The distribution of the interval of the Cox process with shot noise intensity for insurance claims and its moments

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1 The diibuion of he ineval of he Cox poce wih ho noie ineniy fo inuance claim and i momen Angelo Daio, Ji-Wook Jang Depamen of Saiic, London School of Economic and Poliical Science, Houghon See, London WCA AE, Unied Kingdom ( a.daio@le.ac.uk) Acuaial Sudie, Faculy of Commece and Economic, Univeiy of New Souh Wale, Sydney, NSW 5, Aualia ( j.jang@unw.edu.au) Abac. Applying piecewie deeminiic Makov pocee heoy, he pobabiliy geneaing funcion of he Cox poce, incopoaing wih ho noie poce a he claim ineniy, i obained. We alo deive he Laplace anfom of he diibuion of he ho noie poce a claim jump ime, uing aionay aumpion of he ho noie poce a any ime. Baed on hi Laplace anfom and fom he pobabiliy geneaing funcion of he Cox poce wih ho noie ineniy, we obain he diibuion of he ineval of he Cox poce wih ho noie ineniy fo inuance claim and i momen, i.e. mean and vaiance. Key wod: Piecewie deeminiic Makov pocee heoy; maingale; opping ime; he diibuion of he ineval he Cox poce wih ho noie ineniy; inuance claim.. Inoducion In inuance modeling, he Poion poce ha been ued a a claim aival poce. Exenive dicuion of he Poion poce, fom boh applied and heoeical viewpoin, can be found in Camé (93), Cox and Lewi (966), Bühlmann (97), Cinla (975), Gebe (979) and Medhi (98). Howeve hee ha been a ignifican volume of lieaue ha queion he uiabiliy of he Poion poce in inuance modeling (Seal 983 and Bead e al. 984). Fom a pacical poin of view, hee i no doub ha he inuance induy need a moe uiable claim aival poce han he Poion poce ha ha deeminiic ineniy. A an alenaive poin poce o geneae he claim aival, we can employ he Cox poce o a doubly ochaic Poion poce (Cox 955; Bale 963; Haigh 967; Sefozo 97; Gandell 976, 99, 997; Bémaud 98 and Lando 994). An

2 impoan book on Cox pocee i he book by Benning and Koolev, whee he applicaion in boh inuance and finance ae dicued. The Cox poce povide u wih he flexibiliy o allow he ineniy no only o depend on ime bu alo o be a ochaic poce. In a ecen pape (Daio and Jang 3), we demonaed how he Cox poce wih ho noie ineniy could be ued in he picing of caaophe einuance and deivaive. I i impoan o meaue he ime ineval beween he claim in inuance. Thu in hi pape, we examine he diibuion of he ineval of he Cox poce wih ho noie ineniy fo inuance claim. The eul of hi pape can be ued o eaily modified in compue cience/elecommunicaion modeling, elecical engineeing and queueing heoy. We a by defining he quaniy of inee; hi i he doubly ochaic (wih a honoie ineniy) poin poce of claim aival. Then we deive he pobabiliy geneaing funcion of he Cox poce wih ho noie ineniy uing piecewie deeminiic Makov pocee (PDMP) heoy, whoe heoy wa developed by Davi (984). The piecewie deeminiic Makov pocee heoy i a poweful mahemaical ool fo examining non-diffuion model. Fo deail, we efe he eade o Davi (984), Daio (987), Daio and Embech (989), Jang (998, 4), Rolki e al. (999), Daio and Jang (3) and Jang and Kvavych (4). In ecion 3, we deive he Laplace anfom of he diibuion of he ho noie poce a claim ime, uing aionay aumpion of he ho noie poce a any ime. Uing hi Laplace anfom wihin he pobabiliy geneaing funcion of he Cox poce wih ho noie ineniy, we deive he diibuion of he ineval of he Cox poce wih ho noie ineniy fo inuance claim and i momen, i.e. mean and vaiance. In ecion 4 conclude.. The Cox poce and he ho noie poce The Cox poce (o a doubly ochaic Poion poce) can be viewed a a wo-ep andomiaion pocedue. A poce λ i ued o geneae anohe poce N by acing a i ineniy. Tha i, N i a Poion poce condiional on λ which ielf i a ochaic poce (if λ i deeminiic hen N i a Poion poce). Many alenaive definiion of a doubly ochaic Poion poce can be given. We will offe he one adoped by Bémaud (98).

