A Unified Modeling Framework for Life and Non-Life Insurance

Transcription

1 A Unfed Modelng Framework for Lfe and Non-Lfe Insurance Francesca Bagn Yngln Zhang February 21, 218 Absrac In hs paper we propose for he frs me a unfed framework suable for modelng boh lfe and non-lfe nsurance marke, wh nonrval dependence wh he fnancal marke. We nroduce a drec modelng approach, whch generalzes he reduced-form framework for cred rsk and lfe nsurance. We apply hese resuls for prcng nsurance producs n hybrd markes by akng no accoun he role of nflaon under he benchmark approach. Ths framework offers a he same me a general and flexble srucure, as well as explc and reaable prcng formula. JL Classfcaon: C2, G1, G19 Key words: non-lfe nsurance, lfe nsurance, unfed framework, marked pon process, benchmark approach. 1 Inroducon In hs paper we gve a new unfed framework for boh lfe and non-lfe nsurance markes whch are radonally suded n a separaed way. Ths framework ncludes he classc reduced-form framework for cred rsk and lfe nsurance as a specal case and allows reaable prcng formula. In parcular, we use a drec modelng approach o nroduce a dependence srucure beween fnancal and nsurance markes. Ths allows us o consder hybrd markes models, where we sudy he prcng of nsurance clams by also akng no accoun he role of nflaon under he benchmark approach. Hsorcally, he mahemacal modelng of lfe and non-lfe nsurance marke s que asymmerc. Whle here are a lo of recen works concernng lfe nsurance Man afflaon: Deparmen of Mahemacs, LMU Munch, Theresensraße, 39, 8333 Munch, Germany, mal: bagn@mah.lmu.de Secondary afflaon: Deparmen of Mahemacs, Unversy of Oslo, Box 153, Blndern, 316, Oslo, Norway. Deparmen of Mahemacs, LMU Munch, Theresensraße, 39, 8333 Munch, Germany, mal: zhang@mah.lmu.de 1

2 marke modelng, see e.g 26, 27, 1, 13, 15, 2, 7, 8, 6, 9, he case of non-lfe nsurance has been less suded n he mahemacal leraure. In mos of he sudes abou non-lfe nsurance, only dscree me and/or sae space are consdered see e.g. 22, 23, 18. Some models for non-lfe nsurance n connuous me can be found n e.g. 1, 3, 25, 14, 29, 28 and 32. However, hese sengs do no consder a nonrval dependence srucure beween fnancal marke and nsurance marke. In parcular, n e.g. 14, 29 and 28, he nsurance marke s no dsngushed from he fnancal marke, and n e.g. 3 fnancal marke and nsurance marke are assumed o be ndependen. The mporance of consderng a nonrval dependence srucure beween he wo markes s dscussed n 4. The recen nroducon of dervaves based on occurrence nensy ndex, such as moraly dervaves, weaher dervaves ec., makes hs parcularly relevan for lfe nsurance and non-caasrophe non-lfe nsurance. Ths las one, whch ncludes car nsurance, hef nsurance, home nsurance, ec., as opposed o caasrophe non-lfe nsurance 1, covers hgh-probably low-cos evens, and s ofen negleced by he leraure. Ths paper ams o fll hs gap n he leraure, as well as o provde analycal resuls whch can be used for he valuaon of non-lfe non-caasrophe lnked fnancal producs, whch are currenly sll no common bu can be poenally aracve n he fuure. Recen non-lfe nsurance leraure, see e.g. 3, 14, 29, 28 and 32, commonly assume he nsurance nernal nformaon flow as gven by he naural flraon of a marked pon process, whch descrbes he nsurance clam movemen. Prcng and hedgng formulas are hen obaned by usng he compensaor of hs marked pon process. However, as we dscuss n Secon 4, hs approach can no be always followed n he case of mulple flraons wh nonrval dependence. Indeed, wh respec o a generc flraon, s no always rue ha here exss a marked pon process wh a gven compensaor, and he compensaor does no always deermne unquely he law of he process. To overcome hese dffcules, we propose a new framewok, whch uses a drec approach as n Secon 5.1 and of 11 and allows an explc boom-up consrucon o rea more general flraons. We noe ha, whle for lfe nsurance modelng whn he classc reduced-form framework, he compensaor approach and he drec modelng approach concde, see e.g. 11, s no he case for non-lfe nsurance. More precsely, n our new framework we consder an homogeneous nsurance porfolo wh n clams and assume ha he reference flraon F ncludes nformaon relaed o fnancal marke and envronmenal, socal and economc ndcaors. We model he accden mes of he relaed nsurance secures as F-condonally ndependen random varables wh a common F-adaped nensy process µ. very accden even s repored only afer a nonnegave random delay, whch s ypcally he case of non-lfe nsurance, and he furher developmen of he case s modelled by an ndependen marked pon process. Unlke he case of lfe nsurance, he accden mes hemselves and her relaed damages are unknown unl 1 See e.g. 12 for he dsncon beween caasrophe and non-caasrophe nsurance. 2

3 he momen of reporng, and here may be furher updang nformaon afer he frs reporng. We are able o capure hs feaure by provdng a nonrval exenson of he classcal reduced-form framework, whch also nclude he lfe nsurance case. In parcular, hs seng allows o oban analycal valuaon formulas for nsurance producs, whch can be expressed n erm of he accden nensy µ, he delay dsrbuon and he updang dsrbuon, as llusraed n Secon 3. We hen apply hese resuls for prcng of nsurance lables n a hybrd marke under he benchmark approach. The hybrd naure of he combned marke s gven by he presence of dervaves lnked o he nensy process µ on he fnancal marke, whch can be e.g. moraly dervaves or weaher dervaves accordng o he conex. The analycal prcng formulas for nsurance producs can also be useful for fuure desgn of nsurance-lnked dervaves and calculaon of explc hedgng sraeges. Ths paper s organsed as follows. In Secon 2 we consruc a new unfed general framework for lfe and non-lfe nsurance modelng. In Secon 3 we gve some useful valuaon resuls n hs seng. In Secon 4 we dscuss he compensaor approach. In Secon 5 we descrbe he hybrd srucure of he combned marke and derve he real world prcng formula for nsurance secures under he benchmark approach. 2 General framework In hs secon we consruc a unfed general framework whch can be used o model boh lfe and non-lfe nsurance. We consder a flered probably space (Ω, G, G, P, where G : (G, G G, and G s rval. W assume ha G F H, where F : (F and H : (H are flraons represenng respecvely a reference nformaon flow and he nernal nformaon flow only avalable o he nsurance company. Hence G descrbes he global nformaon flow avalable o he nsurance company. The reference flraon F ypcally ncludes nformaon relaed o he fnancal marke, and o envronmenal, polcal and socal ndcaors. Whle we do no specfy he srucure of he reference flraon F, we assume ha he nsurance flraon H s generaed by a famly of marked pon processes, represenng he mes and amouns of losses of he nsurance porfolo. Flraons F and H are no supposed be ndependen. Whou loss of generaly, we assume ha all flraons sasfy he condons of compleeness and rgh-connuy. If no oherwse specfed, all relaons n hs paper hold n he P -a.s. sense. For a dealed background of marked pon processes we refer o e.g. 24, 16 and 2. In he followng we use he classc ermnology of non-lfe nsurance, see e.g. 33 and 3, and specfy he flraon H as follows. We consder an nsurance porfolo wh n polces. For -h polcy wh 1,..., n, he nsurance company s ypcally nformed abou he accden occurred a a random me τ only afer a random delay θ, whch can be very long especally 3

