Optimal Reinsurance/Investment Problems for General Insurance Models


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1 Opimal Reinsurance/Invesmen Problems for General Insurance Models Yuping Liu and Jin Ma Absrac. In his paper he uiliy opimizaion problem for a general insurance model is sudied. he reserve process of he insurance company is described by a sochasic differenial equaion driven by a Brownian moion and a Poisson random measure, represening he randomness from he financial marke and he insurance claims, respecively. he random safey loading and sochasic ineres raes are allowed in he model so ha he reserve process is nonmarkovian in general. he insurance company can manage he reserves hrough boh porfolios of he invesmen and a reinsurance policy o opimize a cerain uiliy funcion, defined in a generic way. he main feaure of he problem lies in he inrinsic consrain on he par of reinsurance policy, which is only proporional o he claimsize insead of he curren level of reserve, and hence i is quie differen from he opimal invesmen/consumpion problem wih consrains in finance. Necessary and sufficien condiions for boh wellposedness and solvabiliy will be given, by modifying he dualiy mehod in finance, and wih he help of he solvabiliy of a special ype of Backward Sochasic Differenial Equaions. Keywords: CramérLundburg reserve model, proporional reinsurance, opimal invesmen, Girsanov ransformaion, dualiy mehod, backward sochasic differenial equaions. 2 Mahemaics Subjec Classificaion: 6H1, Secondary: 34F5, 93E3 Deparmen of Mahemaics, Purdue Universiy, Wes Lafayee, IN ; his auhor is suppored in par by he Sociey of Acuaries CKER gran # Deparmen of Mahemaics, Purdue Universiy, Wes Lafayee, IN ; Sociey of Acuary CKER gran # his auhor is suppored in par by NSF grans #24332, #55427, and he 1
2 1 Inroducion Opimizaion Proporional reinsurance problems have been considered by many auhors in recen years. We refer o he books of Gerber (197) [6], and Bühlmann (197) [3] for he basic idea of proporional reinsurance, and o, say, [7] for he reamen for diffusion models. However, in mos of he previous works he dynamics of he reinsurance problems, ha is, he reserve processes, were usually resriced o he raher simplisic model, such as classical CramérLundberg model, or is simple perurbaions. One of he consequences of such seings was ha he resuls and mehodology used in solving such problems depend, explicily or implicily, on he Markovian naure of he reserve process. he generalizaion of hese resuls o a more realisic environmen is herefore raher difficul. In fac, i is sill raher afresh. his paper is an aemp in his direcion. We shall consider a generalized insurance model as was proposed in MaSun [13]. More precisely, le us consider a risk reserve process, denoed by by X, ha akes he following form: X = x + c s (1 + ρ s )ds + f(s, z)n p (dsdz), (1.1) where c s is he premium rae process, ρ = {ρ is he socalled safey loading process. he las sochasic inegral represens a general claim process in which N p is he couning measure generaed by saionary Poisson poin process p; and f represens he inensiy of he jumps (deailed characerizaions of hese quaniies will be given in 2). Our opimizaion problem is based on he following consideraion: we suppose ha insurance company can manage is reserve, whence risk, in hree ways: invesmen, (proporional) reinsurance, and consumpion. More precisely, we assume ha he insurance company pus is reserve in a financial marke ha conains 1 riskless accoun and some risky asses, and i is allowed o change is invesmen posiions coninuously. Also, we assume ha he insurance company can diver (cede) a fracion of he incoming claims, while yielding a fracion of is premium a he same ime, o a reinsurance company. Finally, he insurance company is also allowed o consume (in he form of dividend, refund, ec.). he goal of he insurance company is hen o opimize cerain uiliy by managing he invesmen porfolio, reinsurance policy, and he consumpion. We should noe ha since a reinsurance policy mus ake values in [, 1], our opimizaion problem seems o resemble he uiliy opimizaion problem wih porfolio consrains. We refer he readers o, e.g., KarazasShreve [1, 11] for he opimal invesmen/consumpion problems wih coninuous models, o Xue [21] for heir jumpdiffusion counerpars, and o CvianicKarazas [4] for 2