3 Definiion. Le ( Ω, F, P) be a pobabiliy pace wih infomaion ucue given by {, [, ]} F = I T. Le N be a poin poce adaped o F. Le λ be a non- negaive poce adaped o F uch ha λ d < almo uely (no exploion). If fo all and u R iu( N ) { } ( ) N λ iu E e I = exp e λ d (.) hen N i called a I -doubly ochaic Poion poce wih ineniy λ whee λ i he σ algeba geneaed by λ up o ime, i.e. I = σ{ λ; }. λ I Equaion (.) give u P exp d d λ λ λ = (.) k! { N N = k ; } and P { τ > λ; } = P{ N N ; } = λ = exp λd (.3) h h whee τ k denoe he lengh of he ime ineval beween he ( k ) and he k claim. Theefoe fom (.3), we can eaily find ha P( τ ) = E λ exp( λ ) d. (.4) k If we conide he poce can alo eaily find ha Λ = λd (he aggegaed poce), hen fom (.) we ( θ )( Λ ) { } N N Λ E( θ ) = E e. (.5) The equaion (.5) ugge ha he poblem of finding he diibuion of N, he poin poce, i equivalen o he poblem of finding he diibuion of, he aggegaed poce. I mean ha we ju have o find he p.g.f. (pobabiliy geneaing funcion) of N o eieve he m.g.f. (momen geneaing funcion) of Λ and vice vea. Λ One of he pocee ha can be ued o meaue he impac of pimay even i he ho noie poce (Cox and Iham, 98, 986 and Klüppelbeg and Mikoch, 995). The ho noie poce i paiculaly ueful wihin he claim aival poce a i 3

4 meaue he fequency, magniude and ime peiod needed o deemine he effec of pimay even. A ime pae, he ho noie poce deceae a moe and moe claim ae eled. Thi deceae coninue unil anohe even occu which will eul in a poiive jump in he ho noie poce. Theefoe he ho noie poce can be ued a he paamee of doubly ochaic Poion poce o meaue he numbe of claim due o pimay even, i.e. we will ue i a a claim ineniy funcion o geneae he Cox poce. We will adop he ho noie poce ued by Cox and Iham (98): λ M = λe + i= Ye i ( Si ) whee λ i iniial value of λ. Y = i a equence of independen and idenically diibued andom { i} i,, vaiable wih diibuion funcion Gy ( ) ( y > ), whee EY ( i ) =. S = i he equence epeening he even ime of a Poion poce { i} i,, M wih conan ineniy ρ. i ae of exponenial decay. We aume ha he Poion poce of each ohe. M and he equence { } i i,, Y = ae independen The Cox poce w ih ho noie ineniy 5 4 No. of poin

5 The geneao of he poce ( Λ, λ, ) acing on a funcion f ( Λ, λ, ) belonging o i domain i given by f f f Α f ( Λ, λ, ) = + λ λ + ρ[ f( Λ, λ+ y, ) dg( ) ( Λ,, )] Λ λ y f λ. (.6) Fo f ( Λ, λ, ) o belong o he domain of he geneao Α, i i ufficien ha f ( Λ, λ, ) i diffeeniable w... Λ, λ, fo all Λ, λ, and ha f(, λ+ y,) dg( y) f(, λ,) <. Le u find a uiable maingale in ode o deive he p.g.f. (pobabiliy geneaing funcion) of a ime (Daio 987). N * Theoem. Le u aume ha Λ and λ evolve up o a fixed ime. Conideing conan k and k ae uch ha k and k k e *, ^ exp( k ) exp { ( k ke Λ + ) λ} exp ρ { gk ( + ke )} d (.7) i a maingale. Poof Define W = Λ + λ and Z = λe, hen he geneao of he poce ( W, Z, ) acing on a funcion f ( wz,, ) i given by f Α f ( wz,, ) = + ρ[ f( w+ yz, + ye, dgy ) ( ) f( wz,, )]. (.8) and f ( wz,, ) ha o aify Α f = fo f ( W, Z, ) o be a maingale. Seing kw kz e e h( ) we ge he equaion ^ h'( ) ρ[ g( k+ ke )] h( ) =. (.9) kw kz e e h( ) belong o he domain of he geneao becaue of ou choice of k, ; he funcion i bounded fo all Solving (.9) h () * whee Κ i an abiay conan. Theefoe k * and ou poce evolve up o ime only. ^ ρ { g ( k+ ke )} d =Κe (.) ^ ρ { g ( k+ ke )} d kw k Z e e e i a maingale and hence he eul follow. 5

