ω ) η t ( B )( ω ) , is such that for every t > , ω ∈ Ω , η ( t, · , ω ) is a probability measure on ( R + , B ( R + )) ,and for every B ∈ B ( R + ) , ( η t ( B )) t > is a G -progressively measurable process.Clearly, we have Λ t = Z t λ s d s for all t > . In particular, we can choose a predictable version of both λ and η , see Section14.3 of [15] for details. The processes λ and η can be interpreted respectively asjump intensity and jump size intensity. We recall that a marked point process ( τ n , X n ) n ∈ N has independent marks if the marks ( X n ) n ∈ N are mutually indepen-dent given N . Proposition 4.3.
The law of a simple marked point process ( τ n , X n ) n ∈ N on (Ω , H ) with finite first moment measure, independent marks and of the form (4.2) isuniquely determined by λ and η . If furthermore λ is H -measurable, then alsothe law of N on (Ω , G ) is uniquely defined.Proof. By Proposition 6.4.IV(a) of [15], the law of marked point process withindependent marks is uniquely determined by the kernel κ and the distribution of N . According to relations (4.1) and (4.2), the kernel κ is given by κ ( B | t, ω ) = η t ( B )( ω ) , ( t, B, ω ) ∈ R + × B ( R + ) × Ω . Corollary 4.8.5 of [20] and Theorem 14.2.IV(c) of [15] show that, if N is simple andof the form (4.2), the process ( E [ λ t |H t ]) t > determines uniquely the distributionof N on (Ω , H ) . If in addition λ is H -adapted, then by Theorem 4.8.1 of [20], alsothe distribution of N on (Ω , G ) is uniquely determined.Nevertheless, Proposition 4.3 requires the jump intensity process λ to be H -adaptedin order to have N uniquely defined in law, which is an unnatural condition in ourcontext.On the contrary, the approach proposed in Section 2 allows to take into accounta dependence structure between the filtrations H and G by directly modeling the F -adapted intensity process µ . Furthermore, this allows to obtain analytical resultsfor valuation formulas as shown in Section 3.23 Pricing of non-life insurance liability cash flow in hy-brid market
In this section, we address the issue of pricing non-life insurance liability cash flowsby applying the results of Section 3. We consider a general structure for a hybridinsurance and financial market. We fix a time horizon T with < T < ∞ , anddenote the inflation index process by I := ( I t ) t ∈ [0 ,T ] , which represents the per-centage increments of the Consumer Price Index (CPI) and follows a nonnegative ( P, F ) -semimartingale. We distinguish real price value, i.e. inflation adjusted, fromnominal price value, which can be converted in real value at any time t ∈ [0 , T ] ,if divided by the inflation index I t . If not otherwise specified, all price values areexpressed in nominal value.We consider d liquidly traded primary assets on the financial market describedby price process vector S := ( S t , ..., S dt ) t ∈ [0 ,T ] , which follows a real-valued ( P, F ) -semimartingale. We assume that there is a publicly accessible index, based on theintensity process µ and modelled by the process L := ( L t ) t ∈ [0 ,T ] with L t := e − Γ t , t ∈ [0 , T ] , see e.g. [13]. This index reflects the underlying systematic risk-factor related tothe 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. traditional financial assets, such as the zero-coupon bond, call and put op-tions, futures etc.;2. inflation linked derivatives, such as inflation linked zero-coupon bond (calledalso zero-coupon Treasury Inflation Protected Security, TIPS), which paysoff I T (equivalent to 1 real unit) at time T , inflation linked call and putoptions, etc.;3. macro risk-factor linked derivatives based on the index L , such as longevitybond which pays off L T at time T , weather index-based derivatives, etc.We denote by L ( S, P, G ) the space of R d -valued G -predictable S -integrable pro-cesses. We call portfolio or value process S δ := ( S δt ) t ∈ [0 ,T ] associated to a tradingstrategy δ := ( δ t ) t ∈ [0 ,T ] in L ( S, P, G ) the following càdlàg optional process S δt − = δ ⊤ t S t = d X i =1 δ it S it , t ∈ [0 , T ] . It is called self-financing if S δt = S δ + Z t δ ⊤ u − d S u = S δ + l X i =1 Z t δ iu − d S iu , t ∈ [0 .T ] . We follow the definitions in [6].