4 n he case of non-lfe nsurance. Once he accden s repored a me τ 1, where τ 1 : τ + θ, (2.1 boh he accden me τ, he reporng delay θ and he mpac sze of he accden, descrbed by a nonnegave random varable X1, become avalable nformaon. Le N + be he se of naural numbers whou zero. We descrbe he -h nsurance polcy movemen by a marked pon process (τj, Θ j j N + wh 2-dmensonal nonnegave marks. Tha s, he sequence (τj j N + s a pon process, where and (Θ j j N + τ j : (Ω, G, P (R +, B(R +, j N +, s a sequence of 2-dmensonal nonnegave random varables, wh Θ j : (Ω, G, P (R 2 +, B(R 2 +, j N +. For every j N +, he random me τj descrbes he reporng me of j-h even relaed o -h polcy. The mark componens Θ j descrbe he reporng delay and he mpac sze of he correspondng even, respecvely, whch are known only f he even s repored. More precsely, we se and τ 1 wh mark Θ 1 (θ, X 1, (2.2 τ j+1 τ 1 + τ j wh mark Θ j+1 (, X j+1 : (, X j, (2.3 for j 1, where ( τ j, X j j N + s an auxlary marked pon process, whch descrbes updang and developmen afer he frs reporng a τ1. Here we assume ha only he frs reporng delay s dfferen from zero, snce n hs paper we focus on modelng he frs accden mes τ and her relaon wh he reference flraon. However our seng can be easly generalzed by consderng non zero random delays n (2.3. We se furhermore ha he marked pon process ( τ j, X j j N + s smple,.e. lm τ j, j and τ j < τ j+1, f τ j <, and sasfes he followng negrably condon 1 { τ j } X j < for all, (2.4 for 1,..., n. In parcular, he random mes (τ j j N + are srcly ordered: τ 1 1 < τ 1 2 < < τ 1 j < τ 1 j+1 <, τ 2 1 < τ 2 2 < < τ 2 j < τ 2 j+1 <,. τ n 1 < τ n 2 < < τ n j < τ n j+1 <. (2.5 4

5 We sress ha every τj may assume nfne value, n such case descrbes an even whch never happens. For he sake of smplcy we assume also he followng. Assumpon Homogeneous delay: he random delays θ, 1,..., n, have he same dsrbuon. 2. Homogeneous developmen: he marked pon processes ( τ j, X j j N +, 1,..., n, have he same dsrbuon. 3. Independen frs mark: he frs marks X1, 1,..., n, are muually ndependen and ndependen from F σ(τ 1... σ(τ n. 4. Independen delay: he random delays θ, 1,..., n, are muually ndependen and ndependen from F σ((τ 1, X σ((τ n, Xn Independen developmen: he marked pon processes ( τ j, X j j N +, 1,.., n are muually ndependen and ndependen from F σ((τ 1 1, θ1, X σ((τ n 1, θn, X n 1. We emphasze ha he above assumpons are general enough. The homogeney assumpons can be sasfed by subdvdng opporunely he nsurance porfolo. The ndependence assumpons reflec he fac ha reporng delays θ, occurrences and sze of he losses afer he frs reporng me, descrbed by ( τ j, X j j N +, are ypcally dosyncrac facors whch are ndependen o each oher and ndependen from he reference nformaon. However, we nroduce a dependence srucure by modelng he occurrence nenses of he accdens, as we wll presen n (2.14 and (2.15. We assume furhermore ha he dsrbuon of delay varables θ, 1,..., n has he followng srucure. Assumpon 2.2. The common cumulave dsrbuon funcon G of θ, 1,..., n, wh G(x : P (θ x, x R, (2.6 sasfes G(x α + x g(xdx, x R, (2.7 where α P (θ P (θ 2, and g s a nonnegave Lebesgue-negrable funcon. Accordng o he above assumpon, he delays may have a mxed dsrbuon. In hs way we cover boh he case of lfe nsurance wh θ,.e. g, and he case of non-lfe nsurance wh non-null delays. 2 The delays θ are assumed o be nonnegave. 5

6 For every 1,..., n, we defne he marked cumulave process N by N (, B(ω : j (ω } 1 {Θ (ω B}, j for every, B B(R 2 +, ω Ω. The process (N defned by N : N (, R 2 + j },, s called ground process assocaed o he marked pon process. A any me, he random varable N couns he number of occurrence of τj up o me. In he leraure, he name marked pon process refers somemes o he process N. Indeed, Lemma of 24 shows ha here s a unque correspondence beween he marked pon process and s marked cumulave process. More precsely, for all and {τ j } {N j}, (2.8 {Θ j B} {τ j < } {N (τ j, B > } (2.9 for all B B(R 2 +. We consder he flraons H,1 : (H,1 H,1 : σ for all, and H,j : (H,j H,j for all. I holds clearly wh ( 1 s} 1 {(θ,x1 B}, s, for all B B(R2 +, j > 1, wh ( : σ j s} 1 {Xj B}, s, for all B B(R + In parcular, n vew of (2.1 we have H,j σ(τ j σ(x j for j > 1. H,1 σ(τ 1 σ((θ, X 1 σ(τ σ((θ, X 1. (2.1 Le H : (H be he naural flraon of he marked cumulave process N, ha s for all, H σ(n (s, B, s, for all B B(R 2 +. The nernal nformaon flow of he nsurance company s descrbed by he flraon H : (H, where,, H : H 1... H n,. (2.11 6