3 he resuls involving porfolio consrains. I is worh noing ha despie he similariy of our problem and he uiliy opimizaion problems in finance, he naure of a reinsurance problem produces various subleies when similar mehodology are applied. For example, in he insurance models he jumps come from he claims, which is independen of he marke, hus i canno be reaed as par of asses like in [21]. Also, since i is no pracical o assume ha he reinsurance policy is proporional o he curren level of reserve, he dynamics of our wealh (reserve) process is no linear and homogeneous, a fundamenal feaure in mos of he exising frameworks for finance problems. As a maer of fac, i is such nonhomogeneiy ha causes he main echnically difficulies in his work. We noe ha in order o avoid overcomplicaing he model we shall relax he admissibiliy requiremens on he porfolio par alhough more consrain can be deal wih in a similar way. In paricular we shall allow shorselling and borrowing wih he same ineres rae, so ha no resricions are need on he bounds and signs of he porfolios. Neverheless, he special naure of he reinsurance and he generaliy of he reserve already provide significan novely in he heory of uiliy opimizaion. o our bes knowledge, such problems have no been fully explored, especially under an acuarial conex. he main resuls of his paper focus on wo aspecs: he wellposedness of he opimizaion problem and he acual resoluion of he opimal sraegy. he firs par of he resuls include he sudy of admissible sraegies, and he acual exisence of such sraegies. Afer a careful sudy of he reinsurance srucure, via he socalled profimargin principle, we derive a reasonable risk reserve model wih reinsurance and invesmen. Such a model is a naural exension of he simples ones as one usually sees in any elemenary acuarial lieraure (wihou diffusion approximaions). he admissibiliy of he porfolio/reinsurance/consumpion riple is hen defined so ha he insurance company does no go defaul over a given planning horizon. Due o he consrain on he reinsurance par, he exisence of such admissible riple becomes a raher echnical issue. In fac, he verificaion of he exisence of admissible sraegy relies on a new resul on he socalled backward sochasic differenial equaions, which is ineresing in is own righ. Our main resul on he exisence of admissible sraegies is hen proved along he lines of he resul of [5], wih some necessary modificaions. We should noe ha our reinsurance policy has o depend on he sizes of he claims. echnically, such a resricion can be removed if he process S has fixed size jumps (cf. e.g., [21]), bu his is no of significan ineres because i will exclude even he simples compound Poisson claim processes. Finally, we would like o poin ou ha our uiliy opimizaion problem are formulaed 3
4 slighly differen from he radiional ones, due o some echnical assumpions needed in order o guaranee he exisence of admissible sraegies. In paricular, we will require ha he uiliy funcion for he erminal reserve o be a runcaed version so ha he erminal wealh of he opimal reserve is bounded. We should noe ha every uiliy funcion can be approximaed by a runcaed sequences, hus an εopimal sraegy could be produced using our resul. Also, i is worh noing ha our final resul rely on he solvabiliy of a special forwardbackward sochasic differenial equaion (FBSDEs for shor, cf., e.g., MaYong [14] for more deails on such equaions). Bu he exisence of he soluion o he presen FBSDE is by no means rivial, and seems o be beyond he scope of all he exising resuls. We will no pursue all hese issues in his paper due o he lengh of he paper, bu we hope o be able o address hem in our fuure publicaions. his paper is organized as follows. In secion 2 we give he necessary preliminaries abou our model. In secion 3 we describe he admissibiliy of he invesmenreinsuranceconsumpion sraegies, and inroduce some equivalen probabiliy measures ha are imporan in our discussion. In secion 4 we inroduce he widersense admissible sraegies and prove he exisence of such sraegies, and in secion 5 we derive a sufficien condiion for he exisence of rue admissible sraegies. he las secion, is devoed o he uiliy opimizaion problem. 