6 Coollay.3 Le ν, ν, ν, θ and be fixed ime. Then ν( Λ Λ ) νλ Ee { e Λ, λ } ( ) exp { ν ( ν ) e } exp [ g{ ν ( ν = ν λ + ρ + ν ) e }] d (.) and ( N N ) νλ E{ θ e N, λ } θ θ ( ) exp { ( ) } exp [ { θ ( θ ν e λ = + ρ g ) }]. + ν e d (.), Poof ν ν * We e k =, k = ( ν ) e, in Theoem. and (.) follow immediaely. (.) follow fom (.) and (.5). Now we can eaily deive he p.g.f. (pobabiliy geneaing funcion) of N and he Laplace anfom of λ uing Coollay.3. Coollay.4 The pobabiliy geneaing funcion of N i given by ( N N ) θ ( ) θ E{ θ λ } = exp { e } exp [ { ( }] λ ρ g e d, (.3) he Laplace anfom of he diibuion of λ i given by ^ νλ { e } = exp( νλ e ) exp ρ { g( νe d E λ )} (.4) and if λ i aympoic (aionay), i i given by ^ νλ ( Ee ) = exp ρ { g( νe )} d (.5) which can alo be wien a ν νλ ρ Ee ( ) = exp G ( udu ) (.6) whee G ( u) = gu u ( ). Poof If we e ν = in (.) hen (.3) follow. (.4) follow if we eihe e ν = in (.) o e θ = in (.). Le in (.4) and he eul follow immediaely. 6

7 If we diffeeniae (.4) and (.6) wih epec o θ and pu θ =, we can eaily obain he fi momen of λ, i.e. ρ ρ E( λ λ ) = + ( λ ) e (.7) and ρ E ( λ ) =. (.8) The highe momen can be obained by diffeeniaing hem fuhe, i.e. ρ Va( λ λ ) = ( e ) (.9) and whee EY ( ) ydg ( y). = = ρ Va ( λ ) = (.) 3. The diibuion of he ineval of he Cox poce wih ho noie ineniy and i momen Le u examine he Laplace anfom of he diibuion of he ho noie ineniy a h claim ime. To do o, le u denoe he ime of he n claim of N by τ n and denoe he value of λ, when N ake he value n fo he fi ime by λ τn. Since a claim occu a ime τ, hi implie ha he ineniy a claim ime, λ τ, hould be highe han he ineniy a any ime λ. Theefoe he diibuion of λ τ hould no be he ame a he diibuion of λ. Le u a wih he following lemma in ode o obain he Laplace anfom of he diibuion of he ho noie ineniy a claim ime. We aume ha he claim and jump (o pimay even) in ho noie ineniy do no occu a he ame ime. Lemma 3. Suppoe ha f ( λ ) i a funcion belonging o i domain and fuhemoe ha i aifie If h( λ ) i uch ha hen lim E{ f( λ )exp( λ d) λ } =. (3.) λ{( h λ) f ( λ)} + Α f ( λ) = (3.) E{( h λ ) λ } = f ( λ ). (3.3) τ 7