7 Smlarly, for 1,.., n, we call Ñ he correspondng marked cumulave processes assocaed o he marked pon processes ( τ j, X j j N + and H he correspondng flraon, respecvely. Smlarly, all oher noaons assocaed o hese las processes wll be denoed by he symbol " ". Lemma 2.3. For every 1,..., n, we have H j N + H,j. Proof. Clearly, we have H H,j. j N + For he oher ncluson, s suffcen o show ha for all s and B B(R 2 +, {τ j s} {Θ j B} H. Indeed, snce he marked pon process (τj, Θ j j N + s smple, from (2.8 and (2.9 we have {τj s} {N s j} {N (s, R 2 + j} H, and {τ j s} {Θ j B} {τ j s} {N (τ j, B > } H. We now nroduce he followng noaons, whch s useful n he sequel. 1,..., n, j N +, we defne For H, j : k j H,k, H, j : k j H,k, smlarly for H,>j and H,<j. In parcular, n he case of j 1, we se H,<1 : {, Ω} for every. The followng corollary s a drec consequence of Lemma 2.3. Corollary 2.4. For every 1,..., n, j N +, we have H H, j H,>j H,<j H, j. Smlarly o he reduced form seng for cred rsk and lfe nsurance, we now model he accden mes τ, 1,..., n and her relaon wh he reference flraon n he followng way. As n Secon of 11, we assume ha all random mes (τj j N, 1,..., n, are no F-soppng mes, and ha accden mes τ, 1,..., n, sasfy he followng condons: for all 1,..., n, τ > P -a.s. and ha for, and s,,, P ( τ > s F P ( τ > s Fs, (2.12 and for l, k 1,.., n wh l k, τ l and τ k are F-condonally ndependen,.e. f, and r, s,,, we have ( ( ( P τ l > r, τ k > s F P τ l > r F P τ k > s F. (2.13 7

8 Remark 2.5. If we defne H, : σ ( s} : s, 1,..., n, hen condon (2.12 s equvalen o X F X F s, for each negrable Hs, -measurable random varable X. Condon (2.13 s equvalen o he F -condonal ndependence beween he σ-algebras H l, and H k,. Furhermore, f F : (F s he F-condonal cumulave funcon of τ, F : P ( τ F,, we assume ha here exss a locally negrable and F-progressvely measurable process µ : (µ, such ha We defne Γ : (Γ as e µ u du 1 F for all. (2.14 Γ : µ udu,. (2.15 The process µ s called nensy process of he random jump me τ and he process Γ s called hazard process of τ. An explc consrucon n xample of 11 shows ha for a gven famly of locally negrable F-progressvely measurable process µ, 1,..., n, s always possble o consruc random mes τ, 1,..., n, such ha Γ s he hazard process of τ for every,..., n, and all he assumpons above are sasfed. For he sake of smplcy, we assume ha he nsurance porfolo s homogeneous. Assumpon 2.6. The accden mes τ, 1,..., n, have he same nensy process. Under he homogeney condon, we denoe he common F-condonal cumulave funcon, hazard process and nensy process respecvely by F, Γ and µ. The above assumpon reflecs he fac ha, whle he polcy developmens may no have drec lnk o he nformaon flow F, he accden occurrences τ, 1,..., n, are nfluenced by some common sysemac rsk-facors, and he common condonal nensy µ s deducble from he reference nformaon flow. We now show how he general framework descrbed above comprehends n a synhec way boh lfe and non-lfe nsurance modelng. 2.1 Lfe nsurance Lfe nsurance polces ypcally do no have reporng delay and depend only on τ, 1,..., n, whch acually represen he decease mes. Ths can be ncluded n our framework by seng θ, τj for all j > 1 and X j 1 for all 8

9 j N +, and nerpreng τ as he decease me of person, where 1,..., n. The flraon G s hence reduced o G F H 1... H n, where H σ ( s}, s,, 1,..., n. In parcular, he F-progressvely measurable process µ s nerpreed as moraly nensy n hs conex. Lfe nsurance whn hs seng s nensvely suded n he leraure, see e.g. 2, 8, 6 and Non-lfe nsurance The framework n Secon 2 n s full generaly descrbes he case of non-lfe nsurance, and ncludes he seng of e.g. 14, 29, 28 and 3 as specal cases. Indeed, non-lfe nsurance polces ypcally have reporng delay,.e. θ, whch can also coun o several years. For -h polcy, we nerpre Xj as paymen amoun a he j-h random mes τj ; he exac accden me τ and frs paymen amoun X1 s known only afer reporng a me τ 1. Furher developmens may occur afer he frs reporng and before he selemen of clam. The oal number of developmens (τj j N + s unknown as well as he amoun of correspondng paymens (Xj j N +. 3 Valuaon formulas In hs Secon, we sae several resuls under he above srucure assumpons, by followng Secon 5.1 of 11 for he presenaon. We sar wh exenson of relaon (2.12 and he F-ndependence (2.13 of τ, 1,..., n. We show ha, f hese relaons hold for he flraons H,, 1,..., n, hen hey hold for he whole flraons H, 1,..., n. Lemma 3.1. For any, and l, k 1,..., n wh l k, he σ-algebras H l and H k are F -ndependen. Proof. In vew of Lemma 2.3, s suffcen o prove ha H k,p and H k,q are F - ndependen for all p, q N +. For noaon smplcy, we only consder p 1 and q 1, snce he oher cases are smlar. Tha s, we wan o show l p s} 1 {Xp B l l } q k r} 1 {Xq k B k } F l p s} 1 {Xp l Bl } F k q r} 1 {Xq k Bk } F, 9

10 where s, r,, 3 and B l, B k B(R +. By (2.2 and (2.3, he above equaly s equvalen o l +θ l + τ p l s}1 { X p l Bl } k+θk + τ q k r}1 { X p k Bk } F l +θ l + τ p l s}1 { X p l Bl } F k +θ k + τ q k r}1 { X p k Bk } F. If we defne he followng deermnsc funcons f l (x : 1 {θ l + τ p s x} 1 l { X p B l l }, f k (x : 1 {θ k + τ p l r x}1 { X q l Bk }, hen f l (x f l (x1 {x s}, f k (x f l (x1 {x r}. In parcular, f l (τ l and f l (τ k are Hl, - and H k, -measurable respecvely. Ths ogeher wh Remark 2.5 and he ndependence condons n Assumpon 2.1 yelds l +θ l + τ p l s}1 { X p l Bl } k+θk + τ q k r}1 { X p k Bk } F l +θ l + τ p l s}1 { X p l Bl } k+θk + τ q k r}1 { X p k Bk } F σ(τ l σ(τ k F 1 {x+θ l + τ p s} 1 l { X p B l l } 1 {y+θ k + τ q k r} 1 xτ { X p k B } k l yτ k F 1 {x+θ l + τ p l s}1 { X p l Bl } 1 xτ l {y+θ k + τ q k r}1 { X p k Bk } yτ k F f l (τf l l (τ k F f l (τ l F f l (τ k F. The same calculaon as above yelds f l (τ l F f l (τ k F l +θ l + τ p s} 1 l { X p B l l} F k +θ k + τ q k r} 1 { X p k B k} F, whch concludes he proof. Lemma 3.2. For any s and 1,..., n, f X s H s-measurable, hen X F X F s. 3 We noe ha may assume he value. 1