24e introduce the following set V + x = { S δ self-financing : δ ∈ L ( S, P , G ) , S δ = x > , S δ > } . Definition 5.1. A benchmark or numéraire portfolio S ∗ := ( S ∗ t ) t ∈ [0 ,T ] is an ele-ment of V +1 , such that S δs S ∗ s > E (cid:20) S δt S ∗ t (cid:12)(cid:12)(cid:12)(cid:12) G s (cid:21) , s, t ∈ [0 , T ] , t > s. We follow the approach of [34] and work under the following assumption.
Assumption 5.2.
There exists a benchmark portfolio S ∗ . In [19], it is shown that Assumption 5.2 is weaker than assuming the existenceof an equivalent martingale measure. As discussed in [4], this weak no-arbitrageassumption is more suitable for modeling a hybrid market as in our case. Given ageneric random variable or process X , we denote by ˆ X := X/S ∗ the benchmarked value of X . The following lemma is proved in [5]. Lemma 5.3.
If the vector process of primary assets S is continuous, then thebenchmarked vector process ˆ S := S/S ∗ is a ( P, G ) -local martingale. For the sake of simplicity, we assume the following conditions similar to the onesin [10].
Assumption 5.4.
The inflation index process I = ( I t ) t ∈ [0 ,T ] and the vector processof primary assets S are continuous. The benchmark portfolio S ∗ = ( S ∗ t ) t ∈ [0 ,T ] iscontinuous, F -adapted, and the benchmarked value process ˆ S := S/S ∗ is an ( F , P ) -local martingale. Inflation linked zero-coupon bond (or TIPS) is a primary asset,i.e. an element of the vector S . The payment stream in real unit of the insurance company towards policy-holders is modelled by a nonnegative ( P, G ) -semimartingale D := ( D t ) t ∈ [0 ,T ] . Wedenote by A := ( A t ) t ∈ [0 ,T ] the nominal benchmarked cumulative payment, namely A t := Z t I u S ∗ u d D u , t ∈ [0 , T ] . (5.1) Definition 5.5.
We call real world pricing formula associated to A the followingformula V t := S ∗ t I t E [ A T − A t | G t ] = S ∗ t I t E "Z ] t,T ] I u S ∗ u d D u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t , (5.2) for t ∈ [0 , T ] . V t in (5.2) is expressed in real value, i.e. inflation adjusted value. Accordingto the benchmark approach of [34], a portfolio’s process is fair, if its benchmarkedvalue process is a P -martingale. The real-world pricing formula (5.2) then providesthe fair portfolio of minimal price among all replicating self-financing portfolios fora given benchmarked claim ˆ H , if ˆ H is hedgeable. In the case of incomplete marketmodels, it corresponds to the benchmarked risk-minimizing price for the paymentprocess A at time t , if A is square integrable, i.e. sup t ∈ [0 ,T ] E (cid:2) A t (cid:3) < ∞ . The relationship between benchmark approach and risk minimization method hasbeen analysed in [34] and [5] for a single payoff. We refer to Appendix A of [10]for the case of dividend payments.
In the setting outlined above, we now apply the results of Section 3 to compute thereal-world pricing formula for non-life insurance claims, under the interpretationof Section 2.2. The cumulative payment at time t related to i -th policy expressedin real value is given by ∞ X j =1 { τ ij t } X ij = N it X j =1 X ij . The nominal benchmarked cumulative payment process A := ( A t ) t ∈ [0 ,T ] is hence A t := Z t I s S ∗ s d D s = n X i =1 N it X j =1 I τ ij S ∗ τ ij X ij , t ∈ [0 , T ] . (5.3)The estimation of A is called reserving problem in the context of non-life insurance,see [1]. Unlike the life insurance case, the risk related to non-life insurance policiesis hence not only related to the accident itself, but also to the first reporting delay(this is the case of incurred but not reported claims, called IBNR claims), to thetime and the size of developments after the first reporting. We now focus on pricingand hedging the nominal remaining payment A T − A t , for t ∈ [0 , T ] . We assumethat the process I/S ∗ is F -conditionally independent from τ i , for all i = 1 , ..., n ,and that the cumulative payments related to marked point processes (˜ τ i , ˜ X ij ) j ∈ N + , i = 1 , ..., n , ˜ N it X j =1 ˜ X ij , t ∈ [0 , T ] , i = 1 , ..., n, are i.i.d. compound Poisson processes, i.e. ˜ N i are mutually independent Poissonprocesses with parameter λ , and ˜ X ij are i.i.d. integrable