2 Preliminaries and Reserve Model Formulaions hroughou his paper we assume ha all uncerainies come from a common complee probabiliy space (Ω, F, lp) on which is defined a ddimensional Brownian moion W = {W :, and a saionary Poisson poin process p. We assume ha W and p are independen, which will represen he randomness from he financial marke and he insurance claims, respecively. For noaional clariy, we denoe F W = {F W : and F p = {F p : o be he filraions generaed by W and p, respecively, and denoe F = F W F p, wih he usual lpaugmenaion such ha i saisfies he usual hypoheses (cf. e.g., Proer [15]). Furhermore, we shall assume ha he poin process p is of class (QL) (cf. [8] or [9]), and denoe is corresponding couning measure by N p (ddz). he compensaor of N p (ddz) is hen ˆN p (ddz) = E(N p (ddz)) = ν(dz)d, where ν(dz) is he Lévy measure of p, saisfying ν( ) <, where = (, ). Le us specify some noaions in his paper. Le le be a generic Euclidean space. Regardless of is dimension we denoe, and o be is inner produc and norm, respecively. he following spaces will be frequenly used: 4
5 C([, ]; le) is he space of all le valued coninuous funcions on [, ]; for any subσfield G F and 1 p <, L p (G; le) denoes he space of all levalued, Gmeasurable random variables ξ such ha E ξ p <. As usual, ξ L (G; le) means ha i is Gmeasurable and bounded. for 1 p <, L p (F, [, ]; le) denoes he space of all levalued, Fprogressively measurable processes ξ saisfying E ξ p d <. he meaning of L (F, [, ]; le) is defined similarly. F p (resp. F 2 p ) denoes he class of all random fields ϕ : Ω, such ha for fixed z, he mapping (, ω) ϕ(, z, ω) is F p predicable, and ha E ϕ(s, z) ν(dz)ds <, (resp. E ϕ(s, z) 2 ν(dz)ds <.) (2.1) Le us now give more specificaions on he claim process + S = f(s, z)n p (dsdz),. (2.2) We noe ha if he inensiy f(s, z) z and ν( ) = λ >, hen S is simply a compound Poisson process. Indeed, in his case one has S = s< s Dp S = k 1 S k 1 {k, wih p = S being a Poisson poin process, D p = { : p = k=1 { k, and P {p k dz = 1 λ ν(dz), for all k 1. Furhermore, N = k=1 1 {k is a sandard Poisson process wih inensiy λ >, and S can be rewrien as S = N k=1 p k. (cf. [8], [9]). In wha follows we shall make use of he following imporan assumpion on he claim densiy f: (H1) he random field f F p, and i is coninuous in, and piecewise coninuous in z. Furhermore, here exis consans < d < L such ha d f(s, z, ω) L, (s, z) [, ), P a.s. (2.3) We remark ha he compac suppor in (H1) reflecs he simple fac in insurance: he deducible and benefi limi, and his is possible because ν( ) <. Alhough mahemaically we can replace such an assumpion by cerain inegrabiliy assumpions on boh f and f 1 agains he Lévy measure ν, or ha f has a cerain compac suppor in z, we prefer wriing i in his simple way because of is pracical meaning. 5
6 A. Reserve model wih reinsurance. Le us now look more closely ino our basic reserve process X. Le us recall (1.1) (we wrie down again for ready reference): X = x + c s (1 + ρ s )ds + f(s, z)n p (dsdz). (2.4) In ligh of he wellknown equivalence principle in acuarial mahemaics (cf. Bowers e al. [2]), we see ha he premium process {c can be quaniaively specified by he following equaion: c = E{ S F p = f(, z)ν(dz),, P a.s. (2.5) Moreover, i is common o require ha he premium and he expenseloading saisfy he following ne profi condiion : essinf ω Ω { c (ω)(1 + ρ (ω)) f(, z, ω)ν(dz) >,. (2.6) We summarize he above ino he following sanding assumpion. (H2) he safey loading process ρ is a bounded, nonnegaive F p adaped process, and he he premium process c is an F p adaped saisfying (2.5). Furhermore, he processes c, ρ, saisfy he ne profi condiion (2.6). We remark ha in he simplified case when f(, z) z, ha is, he claim process is simply a compound Poisson, and ρ is a consan, one has c s = c = zν(dz) = λe[u 1 ], where U 1 = S 1 is he jump size of he claim. In his case (2.6) becomes c(1+ρ) > λe(u 1 ), a usual ne profi condiion (cf. AsmussenNielsen [1]). We now give he definiion of a (generalized) reinsurance policy. Definiion 2.1 A (proporional) reinsurance policy is a random field α : [, ) Ω [, 1] such ha α F p, and ha for each fixed z, he process α(, z, ) is predicable. Given a reinsurance policy α, he par of he claim ha a insurance company reains o iself during any ime period [, + ] is assumed o be [α S] +, where [α S] α(s, z)f(s, z)n p (dzds). In oher words, he par of he claims i cedes o he reinsurer is [(1 α) S] +. Remark 2.2 he dependence of a reinsurance policy α on he spaial variable z amouns o saying ha he proporion can depend on he sizes of he claims, which is no unusual in pracice. Alhough a radiional reinsurance policy as a predicable process α,, migh be simpler o rea from modeling poin of view, i is noed (as we shall see) ha in 6 =