8 Poof Fom (3.) f ( λ ) + [ λ{ h( λ ) f ( λ )}] d i a maingale and ince τ i a opping ime, whee P( τ ) = P( N > ) and N i λ -meauable, we have E τ { f ( λτ λ ) E[ [ λ { h( λ ) f ( λ )}] d λ ] f ( λ ) + =. (3.4) If we now place a condiion on he ealiaion λ v ; v, hen he fi em of he lef-hand ide in (3.4) i E{ f( λτ λ )} = E{ f( λτ ) ; } ( ; ) λv v dp λv v (3.5) and he econd em of he lef-hand ide in (3.4) i τ τ Ω E[ [ λ { h( λ ) f ( λ )}] d λ ] = E[ λ { h( λ ) f ( λ )} d λ ; v ] dp( λ ; v ) v v Ω (3.6) whee dp( λ ; v ) i he pobabiliy diffeenial of a paicula ealiaion in Ω, v he e of all poible ealiaion. Since τ i diibued wih deniy, λ exp( λ d) on (, ) and a ma, exp( and τ λ d) a, condiionally on λ ;, whee N i he Cox poce, we have E{ f ( λτ ) λ ; ν } { f ( λ ) λ exp( λ d)} d f ( λ )exp( λ d) = + (3.7) v E[ λ{ h( λ) f( λ)} d λv; ν ] = [ λ { h( λ ) f ( λ )} d P( τ = )] d + λ{ h( λ ) f ( λ )} d P( τ > = [ λ { h( λ ) f ( λ )} d λ exp( λ d)] d + λ { h( λ ) f ( λ )} d exp( λ d) = [ λ exp( λd) d] λ{ h( λ ) f ( λ )} d + λ{ h( λ ) ) f ( λ )} d exp( λ d) 8

9 = {exp( λd) exp( λd)} λ{ h( λ ) f ( λ )} d + λ{ h( λ ) = f ( λ )} d exp( λ d) λ { h( λ ) f ( λ )}exp( λ d) d. (3.8) Pu = in (3.8), hen we have E[ τ λ { h( λ ) f ( λ )} d λ, ν ] = v λ { h( λ ) f ( λ )}exp( λ d) d. (3.9) Theefoe (3.4) become τ E{ f ( λτ λ ) E[ [ λ { h( λ ) f ( λ )}] d λ ] + ) = [ f( λ ) λ exp( λ d) d+ f( λ )exp( λ d)] dp( λ ; v Ω v + [ λ { h( λ ) f ( λ )}exp( λ d) d] dp( λ ; v ) Ω v = f ( λ )exp( λd) dp( λ ; v ) + [ λh( λ )exp( λd) d] dp( λ ; v ) Ω v v Ω v Ω = E{ f( λ )exp( λ d) λ } + [ h( λ ) λ exp( λ d) d] dp( λ ; v ) (3.) = f ( λ ). Leing in (3.), hen fom (3.), he fi em in he lef-hand ide end o and he econd em o have E{( h λ ) λ } = f ( λ ). τ E{( h λ τ ) λ }, a λ exp( λd) i a deniy. Theefoe we Auming ha he ho noie poce λ i aionay, le u deive he Laplace anfom of he diibuion of he ho noie poce a claim ime, λ τ. Theoem 3. If he ho noie poce λ i aionay, he Laplace anfom of he diibuion of he ho noie poce a claim ime i given by ν νλ G ν ρ τi ( ) = ( ) E e exp G ( u ) du. (3.) Poof Fom Lemma 3., which implie ha if f ( λ ) and h( λ ) ae uch ha 9