11 Proof. The proof of he Lemma s smlar o he one of Lemma 3.1. Indeed, s suffcen o apply Remark 2.5. As a consequence of he above wo lemmas, he G-condonal expecaon can be reduced o F H -condonal expecaon n mos cases. Corollary 3.3. If T <, and Y s an negrable (F T H T -measurable random varable, hen Y G Y F H. Proof. I s enough o prove he saemen for he ndcaor funcons of he form Y 1 A 1 B where A F T and B HT. We noe ha G F H 1... H n. Le C F, D j H j, j 1,..., n. I s suffcen o show ha 1 A 1 B dp 1 A 1 B F H dp. C D 1... D n C D 1... D n Clearly, holds 1 A 1 B dp C D 1... D n C D A B,...,n j C D A B By Lemma 3.1 and Lemma 3.2, we have 1 D j F T H T,...,n j 1 D j F,...,n j,...,n j 1 D jdp,...,n j 1 D j,...,n j 1 D j F T H T dp. 1 D j F T F H. 11

12 I follows, 1 A 1 B dp C D 1... D n C D A B 1 A 1 B C D,...,n j,...,n j 1 A 1 B F H C D C D,...,n j F H dp 1 D j F H dp 1 D j F H dp 1 D j,...,n j 1 D j 1 A 1 B F H dp 1 A 1 B F H C D 1... D n dp. An oher mporan corollary of Lemma 3.1 and Lemma 3.2 s he so called H- hypohess beween flraons F and G,.e. he propery ha every F-marngale s also a G-marngale. Corollary 3.4. The H-hypohess holds beween flraons F and G. Proof. By Lemma of 11, H-hypohess beween wo flraons F G s equvalen o he propery ha for any and any bounded, G -measurable random varable η, holds ha η F η F. I s suffcen o check he above relaon for ndcaor funcons of he form 1 A 1 B B n, where A F, B H, 1,..., n. By applyng Lemma 3.1 and Lemma 3.2, we oban 1 A 1 B B n F 1 A 1 B B n F n 1 A 1 B F 1 n 1 A 1 B F 1 1 A 1 B B n F 1 A 1 B B n F. 12

13 Now we would lke o derve some more explc represenaons. We noe ha for every negrable random varable Y,, 1,..., n and j N +, we have he decomposon Y H F j >} Y H F + j } Y H F. (3.1 In he followng we wll evaluae separaely he wo componens on he rgh-hand sde of (3.1. The followng lemma s mporan for dervng a represenaon of he frs componen. Lemma 3.5. For any, 1,..., n and j N +, we have where H F G,j, { } G,j : A G : C H,<j F, A {τj > } C {τj > }. (3.2 Proof. By Corollary 2.4, holds ha H H,<j H, j. Hence, s suffcen o show ha boh H, j and H,<j F belong o G,j. In he frs case, f > 1 and A {τk s} {X k B} for some k j, s and B B(R, we ake C. Smlarly for 1 and A {τk 1 s} {(θ k, Xk 1 B} for k j, s and B B(R 2 +. In he second case, f A H,<j F we ake C A. The followng Proposon gves wo represenaons of he frs componen on he rgh-hand sde of (3.1. Represenaon (3.3 s analogue o Lemma n 11, represenaon (3.4 s new and wll be used for our furher dscusson. Proposon 3.6. For any, 1,..., n, j N + and any negrable G- measurable random varable Y, we have 1 j >} Y H {τ F 1 j >} Y H,<j F {τ j >} ( P τj (3.3 > H,<j F j >} Y H, j F. (3.4 Proof. qualy (3.3 s equvalen o j >} Y P ( τ > H,<j F H F j >} j >} Y H,<j F. 13

14 We noe ha he rgh-hand sde s (H F -measurable. Hence, suffces o show ha for any A H F, j >} Y P ( τj > H,<j F dp j >} j >} Y H,<j F dp. A By Lemma 3.5, here s an even C H,<j A F such ha A {τ j > } C {τ j > }, hence A j >} Y P ( τ j > H,<j F dp A {τ j >} Y P ( τ j > H,<j F dp C {τ j >} Y P ( τ j > H,<j F dp C C C C j >} Y P ( τj > H,<j F dp j >} Y H,<j F j >} H,<j F dp j >} j >} Y H,<j F H,<j F dp j >} j >} Y H,<j F dp j >} Y H,<j F dp C {τ j >} A {τ j >} A j >} Y H,<j F dp j >} j >} Y H,<j F dp. qualy (3.4 can be proved n he same way. We only need o observe ha { } G,j A G : C H, j F, A {τj > } C {τj > }. Hence, he σ-algebra H,<j proof. n (3.3 can be replaced by H, j. Ths concludes he Now we focus on he second componen on he rgh-hand sde of (3.1. followng lemma gves a slghly more general resul. The 14

15 Lemma 3.7. For any, 1,..., n, j N +, any σ-algebra A G and any negrable G-measurable random varable Y, we have j } Y H, j A j } Y H, j A. Proof. We noe ha he lef-hand sde s (H, j A-measurable. Snce he marked pon process (τj, Θ j j N + s smple,.e. he src monooncy (2.5 holds, f A H, j A A, hen A {τj } H, j A, and j } A {τ Y dp Y dp j } Ths concludes he proof. Remark 3.8. Snce we have A j } Y H, j A H, j σ ( τ h, h 1,..., j, Y H, j A {τj } dp. A dp Lemma 3.7 shows ha, f τj has occurred before me, hen paral nformaon abou τj up o s equvalen o full nformaon abou all he random mes τ h, h 1,..., j. In parcular, f Y s a funcon of τ1,..., τ j,.e. Y f(τ 1,..., τ j, hen he condonal expecaon s smply j } Y H, j A j } Y. We summarze he above resuls n he followng represenaon heorem. Theorem 3.9. For any, 1,..., n, j N + and any negrable G-measurable random varable Y, we have Y H F 1{τ j } Y H, j If furhermore Y s (H T F T -measurable, hen Y G j } Proof. Snce Y H, j Y H F j } Y H F + H,>j F + j >} Y H, j F. H,>j F + j >} Y H, j F. j >} Y H F, he frs par s a sraghforward consequence of Proposon 3.6 and Lemma 3.7. For he second par, suffces o apply Corollary