7 general one may no be able o find an opimal sraegy in such a form, unless S has fixed size jumps (i.e., ν(dz) is a discree measure). Bu such a case is obviously no of significan ineres because i will even exclude he general compound Poisson claim processes. We now give a heurisic derivaion of he reserve equaion wih reinsurance using he socalled profi margin principle. Le us denoe he safey loading or he reinsurance company by ρ r, and he modified safey loading of he original (ceden) company afer reinsurance by ρ α. Consider an arbirary small inerval [, + ], and denoe E p { = E{ F p. hen he following ideniy, which we call he profi margin principle, should hold: (1 + ρ )E p {[1 S]+ (1 + ρ r )E p {{ original premium {[(1 α) S]+ {{ premium o he reinsurance company = (1 + ρ α )E p {[α S]+ {{ modified premium. (2.7) Now assume ha during his inerval he reinsurance policy does no change in ime. Using he assumpion (H1) on f, one shows ha, for any β F p, E {[β S] + = + β(, z)f(s, z)ν(dz)ds = β(, z)f(, z)ν(dz) + o( ). Now, approximaing E p {[β S]+ by β(s, z)f(s, z)ν(dz) wih β = 1, α, 1 α, respecively, in (2.7) and recalling (2.5), we obain ha (1 + ρ )c (1 + ρ r ) (1 α(, z))f(, z)ν(dz) = (1 + ρ α ) α(, z)f(, z)ν(dz). (2.8) herefore, during [, + ] he reserve changes as follows X + X = c (1 + ρ ) (1 + ρ r ) (1 α(, z))f(, z)ν(dz) + α(, z)f(s, z)n p (dzds) (2.9) + = (1 + ρ α ) α(, z)f(s, z)ν(dz) α(, z)f(s, z)n p (dzds). For noaional simpliciy from now on le us denoe S α = α(, z)f(s, z)n p (dzds), m(, α) = α(, z)f(s, z)ν(dz), (2.1) hen (2.9) leads o he following equaion for he reserve process: X = x + (1 + ρ α s )m(s, α)ds S α = x + (1 + ρ α s )m(s, α)ds α(s, z)f(s, z)n p (dsdz). (2.11) 7
8 Remark 2.3 In he case when he reinsurance policy α is independen of z, we have S α = α(s) f(s, z)n p (dzds) = α(s)ds s and m(, α) = α(s) f(s, z)ν(dz) = α(s)c s, as we ofen see in he sandard reinsurance framework. Also, we noe ha if ρ r = ρ α (hence equal o ρ!), hen he reinsurance is called cheap. Bu under he profi margin principle, we see ha wheher a reinsurance is cheap does no change he form of he reserve equaion (2.11). From now on we shall drop he superscrip α from ρ a for simpliciy, even when noncheap reinsurance is considered. B. Reserve model wih reinsurance and invesmen. We now consider he scenario when an insurance company is allowed o inves par or all of is reserve in a financial marke. We assume ha he marke has k risky asses (socks) and 1 riskless asse (bond or money marke accoun). We model he dynamics of he marke prices of he bond and socks, denoed by P, P i, respecively, where and i = 1, 2,, k, which are described by he following sochasic differenial equaions: dp = r P d, dp i = P i [µ i d + k j=1 σ ij dw j ], i = 1, 2,, k. [, ], (2.12) where {r is he ineres rae, µ = (µ 1, µ 2,, µ k ) is he appreciaion rae, and σ = (σ ij )k i,j=1 is he volailiy marix. We shall make he following assumpions for he marke parameers: (H3) he processes r, µ, and σ are F W adaped and bounded. Furhermore, he process σ is uniformly nondegenerae: i.e., here exiss δ >, such ha σ σ δi, [, ], P a.s. As usual, we assume ha he marke is liquid and he insurance company can rade coninuously. Suppose ha he insurance company s oal reserve a each ime is X. We shall denoe he invesmen porfolio of he insurance company a each ime by π ( ) = ( ) π 1,, π k, where π i represens he fracion of is reserve X allocaed o he i h sock (hence he amoun of money ha i pus ino he ih sock would be πx i, i = 1, 2,, k); and i pus he res of he money, X k i=1 π i X = (1 k i=1 π i )X, ino he money marke accoun. Furhermore, we assume ha he insurance company also has he righ of consumpion, which may include dividend/bonus, ec. Denoe rae of he consumpion o be an adaped process D = {D :. We should noe ha in his paper we shall firs allow shor selling and borrowing (wih same ineres rae). ha is, we do no require ha π i and k i=1 π i 1,. However, we do need some consrains on he porfolio process π for echnical reasons. 8