10 λ{( h λ) f( λ)} λ f '( λ) + ρ{ f( λ+ y) dg() y f( λ)} = (3.) and (3.) i aified, we have { λ λ } E h( τ ) ( i τ = f λ ) + i τ (3.3) i g '( ν ) by aing he poce fom τ i. Employing f ( λ) = λ e g( ν ) f ( λ ) clealy aifie (3.) and ubiuing ino (3.), hen we have νλ, he funcion g'( ν) g'( ν) λ h( λ) λe + e νλ λ e λe + g( ν) g( ν) νλ g'( ν) g'( ν) ρ = e λg'( ν) + g'( ν) g( ν) λ e g( ν) g( ν) νλ νλ νλ νλ { ν } νλ =ρλe g( ). Divide by λ and implify hen we have g'( ν) νλ νλ νλ νλ h( λ) = λe ( ν) + e ( ν) e + ρe { g( ν) }. (3.4) g( ν) Fom (3.3), i i given ha E{ h( λ )} = E E{ h( λ ) λ } = E{ f( λ ) i+ i+ i τi So pu (3.4) ino (3.5), hen νλ τ τ τ } g '( ν ) i+ i+ i+ g ( ) i+ i+. (3.5) E[ λ exp( νλ )( ν ) + exp( νλ ) ( ν ) exp( νλ ) + ρ exp( νλ ){ g( ν )}] τ τ τ τ τ ν g '( ν ) = E{ λτ exp( νλ ) exp( )}. i τ νλ i+ τi+ g( ν ) (3.6) When he poce λ i aionay, λ τi+ and λ τi have he ame diibuion whoe i Laplace anfom we denoe by H( ν ) E( e νλ ) τ =. Theefoe fom (3.6), we have g '( ν ) ( ν) H'( ν) ( ν) H( ν) + [ + ρ{ g( ν)}] H( ν) g( ν ) g '( ν ) =H'( ν ) H( ν ). g( ν ) (3.7) Divide boh ide of (3.7) by ν, hen we have

11 g'( ν) ρ g( ν) H'( ν ) + H( ν) + { + } H ( ν ) =. (3.8) g( ν) ν ν Solving (3.8), ubjec o H( ) = (3.9) hen he Laplace anfom of a diibuion of he ho noie poce a claim ime i given by g( ν) ρ H( ) K exp ν ν = G ( u) du ν whee K i a conan. Theefoe fom (3.9), K = and g( ν ) ρ ν G( ν ) ρ ν H ( ν ) = exp G ( u) du = exp G( u) du. (3.) ν Equaion (3.) povide u wih a vey ineeing eul ha hi i he diibuion of he um of wo andom vaiable; one having he aionay diibuion of λ (ee Coollay.4) and he ohe, having deniy G( y), whee G( y) = G( y). Compaing i wih he diibuion of he ho noie poce, λ a any ime, we can eaily find ha G( ν) ρ ν ρ ν exp G ( udu ) > exp G ( udu ). I i heefoe he cae ha λ τ i ochaically lage han λ. In ohe wod, he ineniy a claim ime i highe han he ineniy a any ime. Now le u deive he diibuion of he ineval of he Cox poce wih ho noie ineniy fo inuance claim uing Theoem 3.. Coollay 3.3 Aume ha i he ime a which a claim of N ha occued and he aionay of λ ha been achieved. Then he ail of he diibuion of he ineval of he Cox poce wih ho noie ineniy i given by G( e ) ρ P( τ > ) = exp ( e ) G d. (3.)

12 Poof Fom (.3), he pobabiliy geneaing funcion of N i given by N θ θ E( θ λ) = exp { ( e ) λ} exp ρ [ g{ ( e )}] d. (3.) Se θ = in (3.) and ake expecaion, hen he ail of he diibuion of τ i given by { ( e ) e P( τ > ) = exp ρ { g( )} d E exp λ. (3.3) } Subiue (3.) ino (3.3), hen he eul follow immediaely a i he ime a which a claim ha occued and λ i aionay. Coollay 3.4 The expecaion and vaiance of he ineval beween claim ae given by E( τ) = P( τ > ) d = (3.4) ρ and u u G( e ) ρ ( τ ) [ exp{ ( ) }] G Va = u e d du ρ. (3.5) Poof Inegae (3.), hen (3.4) follow. (3.5) i obained fom u u G( e ) ρ = = E( τ ) f ( ) d [ u exp{ G ( e ) d}] du. An ineeing eul we can find fom (3.4) and (.8) i ha he expeced ineval beween claim i he invee of he expeced numbe of claim, whee he numbe of claim follow he Cox poce wih ho noie ineniy, which i alo he cae fo a Poion poce. 4. Concluion We aed wih deiving he pobabiliy geneaing funcion of he Cox poce wih ho noie ineniy, employing piecewie deeminiic Makov pocee heoy. I wa neceay o obain he diibuion of he ho noie poce a claim ime a i i no he ame a he diibuion of he ho noie poce a any ime, i.e. he ineniy