16 We now show some resuls whch wll play a key role for he reserve esmaon problem n Secon 2.2. Le T < and Z : (Z,T be a connuous, bounded and F-adaped process. For 1,..., n, we now consder and compue Y N T jn X jz τ j Y G 1 {<τ j T} X jz τ, (3.5 j N T jn XjZ τ j G. (3.6 In parcular, smlarly o before, we sudy separaely he wo componens of he decomposon of (5.3 wh respec o he frs reporng me τ1,.e. N T N XjZ τ j G T N >} XjZ τ j G T + } XjZ τ j G, jn jn jn (3.7 and derve more explc formulas n erms of he nensy process µ, he dsrbuon of delay θ, and he dsrbuon of developmen N afer he frs reporng. We sar wh he F-condonal expecaon of τ 1. Lemma 3.1. For any 1,..., n and, we have and P ( τ 1 > F e µudu + P ( τ 1 F Ḡ( ue u µvdv µ u du, (3.8 G( ue u µvdv µ u du, (3.9 where G s he cumulave dsrbuon funcon of θ defned n (2.6 and Ḡ(x : 1 G(x P (θ > x, x R. (3.1 Proof. We prove only equaly (3.8, snce equaly (3.9 drecly follows. Noe ha by Assumpon 2.1, θ s ndependen from F σ(τ. Furhermore, boh θ and τ are P -a.s. nonnegave. Therefore, we have P ( τ1 > F +θ >} F >} + } +θ >} F e µudu + } +θ >} F σ(τ F e µudu + } 1 {θ > x} xτ F e µudu + }Ḡ( τ F. 16

17 To conclude we only need o show }Ḡ( τ F Ḡ( ue u µvdv µ u du. (3.11 Ths can be done n he same way as for Proposon of 11, n vew of relaon (2.12 and he fac ha G s connuous accordng o Assumpon 2.2. Remark Noe ha (3.11 s he condonal probably ha he accden has ncurred, bu no ye repored (IBNR evens n he ermnology used n he nsurance secor. In expresson (3.9 of Lemma 3.1, he parameer appears also n he negrand. The followng corollary mproves relaon (3.9 and shows ha he process of condonal expecaon (P ( τ 1 F s absoluely connuous wh respec o he Lebesgue measure. Corollary For any 1,..., n, we have P ( τ 1 F ( α e s µvdv µ s + where α and g are defned n (2.7. s g(s ue u µvdv µ u du ds, (3.12 Proof. Ths follows mmedaely from Assumpon 2.2, relaon (3.9 and Lebnz negral rule. Lemma If he process 4 Z : (Z u u,t s lef-connuous and bounded 5 and Z s F T -measurable for all, hen we have T 1 {<τ 1 T} Z τ1 F for 1,..., n and, T. Z u dp ( τ 1 u F u F, Proof. The proof s smlar o he one n Proposon of 11. We assume frs ha Z s sepwse consan,.e. we assume whou loss of generaly ha Z u n Z j 1 {j <u j+1 }, j 4 We do no assume ha Z s F-adaped, see Remark We emphasze ha he boundedness condon can be generalzed. 17

18 for < u T, where <... < j+1 T, Z j s F T -measurable for all j,..., n. Lemma 3.2 yelds 1 {<τ 1 T} Z τ1 F n Z j 1 {j <τ1 j+1} j F n Z j 1 {j <τ1 j+1} F T j F n Z j ( 1 j+1} F j+1 1 j} F j j F (3.13 T Z u dp ( τ1 u Fu F. (3.14 In he general case, s suffcen o fnd a sepwse consan approxmaon for Z. Snce Z s bounded, we have he convergence of he Remann sum under he sgn of condonal expecaon n (3.13 o he Lebesgue-Seljes negral n expresson (3.14. Hence also he convergence of he condonal expecaon follows. Remark We sress ha he above Lemma nvolves only Lebesgue-Seljes negral n (3.14, whch concdes wh Lebesque negral n vew of Corollary Hence, s no necessary o assume ha Z s F-adaped. Now we are able o calculae he frs componen on he rgh-hand sde of (3.7. We defne Ñ m( : X j, f, (3.15 m( :, f <, where Ñ denoes he ground process of ( τ j, X j j N +,.e. Ñ : 1 { τ j },. (3.16 Noe ha m does no depend on because of Assumpon 2.1 (2. Proposon Le Z : (Z,T be a connuous, bounded and F-adaped 6 6 Noe ha he resul also holds under dfferen negrably and measurably condons. 18

19 process and Y be as n (3.5, hen for any, T, 1 >} Y H F ( T X1 Z u + T u Z vd m(v u dp ( τ1 u F u F 1 >} P ( τ1 > F where m s defned n (3.15. Proof. By applyng (3.4 n Proposon 3.6 o Y defned n (3.5, we ge 1 >} Y H F 1 >} Y H,1 F 1 >} j T} X jz τ j F H,1 1 >} 1 T} X 1Z τ 1 H,1 F + 1 >} j T} X jz τ j j2 H,1 F. (3.17 For he frs componen of (3.17, s suffcen o use (3.3 n Proposon 3.6 and an argumen smlar o Proposon of 11, akng no accoun he ndependence condon n Assumpon 2.1 (3 and Lemma 3.2. We have hence 1 >} 1 T} X 1Z τ H,1 1 F 1 {<τ 1 T} X 1Z τ H,1 1 F 1 {<τ 1 1 T} X 1 Z τ F 1 {τ 1 >} P ( τ1 > F T X1 Z udp ( τ1 u F u F 1 >} P ( τ1 >. F Now we focus on he second componen of (3.17. We assume frs ha resrced on he nerval, T, Z s a bounded, sepwse, F-predcable process,.e. Z u n Z 1 { <u +1 }, (3.18 for < u T, where <... < n+1 T and Z s F -measurable for all 19

20 ,..., n. In such case, we have 1 >} j T} X jz τ j j2 H,1 1 >} 1 {<τ 1 + τ j T} X jz τ j F H,1 n 1 >} 1 { <τ1 + τ j +1} X jz F H,1 n 1 >} Z 1 { <τ1 + τ j +1} X j 1 >} n Z 1 { <x+ τ j +1} X j 1 >} n F H,1 F σ(τ1 xτ 1 H,1 F H,1 F Z ( m(+1 τ 1 m( τ 1 H,1 F, (3.19 where n he second las equaly we use he ndependence beween he marked pon process ( τ j, X j j N + and he σ-algebra H,1 F n Assumpon 2.1. Ths shows ha for any bounded, sepwse, F-predcable process Z, we have 1 >} T j T} X jz τ j F 1 >} Z u d m(u τ1 H,1 F. j2 H,1 A connuous bounded process Z can be approxmaed by a sequence of bounded, sepwse and F-predcable processes,.e. here s a sequence Z n of he form (3.18 such ha Z n Z and Z n M, wh M >. Snce m s rgh-connuous and monoone, he Lebesgue Seljes negral T s well defned. I holds by Lebesgue Theorem T Z n u d m(u τ 1 Z u d m(u τ 1 (3.2 T Z u d m(u τ 1. Furhermore, T Zu n d m(u τ1 T M d m(u τ1 M m(t τ 1 m( τ1. (3.21 2