9 Definiion 2.4 A porfolio process is an IR k valued, Fadaped process π such ha { { k E π s X s 2 ds = E where X is he oal reserve a ime. i=1 πsx i s 2 ds <, (2.13) A consumpion (rae) process is an Fpredicive nonnegaive process D saisfying { E D s ds <. (2.14) Following he same idea as ha in par A, if we assume ha during a small ime duraion [, + ] he porfolio π, reinsurance policy α and he consumpion rae D, as well as all he parameers are freezed a heir values a ime, hen i is easy o see ha he reserve change during [, + ] should be k π X + = X + X i P i P i + (1 k i=1 π)x i i=1 P P {{ invesmen gain + + (1 + ρ )m(, α) α(, z)f(s, z)n p (dzd) {{ premium income {{ claim D {{ consumpion (2.15) Leing, and using he price equaions (2.12), we see ha he reserve process X should follow he SDE: dx = Or in he inegral form: X = x + { + [ ] r + π, µ r 1 X d + X π, σ dw + (1 + ρ )m(, α)d α(, z)f(, z, )N p (ddz) D d, [, ]. (2.16) X s [r s + π s, µ s r s 1 ] + (1 + ρ s )m(s, α) ds + α(s, z)f(s, z)n p (dsdz) X s π s, σ s dw s D s ds, [, ]. (2.17) We ofen call a porfolio/reinsurance pair (π, α) is Dfinancing (see, for example, [11]) if he risk reserve X saisfy (2.17).. 3 Admissibiliy of Sraegies In his secion we analyze some naural consrains on he invesmen and reinsurance sraegies. We have already menioned ha he consrain α [, 1] is inrinsic in order 9
10 o have a sensible reinsurance problem. Anoher special, fundamenal consrain for an insurance company is ha he reserve should (by governmen regulaion) be alof, ha is, a any ime, he reserve should saisfy X x,π,α,d C for some consan C > a all ime. Mahemaically, one can always ake C = (or by changing x o x C ). We henceforh have he following definiion of he admissibiliy condiion. Definiion 3.1 For any x, a porfolio/reinsurance/consumpion riple (π, α, D) is called admissible a x, if he risk reserve process saisfies X x,π,α,d = x; X x,π,α,d, [, ], P a.s. We denoe he oaliy of all sraegies admissible a x by A(x). We shall firs derive a necessary condiion for an admissible sraegy. In ligh of he sandard approach in finance (α in our case) (see, e.g., KarazasShreve ([1], [11]), we denoe he risk premium of he marke by θ = σ 1 (µ r 1), and he discoun facor by γ = exp{ r sds,. Define = W + W { Z = exp { Y = exp θ s ds (3.1) θ s, dw s 1 2 ln(1 + ρ s )N p (dsdz) ν( ) θ s 2 ds, (3.2) Finally, he socalled saepricedensiy process is defined as H = γ Y Z. ρ s ds. (3.3) We now give wo lemmas concerning he GirsanovMeyer ransformaions ha will be useful in our discussion. Consider he following change of measures on he measurable space (Ω, F ): dq Z = Z dp ; dq = Y dq Z = Y Z dp. (3.4) hen by he Girsanov heorem (cf. e.g., [1]) we know ha he process W is a Brownian moion under measure Q Z. lemma We collec some less obvious consequences in he following Lemma 3.2 Assume (H2). hen, under probabiliy measure P, he process {Y is a squareinegrable maringale. Furhermore, define dq Y = Y dp on F, hen (i) he process Z is a squareinegrable Q Y maringale; (ii) for any reinsurance policy α, he process N α = (1 + ρ s )m(s, α)ds + 1 α(s, z)f(s, z)n p (dsdz) (3.5)
11 is a Q Y local maringale. (iii) he process ZN α is a Q Y local maringale. Proof. Le ξ = ln(1 + ρ s )N p (dsdz) Λ ρ sds, where Λ = ν( ). hen Y = exp{ξ. Applying Iô s formula (cf. e.g., [8]) we have + { Y = 1 Λ Y s ρ s ds + exp{ξ s + ln(1 + ρ s ) exp{ξ s N p (dsdz) = 1 Λ + Y s ρ s ds + + Y s [exp{ln(1 + ρ s ) 1]N p (dsdz) (3.6) = 1 + Y s ρ s Ñ p (dsdz). ha is, Y is a local maringale. On he oher hand, noe ha { Y 2 + = exp 2 ln(1 + ρ s)n p (dsdz) Λ 2ρ s ds { + = exp ln(1 + ρ s) 2 N p (dsdz) Λ = Ỹe Λ ρ2sds, [( ] 1 + ρ s ) 2 1 ds e Λ ρ2 s ds where Ỹ is he same as Y bu wih ρ being replaced by (1 + ρ)2 1. hus, repeaing he previous argumens one shows ha Ỹ saisfy he following Sochasic differenial equaion + Ỹ = 1 + Ỹ s [(1 + ρ s ) 2 1]Ñp(dsdz), (3.7) hence Ỹ is a local maringale as well. Since Ỹ is posiive, i is a supermaringale. herefore EỸ EỸ = 1 for all. he boundedness of ρ hen leads o ha EY 2 EỸe Λ ρsds <. hus Y is indeed a rue maringale. Now consider processes Z and N α under he probabiliy measure Q Y. Since ρ is F p  adaped, i is independen of W (under P ). hus Y and Z are independen under P. Noe ha Z saisfies he SDE: Z = 1 θ s Z s dw s, (3.8) i is a squareinegrable maringale under P, whence under Q Y, proving (i). o see (ii) we need only show ha he process N α Y is a P local maringale for any reinsurance policy α. Indeed, applying Iô s formula, noing ha Y saisfies he SDE (3.6), and recalling he definiion of m(, α) (2.1), we have + N α Y = N α s Y s ρ s Ñ p (dsdz) + Y s (1 + ρ s )m(s, α)ds IR + + Y s α(s, z)f(s, z)n p (dsdz) α(s, z)f(s, z)y s ρ s ν(dz)ds + + = N α s Y s ρ s Ñ p (dsdz) Y s α(s, z)f(s, z)ñp(dsdz). (3.9) 11