13 of claim ime ae highe han he ineniy a any ime. Auming ha he ho noie poce i aionay, we deived he diibuion of he ineval of he Cox poce wih ho noie ineniy fo inuance claim and i momen fom i pobabiliy geneaing funcion. The eul of hi pape can be ued o eaily modified in compue cience/elecommunicaion modeling, elecical engineeing and queueing heoy a an alenaive couning poce of a Poion poce. Refeence Bale, M. S. (963) : The pecal analyi of poin pocee, J. R. Sa. Soc., 5, Bead, R.E., Penikainen, T. and Peonen, E. (984) : Rik Theoy, 3 d Ediion, Chapman & Hall, London. Bening, E. and Koolev, V. Y. () : Genealied Poion Model and hei Applicaion in Inuance and Finance, VSP, Uech. Bémaud, P. (98) : Poin Pocee and Queue: Maingale Dynamic, Spinge- Velag, New Yok. Bühlmann, H. (97) : Mahemaical Mehod in Rik Theoy, Spinge-Velag, Belin-Heidelbeg. Cinla, E. (975) : Inoducion o Sochaic Pocee, Penice-Hall, Englewood Cliff. Cox, D. R. (955) : Some aiical mehod conneced wih eie of even, J. R. Sa. Soc. B, 7, Cox, D. R. and Iham, V. (98) : Poin Pocee, Chapman & Hall, London. Cox, D. R. and Iham, V. (986) : The viual waiing ime and elaed pocee, Adv. Appl. Pob. 8, Cox, D. R. and Lewi, P. A. W. (966) : The Saiical Analyi of Seie of Even, Meheun & Co. Ld., London. Camé, H. (93) : On he Mahemaical Theoy of Rik, Skand. Jubilee Volume, Sockholm. Daio, A. (987) : Inuance, Soage and Poin Poce: An Appoach via Piecewie Deeminiic Makov Pocee, Ph. D Thei, Impeial College, London. Daio, A. and Embech, P. (989) : Maingale and inuance ik, Commun. Sa.-Sochaic Model, 5(), 8-7. Daio, A. and Jang, J. (3) : Picing of caaophe einuance & deivaive uing he Cox poce wih ho noie ineniy, Finance & Sochaic, 7/, Davi, M. H. A. (984) : Piecewie deeminiic Makov pocee: A geneal cla of non diffuion ochaic model, J. R. Sa. Soc. B, 46,

14 Gebe, H. U. (979) : An Inoducion o Mahemaical Rik Theoy, S. S. Huebne Foundaion fo Inuance Educaion, Philadelphia. Gandell, J. (976) : Doubly Sochaic Poion Pocee, Spinge-Velag, Belin. Gandell, J. (99) : Apec of Rik Theoy, Spinge-Velag, New Yok. Gandell, J. (997) : Mixed Poion pocee, Chapman and Hall, London. Jang, J. (998) : Doubly Sochaic Poin Pocee in Reinuance and he Picing of Caaophe Inuance Deivaive, Ph. D Thei, The London School of Economic and Poliical Science. Jang, J. (4) : Maingale appoach fo momen of dicouned aggegae claim, Jounal of Rik and Inuance, 7/, -. Jang, J. and Kvavych, Y. (4); Abiage-fee pemium calculaion fo exeme loe uing he ho noie poce and he Eche anfom, Inuance: Mahemaic & Economic, 35/, 97-. Klüppelbeg, C. and Mikoch, T. (995) : Exploive Poion ho noie pocee wih applicaion o ik eeve, Benoulli,, Lando, D. (994) : On Cox pocee and cedi iky bond, Univeiy of Copenhagen, The Depamen of Theoeical Saiic, Pe-pin. Medhi, J. (98) : Sochaic Pocee, Wiley Eaen Limied, New Delhi. Rolki, T., Schmidli, H., Schmid, V. and Teugel, J. (998) : Sochaic Pocee fo Inuance and Finance, John Wiley & Son, UK. Seal, H. L. (983) : The Poion poce: I failue in ik heoy, Inuance: Mahemaic and Economic,, Sefozo, R. F. (97) : Condiional Poion pocee, J. Appl. Pob., 9,

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