21 The rgh-hand sde of (3.21 s unformly bounded by (3.15 and (2.4. By applyng agan Lebesgue Theorem, we have also he convergence of he condonal expecaons T 1 >} Zu n d m(u τ1 T F Z u d m(u τ1 F. H,1 We noe ha m(u for u <, hence, 1 >} j T} X jz τ j j2 H,1 F T 1 >} Z u d m(u τ1 H,1 F T 1 {<τ 1 T} Z u d m(u τ1 H,1 F. H,1 By applyng agan (3.3 n Proposon 3.6 o he above expresson, we ge 1 >} j T} X jz τ j F 1 >} j2 H,1 T 1 {<τ 1 T} Z u d m(u τ1 F P ( τ1 >. F Le Z s : T Z u d m(u s, s, T. We noe ha m s rgh-connuous and monoone. On one hand, for fxed s, T, he funcon d s (u : m(u s, u, T, s also rgh-connuous and monoone and defnes he cumulave dsrbuon funcon of a fne posve measure, n vew of (2.4. One he oher hand, for fxed u, T, he funcon m(u s, s, T, s lef-connuous n s,.e. for every seres s n s, we have he ponwse convergence lm d s n (u d s (u for all u, T s n s of he cumulave dsrbuon funcons, equvalen o he convergence n dsrbuon or weak convergence n measure 7. Ths yelds he convergence Z sn Z s, P a.s., 7 A seres of posve fne measures (ν n n N converges weakly o a posve fne measure ν, f for all bounded connuous funcons f, holds fdν n fdν 21

22 ha s, Z s : T Z u d m(u s, s, T, s lef-connuous. Fuhermore, s also bounded. Now we apply Lemma 3.13 and oban T 1 {<τ 1 1 T} Z u d m(u τ 1 F {τ 1 >} P ( τ1 > F 1 {<τ 1 T} Z τ F 1 1 >} P ( τ1 > F T Z u dp ( τ1 u Fu F 1 >} P ( τ1 > F ( T T Z v d m(v u dp ( τ1 u Fu F 1 >} P ( τ1 >. F As he las sep, we noe ha for u < s, T m(u s. Ths concludes he proof. Z u d m(u s T s Z ud m(u s snce Remark The proof of Proposon 3.15 reles on assumpon (3.15. Anoher suffcen condon would be he connuy of m, such as n he case of a compound Posson process or a Cox process wh connuous nensy process and negrable marks. Indeed, snce m(u for u <, T Zu n d m(u τ1 T 1 >}M d m(u τ1 1 >} 1 {<τ 1 T} M m(t τ 1 m( τ 1, and he rgh-hand sde s unformly bounded f m s connuous. The followng proposon gves a represenaon of he second componen on he rgh-hand sde of (3.7. Proposon Under ( he same assumpons of Proposon 3.15, f for each Ñ 1,..., n, he process, where Ñ s defned n (3.16, s of X j,t ndependen ncremens wh respec o s naural flraon H, hen for, T and Y as n (3.5, holds T 1 } Y H F 1 } Z u d m(u x H,1 H x F, xτ 1 for 1,..., n. Proof. I follows from Lemma 3.7 ha }Y H F 1 } Y H,1 F. 22

23 Smlarly o he proof of Proposon 3.15, we assume frs Z of he form (3.18. In such case, we have 1 } Y H,1 H,>1 F 1 } 1 {<τ 1 + τ j T} X jz τ j H,>1 F H,1 n 1 } 1 { <τ1 + τ j +1} X jz n 1 } 1 { <x+ τ j +1} X jz n 1 } 1 { <x+ τ j +1} X jz H,1 H,1 H,1 H,>1 F H,>1 F xτ 1 H x F, xτ 1 where he las sep follows from he defnons of he flraons. By usng ower propery, he ndependence beween he marked pon process ( τ j, X j j N + and F H,1 (see ( Assumpon 2.1, and he ndependence of ncremens of he process Ñ he process, we ge furhermore X j,t 1 } Y H,1 H,>1 F n 1 } Z 1 { τ j +1 x} X j 1 { τ j x} X j H,1 H,1 H x F H x F H,1 H x F n 1 } Z 1 { τ j +1 x} X j H x 1 { τ j x} X j H x H,1 H x F xτ 1 Ñ n x Ñ 1 } x m( +1 x X j m( x + Z n 1 } Z ( m( +1 x m( x 23 H,1 F xτ 1 X j xτ 1 H,1 H x F (3.22 xτ 1

24 Ths yelds ha for any bounded, sepwse, F-predcable process Z, we have T 1 } Y H F 1 } Z u d m(u x H,1 F. xτ 1 If Z s connuous, bounded and F-adaped, hen Z can be approxmaed by a sequence of bounded, sepwse and F-predcable processes. Ths ogeher wh he fac ha m s rgh-connuous and monoone guaranees ha he Remann sum n (3.22 under he sgn of condonal expecaon converges o Lebesgue- Seljes negral, by usng he same argumens of Proposon We summarze he resuls n he followng heorem, whch gves an explc represenaon of G-condonal expecaon wh respec o he frs reporng me τ 1. Theorem Le Z : (Z,T be a connuous, bounded and F-adaped ( process 8 Ñ, Y be of he form (3.5. If he process, has ndependen ncremens and m s defned n (3.15, hen T Y G 1 } Z u d m(u x + 1 >} for 1,..., n, where and P ( τ 1 F T H,1 X j,t H x F xτ 1 dp ( τ1 u Fu F ( X 1 Z u + T u Z vd m(v u ( α e u µvdv µ u + P ( τ 1 > F e µudu + P ( τ 1 > F, u wh α and g defned n (2.7 and Ḡ defned n (3.1. g(u ve v µsds µ v dv du, Ḡ( ue u µvdv µ u du, Proof. I s enough o combne Corollary 3.3, Lemma 3.1, Corollary 3.12, Proposon 3.15 and Proposon Compared o Theorem 3.9, Theorem 3.18 s more explc and has he advanage ha he represenaon s expressed as funcon of µ, he dsrbuon of θ and he dsrbuon of ( τ j, X j j N +. Ths wll be useful for our furher dscusson. 8 Noe ha he resul of Theorem 3.18 also holds under dfferen negrably and measurably condons. 24