12 hus N α Y is P local maringale. Finally, (iii) follows from an easy applicaion of Iô s formula. he proof is complee. A direc consequence of Lemma 3.2 is he following corollary. Corollary 3.3 Assume (H2) and (H3). he process W is also a QBrownian moion, and N α is a Qlocal maringale. Consequenly, N α W is a Qlocal maringale. Proof. We firs check W. Noe ha W () is sill a coninuous process under Q, and for s, one has E Q {W W s F s = 1 Y s Z s E{Y Z (W W s ) F s = 1 Y s E{Y F s 1 Z s E{Z (W W s ) F s (3.1) = E Q Z {W W s F s =. In he above he firs equaliy is due o he Beyes rule (cf. [1]), he second equaliy is due o he independence of Y and Z, and in he hird equaliy we used he Beyes rule again, ogeher wih he facs ha Y is a P maringale and W is a Q Z Brownian moion. Similarly, one can show ha E Q {(W Ws ) 2 F s = E Q Z {(W Ws ) 2 F s = I d ( s), (3.11) where I d is he d d ideniy marix. Applying Lévy s heorem we see ha W is a Brownian moion under Q. o see ha N α is a Qlocal maringale we mus noe ha he reinsurance policy α is assumed o be Fadaped, hence N α is neiher independen of Y, nor of Z. We proceed wih a slighly differen argumen. Firs noice ha by an exra sopping if necessary, we may assume ha N α is bounded, whence a Q Y maringale by Lemma 3.2(ii). Also, in his case he conclusion (iii) of Lemma 3.2 can be srenghened o ha N α Z is a Q Y maringale as well. Bearing hese in mind we apply Beyes rule again and use Lemma 3.2(i) o ge E Q {N α Ns α F s = 1 E Q Y {Z (N α Ns α ) F s (3.12) Z s = 1 E Q Y {{E Q Y {Z F N α E Q Y {Z F s Ns α F s Z s = 1 E Q Y {Z N α Z s Ns α F s =. Z s hus N α is a Qmaringale. he las claim is obvious. he proof is complee. 12
13 he following necessary condiion for he admissible riple (π, α, D), also known as he budge consrain, is now easy o derive. heorem 3.4 Assume (H2) and (H3). hen for any (π, α, D) A(x), i holds ha { E H s D s ds + H X x,α,π,d x, (3.13) where H = γ Y Z, γ = exp{ r s ds. (3.14) Proof. For simpliciy we denoe X = X x,π,α,d. Recall he reserve equaion (2.17) and rewrie i as X = x + {r s X s + X s π s, σ s (θ s ds + dw s ) (3.15) (1 + ρ s )m(s, α)ds α(s, z)f(s, z)n p (dsdz) D s ds = x + {r s X s + X s π s, σ s dw s + N α D s ds, where θ = σ 1 (µ r ) is he risk premium. Clearly, applying Iô s formula we can hen wrie he discouned reserve, denoed by X = γ X,, as follows X = x + X s π s, σ s dws + γ s dns α γ s D s ds,. (3.16) herefore, under he probabiliy measure Q defined by (3.4), he process X + γ s D s ds = x + X s π, σ s dws γ s dns α is a local maringale. Furher, he admissibiliy of (π, α, D) implies ha he lef hand side is a posiive process, hence i is a supermaringale under Q. I follows ha proving he heorem. x E Q { X + γ s D s ds = E{H X + H s D s ds, (3.17) We remark ha he budge consrain (3.13) akes he same form as hose ofen seen in he pure finance models wihou claims (cf., e.g., [11]). he difference is ha he discouning is accomplished by a differen saepricedensiy process H. 13
14 4 Widersense Sraegies and he Auxiliary Marke In his and he nex secion we shall sudy he exisence of admissible sraegies. In oher words, we shall prove ha he se of admissible sraegies, A(x), is indeed nonempy for any iniial endowmen x. We begin in his secion by inroducing he noion of widersense sraegies. Definiion 4.1 We say ha a riple of Fadaped processes (π, α, D) a widersense sraegy if π and D saisfy (2.13) and (2.14), respecively; and α Fp 2 (see (2.1)). Moreover, we call he process α in a widersense sraegy a pseudoreinsurance policy. he following lemma gives he exisence of he widersense sraegies. Lemma 4.2 Assume (H1) (H3). hen for any consumpion process D and any F  measurable nonnegaive random variable B such ha E(B) > and { E H s D s ds + H B = x, (4.1) here exis a Dfinancing porfolio process π and a pseudoreinsurance policy α, such ha he soluion X x,π,α,d o he SDE (2.17) saisfies X x,π,α,d >, ; and X x,π,α,d = B, P a.s. Proof. Le he consumpion rae process D be given. Consider he following Backward Sochasic Differenial Equaion (BSDE) on he probabiliy space (Ω, F, P ): X = B + {r s X s + ϕ s, θ s D s + ρ s ψ(s, z)ν(dz) ds ϕ s, dw s ψ(s, z)ñp(dsdz). (4.2) Exending he resuls of BSDE wih jumps by Siu [17] and using he Maringale Represenaion heorem involving random measures (cf. [9] or Lemma 2.3 in [19]), i can be shown ha he BSDE (4.2) has a unique (Fadaped) soluion (X, ϕ, ψ) saisfying { E X s 2 + ϕ s 2 + ψ(s, z) 2 ν(dz) ds <. Le us define α(, z) = ψ(,z) f(,z), for all (, z) [, ), P a.s. hen by (H1) we see ha α F 2 p, hus i is a pseudoreinsurance policy. We claim ha X >, for all, 14