nonnegative randomvariables independent from ˜ N i with expectation E [ ˜ X ij ] = m . In this case, we have ˜ m ( t ) = λmt, t ∈ [0 , T ] , ˜ m is defined in (3.11).In view of the above assumptions, all conditions in Theorem 3.16 are satisfiedin the case of Y = A T − A t , for t ∈ [0 , T ] . Let R t be the number of reported claimsat time t , i.e. R t := n X { τ i t } , t ∈ [0 , T ] . The real world pricing formula (5.2) together with Corollary 3.2, Theorem 3.16and Assumption 5.4 yields V t I t S ∗ t = E [ A T − A t | G t ] = E n X i =1 N iT X j = N it I τ ij S ∗ τ ij X ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G t = n X i =1 E N iT X j = N it I τ ij S ∗ τ ij X ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t ∨ H it = λmR t E (cid:20)Z Tt I u S ∗ u d u (cid:12)(cid:12)(cid:12)(cid:12) H i, ∞ ∨ F t (cid:21) + ( n − R t ) E hR Tt (cid:16) E [ X i ] I u S ∗ u + λm R Tu I v S ∗ v d v (cid:17) d P (cid:0) τ i u (cid:12)(cid:12) F u (cid:1)(cid:12)(cid:12)(cid:12) F t i e − R t µ u d u + R t ¯ G ( t − u ) e − R u µ v d v µ u d u = λmR t Z Tt E (cid:20) I u S ∗ u (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) d u + ( n − R t ) E hR Tt (cid:16) E [ X i ] I u S ∗ u + λm R Tu I v S ∗ v d v (cid:17) d P (cid:0) τ i u (cid:12)(cid:12) F u (cid:1)(cid:12)(cid:12)(cid:12) F t i e − R t µ u d u + R t ¯ G ( t − u ) e − R u µ v d v µ u d u = λmR t ( T − t ) I t S ∗ t + ( n − R t ) E hR Tt (cid:16) E [ X i ] I u S ∗ u + λm R Tu I v S ∗ v d v (cid:17) d P (cid:0) τ i u (cid:12)(cid:12) F u (cid:1)(cid:12)(cid:12)(cid:12) F t i e − R t µ u d u + R t ¯ G ( t − u ) e − R u µ v d v µ u d u , (5.4)where the conditional probability function P (cid:0) τ i t (cid:12)(cid:12) F t (cid:1) is given in (3.10), i.e. P (cid:0) τ i t (cid:12)(cid:12) F t (cid:1) = Z t (cid:18) α e − R s µ v d v µ s + Z s g ( s − u ) e − R u µ v d v µ u d u (cid:19) d s. The first component on the left-hand side of (5.4) λmR t ( T − t ) I t S ∗ t corresponds to already reported claims. We observe that the valuation of this partdoes not involve any more the updating information after the first reporting. The27econd component on the right-hand side of (5.4) ( n − R t ) E hR Tt (cid:16) E [ X i ] I u S ∗ u + λm R Tu I v S ∗ v d v (cid:17) d P (cid:0) τ i u (cid:12)(cid:12) F u (cid:1)(cid:12)(cid:12)(cid:12) F t i e − R t µ u d u + R t ¯ G ( t − u ) e − R u µ v d v µ u d u , (5.5)which can be further explicitly computed, corresponds to not reported claims andincludes both cases of incurred but not reported (IBNR) claims as well as notyet incurred claims. The standard literature of non-life insurance is mainly fo-cused on IBNR claims. However, for the pricing problem it is more appropriateto consider the entire expression (5.4). As already mentioned in at the beginningof this section, this price equals the benchmarked risk-minimizing price, if we as-sume square integrability of the claim. In particular, using the same argumentsof Proposition 4.11 in [2] and Section 4.1 of [10], we can calculate the associatedbenchmarked risk-minimizing strategy. The form of V suggests how to designderivatives which can be used to hedge risks in this market model. In particular,since V is expressed in terms of the intensity process µ , the distribution of θ i andthe distribution of (˜ τ ij , ˜ X ij ) j ∈ N + , the benchmarked risk-minimizing strategy can beexplicitly calculated. For further details on the benchmarked risk-minimizationmethod for non-life insurance liabilities, we refer to [38]. One method to derivethe distribution of µ can be found in [10]. In this paper, we introduce a general framework for modeling an insurance liabilitycash flow in continuous time by extending the reduced-form setting. This frame-work allows to consider a nontrivial dependence between the reference informationflow and the internal insurance information flow. In this setting, we compute ex-plicit valuation formulas, which can be used for pricing non-life insurance productsunder the benchmark approach.
Acknowledgements
The authors would like to thank Irene Schreiber for interesting discussions aboutproperty and casualty insurance.
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