25 4 Comparson wh he compensaor approach In hs secon, we compare our framework wh he compensaor approach for non-lfe nsurance n he exsng leraure. Whn hs secon, he flraon H denoes he naural flraon of a marked pon process (τ n, X n n N +, wh marked cumulave process N, and G s a generc enlargemen of H. We se H : H and G : G. In mos of he curren leraure, e.g. 3, 14, 29, 28 and 32, he sudy of non-lfe nsurance conracs s based on modelng he G-compensaor of N, snce he G-compensaor s nvolved n he prcng formula and n he calculaon of he hedgng sraegy. Whle n he case of lfe nsurance, he drec modelng approach and he compensaor approach concde, see e.g. 11 and 9, s however no he case for non-lfe nsurance as we explan n he followng. Defnon 4.1. The G-mark-predcable σ-algebra on he produc space R + B(R + Ω s he σ-algebra generaed by ses of he form (s, B A where < s <, B B(R + and A G s. Defnon 4.2. The G-compensaor of a marked pon process (τ n, X n n N + s any G-mark-predcable, cumulave process Λ(, B, ω such ha, (Λ(, B wh Λ(, B( : Λ(, B, s he G-compensaor of he pon process (N(, B. We use he noaon (Λ, Λ : Λ(, R +, o denoe he G-compensaor of he ground process (N. Theorem 14.2.IV(a of 16 shows ha gven a marked pon process (τ n, X n n N + wh fne frs momen measure, s G-compensaor Λ always exss and s (l P - a.e. unque, where l denoes he Lebesgue measure on R +. In parcular, for all (, B, ω R + B(R + Ω, he followng relaon holds Λ(, B, ω κ(b s, ωλ(ds, ω, (4.1 where κ(b s, ω, B B(R +, s, ω Ω, s he unque predcable kernel such ha for all A G s, < s <, B B(R +, N(u, B(ωduP (dω κ(b u, ωn u (ωdup (dω. A s A However, under general condons s no always rue ha gven a G-markpredcable and cumulave process Λ, here exss a marked pon process (τ n, X n n N + wh G-compensaor Λ. The problem s frs menoned n 21, where he case wh G H s solved. An exenon of he exsence heorem o he case of G F H,.e. when he flraons F and H are ndependen, s provded n 17. Furhermore whle he law of N s unquely deermned by he H-compensaor, hs s no rue for he G-compensaor. See dscusson n 21 and Secon 4.8 of 2. Consequenly, he leraure wh he compensaor approach s mosly lmed o he cases of G H, see e.g. 14, 29, 28, or G F H, see e.g s

26 In he followng we provde a suffcen condon n he general case of G F H, such ha he law of N s unquely deermned by Λ. Smlarly o e.g. 14, 29 and 28, we assume ha he G-compensaor of (τ n, X n n N + has he followng form Λ(, B λ s η s (dxds for all, B B(R +, (4.2 B where λ : (λ s a G-progressvely measurable process and he mappng η η : R + B(R + Ω (R +, B(R + (, B, ω η (B(ω, s such ha for every, ω Ω, η(,, ω s a probably measure on (R +, B(R +, and for every B B(R +, (η (B s a G-progressvely measurable process. Clearly, we have Λ λ s ds for all. In parcular, we can choose a predcable verson of boh λ and η, see Secon 14.3 of 16 for deals. The processes λ and η can be nerpreed respecvely as jump nensy and jump sze nensy. We recall ha a marked pon process (τ n, X n n N + has ndependen marks f he marks (X n n N are muually ndependen gven N. Proposon 4.3. The law of a smple marked pon process (τ n, X n n N + on (Ω, H wh fne frs momen measure, ndependen marks and of he form (4.2 s unquely deermned by λ and η. If furhermore λ s H-measurable, hen also he law of N on (Ω, G s unquely defned. Proof. By Proposon 6.4.IV(a of 16, he law of marked pon process wh ndependen marks s unquely deermned by he kernel κ and he dsrbuon of N. Accordng o relaons (4.1 and (4.2, he kernel κ s gven by κ(b, ω η (B(ω, (, B, ω R + B(R + Ω. Corollary of 2 and Theorem 14.2.IV(c of 16 show ha, f N s smple and of he form (4.2, he process (λ H deermnes unquely he dsrbuon of N on (Ω, H. If n addon λ s H-adaped, hen by Theorem of 2, also he dsrbuon of N on (Ω, G s unquely deermned. Neverheless, Proposon 4.3 requres he jump nensy process λ o be H- adaped n order o have N unquely defned n law, whch s an unnaural condon n our conex. On he conrary, he approach proposed n Secon 2 allows o ake no accoun a dependence srucure beween he flraons H and G by drecly modelng he F-adaped nensy process µ. Furhermore, hs allows o oban analycal resuls for valuaon formulas as shown n Secon 3. 26

27 5 Prcng n hybrd marke In hs secon we explan n deals he hybrd srucure of he combned marke. We fx a me horzon T wh < T <, and denoe he nflaon ndex process by I : (I,T, whch represens he percenage ncremens of he Consumer Prce Index (CPI and follows a nonnegave (P, F-semmarngale. We dsngush real prce value,.e. nflaon adjused, from nomnal prce value, whch can be convered n real value a any me, T, f dvded by he nflaon ndex I. If no oherwse specfed, all prce values are expressed n nomnal value. We consder d lqudly raded prmary asses on he fnancal marke descrbed by prce process vecor S : (S 1,..., S d,t, whch follows a real-valued (P, F- semmarngale. We assume ha here s a publcly accessble ndex, based on he nensy process µ and modelled by he process L : (L,T wh L : e Γ,, T, see e.g. 13. Ths ndex reflecs he underlyng sysemac rsk-facor relaed o he nsurance porfolo, such as moraly rsk, weaher rsk, car accden rsk, ec. We dsngush hree knds of prmary asses as elemens of he vecor S: 1. radonal fnancal asses, such as he zero-coupon bond, call and pu opons, fuures ec.; 2. nflaon lnked dervaves, such as nflaon lnked zero-coupon bond (called also zero-coupon Treasury Inflaon Proeced Secury, TIPS, whch pays off I T (equvalen o 1 real un a me T, nflaon lnked call and pu opons, ec.; 3. macro rsk-facor lnked dervaves based on he ndex L, such as longevy bond whch pays off L T a me T, weaher ndex-based dervaves, ec. We denoe by L(S, P, G he space of R d -valued G-predcable S-negrable processes. We call porfolo or value process S δ : (S δ,t assocaed o a radng sraegy δ : (δ,t n L(S, P, G he followng càdlàg adaped process S δ δ S d δs,, T. 1 I s called self-fnancng f S δ S δ + δ u ds u S δ + l 1 δ uds u,.t. We nroduce he followng se V x + {S δ self-fnancng : δ L(S, P, G, S δ x >, S δ > }. 27