15 P a.s. Indeed, noe ha = = we see ha (4.2) can be wrien as or in differenial form: ρ s ψ(s, z)ν(dz)ds + ψ(s, z)ñp(dsdz) ρ s α(s, z)f(s, z)ν(dz)ds + α(s, z)f(s, z)ñp(dsdz) (1 + ρ s )m(s, α)ds + α(s, z)f(s, z)n p (dsdz), { X = B r s X s + ϕ s, θ s D s + (1 + ρ s )m(s, α) ds ϕ s, dw s + = B {r s X s D s ds α(s, z)f(s, z)n p (dsdz) (4.3) ϕ s, dw s + N α N α, dx = {r X D d + ϕ, dw dn α. (4.4) where W and N α are defined as before. Recall from Corollary 3.3 ha W is a QBrownian moion and N α is a Qlocal maringale. Le {τ n be a sequence of sopping imes such ha τ n and for each n, N α,n = N τ α n, is a maringale. Now for any [, ], and any n 1 we apply Iô s formula o ge τn τn τn γ τn X τn + γ s D s ds = γ τn X τn + γ s ϕ s, dws γ s dns α,n. τ n τ n τ n aking condiional expecaions E Q { F τn on boh sides and noing ha he wo sochasic inegrals are all Qmaringales we obain from he opional sampling heorem ha γ τn X τn = E Q{ τn γ τn X τn + τ n γ s D s ds F τn. Leing n and applying he Monoone Convergence heorem we hen have γ X = E Q{ γ B + γ s D s ds F E Q {γ B F >, [, ], P a.s., (4.5) since E(B) > and D is nonnegaive by assumpion. In oher words, we have proved ha P {X >, ; X = B = 1. Le us now define π = (σ ) 1 ϕ X, [, ]. (4.6) 15
16 hen, we see ha π saisfies (2.13), hanks o (H3), and ha (4.4) can now be wrien as dx = {r X D d + X π, σ dw dn α. (4.7) Comparing (4.7) o he reserve equaion (3.16) and noing he fac ha X = B, we see ha X = X x,π,α,d holds if we can show ha X = x. Bu seing = in (4.5) and using he assumpion (4.1) we have his proves he lemma. X = E Q{ { γ X + γ s D s ds = E H X + H s D s ds = x. We remark ha in Lemma 4.2 α is only a pseudoreinsurance policy. In he res of he secion we will modify he widersense sraegy obained above o consruc an admissible sraegy. We shall follow he idea of he socalled dualiy mehod inroduced by Cvianic & Karazas [4] o achieve his goal. We begin by recalling he suppor funcion of [, 1] (see, [4], [16]): δ(x) = δ(x [, 1]) =, x, x, x <. and we define a subspace of F 2 p : (4.8) D = {v Fp 2 : sup [,R] v(, z)ν(dz) < C R, P a.s., R >. (4.9) Recall also he linear funcional m(, ) : [, ] F 2 p IR defined by m(, α) = α(, z)f(, z)ν(dz), α Fp 2. Le v D be given. We consider a ficiious marke in which he ineres rae and appreciaion rae are perurbed in such a way ha he asse prices follow he SDE: dp v, = P v, {r + m(, δ(v))d, dp v,i = P v,i {(µ i + m(, δ(v))d + k j=1 Nex, we rewrie he reserve equaion (3.15) as follows: X = x + + r X d + X π, σ dw + α(s, z)f(s, z)n p (dsdz) 16 σ ij dw j, i = 1,, k. (4.1) (1 + ρ s )α(s, z)f(s, z)ν(dz)ds D s ds. (4.11)
17 Now corresponding o he auxiliary marke we define a general (ficiious) expense loading funcion ρ v (s, z, x) = ρ s +v(s, z)x. Using such a loading funcion and repeaing he previous argumen one shows ha he reserve equaion (4.11) will become X v = x = x + + = x + X v s [ ] r s + m(s, δ(v)) ds + Xs v π s, σ s dws [1 + ρ s + v(s, z)x v s ]α(s, z)f(s, z)ν(dz)ds [ Xs v α(s, z)f(s, z)n p(dsdz) D s ds (4.12) ] r s + m(s, αv + δ(v)) ds + (1 + ρ s )m(s, α)ds + Xs v π s, σ s dws α(s, z)f(s, z, )N p(dsdz) D s ds Xs v rs α,v ds + Xs v π s, σ s dws + N α D s ds, where r α,v = r + m(, αv + δ(v)) could be hough of as a ficiious ineres rae. We observe ha for any pseudoreinsurance sraegy α, he definiion of δ( ) implies ha (suppressing variables (, z)): αv + δ(v) = αv1 {v + (αv v)1 {v< = v {α1 {v + (1 α)1 {v<. (4.13) and r α,v reduces o he original ineres rae if and only if m(, αv +δ(v)) =. In paricular, if α is a (rue) reinsurance policy (hence α 1), hen i holds ha α(, z)v(, z) + δ(v(, z)) v(, z), (, z) [, ), P a.s. (4.14) he following modified widersense sraegies will be useful in our fuure discussion. Definiion 4.3 Le v D. A widersense sraegy (α, π, D) is called vadmissible if (i) m(, av + δ(v)) d <, P a.s. (ii) denoing X v = X v,x,π,α,d, hen X v, for all, P a.s. We denoe he oaliy of widersense vadmissible sraegies by A v (x). We remark ha if v D and (α, π, D) A v (x) such ha α(, z) 1; δ(v(, z)) + α(, z)v(, z) =, d ν(dz)a.e., P a.s. (4.15) hen α is a (rue) reinsurance policy and r α,v = r,. Consequenly, X v = X and (α, π, D) A(x). o ake a furher look a he se A v (x), le us define, for any v D and 17