28 Defnon 5.1. A benchmark or numérare porfolo S : (S,T s an elemen of V 1 +, such ha Ss δ S δ Ss G s, s,, T, s. S We follow he approach of 31 and work under he followng assumpon. Assumpon 5.2. There exss a benchmark porfolo S. In 19, s shown ha Assumpon 5.2 s weaker han assumng he exsence of an equvalen marngale measure. As dscussed n 4, hs weak no-arbrage assumpon s more suable for modelng a hybrd marke as n our case. Gven a generc random varable or process X, we denoe by ˆX : X/S he benchmarked value of X. The followng lemma s proved n 5. Lemma 5.3. If he vecor process of prmary asses S s connuous, hen he benchmarked vecor process Ŝ : S/S s a (P, G-local marngale. For he sake of smplcy, we assume he followng condons smlar o he ones n 9. Assumpon 5.4. The nflaon ndex process I (I,T and he vecor process of prmary asses S are connuous. The benchmark porfolo S (S,T s connuous, F-adaped, and he benchmarked value process Ŝ : S/S s an (F, P -rue marngale. Inflaon lnked zero-coupon bond (or TIPS s a prmary asse,.e. an elemen of he vecor S. The dvdend paymen n real un of he nsurance company owards polcyholders s modelled by a nonnegave (P, G-semmarngale D : (D,T. We denoe by A : (A,T he nomnal benchmarked cumulave paymen, namely A : I u Su dd u,, T. (5.1 Defnon 5.5. We call real world prcng formula assocaed o A he followng formula V : S A T A G S I u dd u G, (5.2 I I for, T.,T Su The value of V n (5.2 s expressed n real value,.e. nflaon adjused value. In parcular, we noe ha corresponds o he benchmarked rsk-mnmzng prce for he paymen process A a me, f A s square negrable,.e. sup A 2 <.,T Ths can be shown n he same way as n Appendx A of 9. 28

29 5.1 Prcng non-lfe nsurance clams In he seng oulned above, we now apply he resuls of Secon 3 o compue he real-world prcng formula for non-lfe nsurance clams, under he nerpreaon of Secon 2.2. The cumulave paymen a me relaed o -h polcy expressed n real value s gven by N j } X j Xj. The nomnal benchmarked cumulave paymen process A : (A,T s hence A : I s Ss dd s N n I τ j S 1 τj X j,, T. (5.3 The esmaon of A s called reserve problem n he cones of non-lfe nsurance, see 1. Unlke he lfe nsurance case, he rsk relaed o non-lfe nsurance polces s hence no only relaed o he accden self, bu also o he frs reporng delay (hs s he case of ncurred bu no repored clams, called IBNR clams, o he me and he sze of developmens afer he frs reporng. We now focus on prcng and hedgng he nomnal remanng paymen A T A, for, T. We assume ha he process I/S s F-condonally ndependen from τ1, for all 1,..., n, and ha he cumulave paymens relaed o marked pon processes ( τ, X j j N +, 1,..., n, Ñ X j,, T, 1,..., n, are..d. compound Posson processes,.e. Ñ are muually ndependen Posson processes wh parameer λ, and X j are..d. negrable nonnegave random varables ndependen from Ñ wh expecaon X j m. In hs case, we have m( λm,, T, where m s defned n (3.15. In vew of he above assumpons, all condons n Theorem 3.18 are sasfed n he case of Y A T A, for, T. Le R be he number of repored clams a me,.e. R : n 1 1 },, T. The real world prcng formula (5.2 ogeher wh Corollary 3.3, Theorem

30 and Assumpon 5.4 yelds I n V S A T A G 1 N n T I τ j S X j 1 jn τ F H j T I u λmr Su du H,1 F T + (n R λmr T + (n R Iu λmr (T I + (n R S u T S T N T I τ j S Xj jn τ j ( X 1 Iu S u + λm T u F G I v S dv dp ( τ v 1 u F u F e µudu + Ḡ( u ue µvdv µ u du du ( X 1 Iu S u + λm T u I v S dv dp ( τ v 1 u F u F e µudu + Ḡ( ue u µvdv µ u du ( X 1 Iu S u + λm T u I v S dv v dp ( τ 1 u F u F e µudu + Ḡ( ue u µvdv µ u du, (5.4 where he condonal probably funcon P ( τ 1 F s gven n (3.12,.e. P ( τ 1 F ( α e s µvdv µ s + s The frs componen on he lef-hand sde of (5.4 λmr (T I S g(s ue u µvdv µ u du ds. corresponds o already repored clams. We observe ha he valuaon of hs par does no nvolve any more he updang nformaon afer he frs reporng. The second componen on he rgh-hand sde of (5.4 (n R T ( X 1 Iu S u + λm T u I v S dv v dp ( τ 1 u Fu F e µudu + Ḡ( ue u µvdv µ u du, (5.5 whch can be furher explcly compued, corresponds o no repored clams and ncludes boh cases of ncurred bu no repored (IBNR clams as well as no ye ncurred clams. The sandard leraure of non-lfe nsurance s manly focused on 3

31 IBNR clams. However, for he prcng problem s more approprae o consder he enre expresson (5.4. As already menoned n Secon 5, hs prce equals he benchmarked rsk-mnmzng prce, f we assume square negrably of he clam. In parcular, usng he same argumens of Proposon 4.11 n 2 and Secon 4.1 of 9, we can calculae he assocaed benchmarked rsk-mnmzng sraegy. The form of V suggess how o desgn dervaves whch can be used o hedge rsks n hs marke model. In parcular, snce V s expressed n erms of he nensy process µ, he dsrbuon of θ and he dsrbuon of ( τ j, X j j N +, he benchmarked rsk-mnmzng sraegy can be explcly calculaed. For furher deals on he benchmarked rsk-mnmzaon mehod for non-lfe nsurance lables, we refer o 34. The mehod o derve he dsrbuon of µ can be found n 9. 6 Concluson In hs paper, we nroduce a new framework whch can be used boh for lfe and non-lfe nsurance modelng by generalzng he reduced-form seng. Ths overcomes he dffculy of nroducng a nonrval dependence beween he reference nformaon flow and he nernal nsurance nformaon flow. In parcular, n hs framework we are able o model he accden mes and her furher reporng, and o oban a he same me analycal valuaon formulas for nsurance producs. We apply hese resuls o he prcng of non-lfe nsurance lables n hybrd markes, by akng no accoun he role of nflaon and under a very weak no-arbrage condon on he combned fnancal and nsurance markes. Acknowledgemens The auhors would lke o hank Irene Schreber for neresng dscussons abou propery and casualy nsurance. References 1. Arjas. The clams reservng problem n non-lfe nsurance: some srucural deas. Asn Bullen, 19(2: , J. Barbarn. Rsk-mnmzng sraeges for lfe nsurance conracs wh surrender opon, 27. Avalable a SSRN: hp://ssrn.com/absrac or hp://dx.do.org/1.2139/ssrn J. Barbarn, T. De Launos, and P. Devolder. Rsk mnmzaon wh nflaon and neres rae rsk: applcaons o non-lfe nsurance. Scandnavan Acuaral Journal, 2: ,

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