18 any vadmissible sraegy (π, α, D), γ α,v H α,v { = exp { rs α,v ds = exp [r s + m(αv + δ(v)]ds, = γ α,v Y Z, [, ]. (4.16) We have he following resul. Proposiion 4.4 Assume (H1) (H3). hen, (i) for any v D, and (π, α, D) A v (x), he following budge consrain sill holds { E H α,v s (ii) if (π, α, D) A(x), hen for any v D i holds ha D s ds + H α,v Xv x; (4.17) X v,x,α,π,d () X x,α,π,d (),, a.s. (4.18) In oher words, A(x) A v (x), v D. Proof. (i) Recall from (3.3) and (3.2) he P maringales Y and Z, as well as he change of measure dq = Y Z dp. Since W is a QBrownian moion and N α is a Qlocal maringale, by he similar argumens as hose in heorem 3.4 one shows ha proving (i). { E H α,v s D s ds + H α,v Xv = E Q{ γ α,v s D s ds + γ α,v Xv x, (ii) Le x be fixed and le (α, π, D) A(x). Since α is a (rue) reinsurance policy, we have α(, z) [, 1], for d ν(dz)a.s. hus < m(s, αv + δ(v))ds < v(s, z) ds <, hanks o (4.14). hus denoe X = X x,α,π,d, X v = X v,x,α,π,d, and δx = X v X. hen, combining (2.17) and (4.12) we obain ha δx = = (X v s r α,v s X s r s )ds + δx s r α,v s ds + δx s π s, σ s dw s δx s π s, σ s dw s + X s m(s, αv + δ(v))ds. (4.19) Viewing (4.19) as an linear SDE of δx wih δx =, we derive from he variaion of parameer formula ha δx = E α,v [E α,v s ] 1 X s m(s, αv + δ(v))ds, 18
19 where E α,v = E(ξ α,v ) is he DoléansDade sochasic exponenial of he semimaringale ξ = rα,v s ds + π s, σ s dw s, defined by { ξ = exp [ rs α,v π sσ s 2] ds π s, σ s dws (4.2) (cf. e.g., [1]). Noe ha (π, α, D) A(x) implies ha X for all and ha m(, αv + δ(v)), hanks o (4.14). Consequenly, δx as well, for all, P a.s. he proof is now complee. 5 Exisence of Admissible Sraegies We are now ready o prove he exisence of admissible sraegies. Recall ha Lemma 4.2 shows ha he budge equaion (4.1) implies he exisence of a Dfinancing wider sense sraegy. We will now look a he converse. o begin wih, le us make some observaions. Noe ha he perurbed reserve equaion (4.12) can be rewrien as X v = x + + = x + { [r s + m(s, δ(v) + αv) + π s, σ s θ s ]Xs v D s + ρ s m(s, α) ds + Xs v π s, σ s dw s α(s, z)f(s, z)ñp(dsdz) (5.1) {[r s + m(s, δ(v))]xs v + m(s, αv)xs v D s ds + Xs v π s, σ s dws α(s, z)f(s, z)ñ p (dsdz), + where Ñ p (ds, dz) = Ñp(ds, dz) ρ s ν(dz)ds. o simplify noaions, we shall now denoe, for any v D and η Fp 2, m v (, η) = η(, z)v(, z)ν(dz) = η v. (5.2) hus, we have m(, η) = m f (, η), and we denoe m 1 (, η) = η. Le us now define ϕ v = X v σ π ; ψ v (, z) = α(, z)f(, z), (, z) [, ]. (5.3) hen (5.1) becomes X v = x + + {[r s + m(s, δ(v))]xs v + ψ v sxs v D s ds + ϕ v s, dw s ψ v (s, z)ñ p (dsdz). (5.4) 19
20 Recall ha W is a Brownian moion and Ñ is a compensaed Poisson random measure under he probabiliy measure Q, our analysis will depend heavily on he following BSDE deduced from (5.4): for any B L 2 (Ω; F ), and v F 2 p, y = B {rsy v s + ψ v sy s D s ds ϕ s, dws + ψ s Ñp (dsdz). (5.5) where r v = r + m(, δ(v)),. We should noe here ha his seemingly sandard BSDE is raher illbehaved. In fac, here has been no exising resul on he exisence and uniqueness of his BSDE in he lieraure. he main obsacle is he erm ψ v sy s, which is neiher Lipschiz nor linear growh in (y v, ψ v ). However, he following resul can be found in Liu [12]. Lemma 5.1 Assume (H1) (H3). Assume furher ha processes r and D are all uniformly bounded. hen for any v D and B L (Ω; F ), he BSDE (5.5) has a unique adaped soluion (y v, ϕ v, ψ v ). We remark ha for any v F 2 p, we can define a porfolio/pseudoreinsurance policy pair from he soluion (y v, ϕ v, ψ v ) as π v = [σ ] 1 ϕv y v ; α v (, z) = ψv (, z) f(, z). Clearly, a necessary condiion for α v o be a rue insurance policy is ha ψ v (, z) α v (, z) f(, z) f(, z) L, ha is, ψ v L sup [, ] v(, z) ν(dz) = L v,ν, where L is he consan in (H1). In wha follows we shall call he pair (π v, α v ) he porfolio/pseudoreinsurance pair associaed o v. Wih he help of Lemma 5.1, we now give a sufficien condiion for he exisence of he admissible sraegy. he proof of his heorem borrows he idea of heorem 9.1 in CvianicKarazas [4], modified o fi he curren siuaion. heorem 5.2 Assume (H1) (H3). Le D be a bounded consumpion process, and B be any nonnegaive, bounded F measurable random variable such ha E(B) >. Suppose ha for some u D whose associaed porfolio/pseudoreinsurance pair, denoed by (π, α ), saisfies ha { E H α,v B + H α,v s D s ds E {H α,u B + 2 H α,u s D s ds = x, v D,
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