Optimal Reinsurance/Investment Problems for General Insurance Models

Transcription

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 non-markovian 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 claim-size 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 well-posedness 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: Cramér-Lundburg 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 ; [email protected]. his auhor is suppored in par by he Sociey of Acuaries CKER gran # Deparmen of Mahemaics, Purdue Universiy, Wes Lafayee, IN ; [email protected]. 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 Cramér-Lundberg 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 Ma-Sun [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 so-called 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., Karazas-Shreve [1, 11] for he opimal invesmen/consumpion problems wih coninuous models, o Xue [21] for heir jump-diffusion counerpars, and o Cvianic-Karazas [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 over-complicaing 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 shor-selling 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 well-posedness 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 so-called profi-margin 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 so-called 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 ε-opimal sraegy could be produced using our resul. Also, i is worh noing ha our final resul rely on he solvabiliy of a special forward-backward sochasic differenial equaion (FBSDEs for shor, cf., e.g., Ma-Yong [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 invesmen-reinsuranceconsumpion sraegies, and inroduce some equivalen probabiliy measures ha are imporan in our discussion. In secion 4 we inroduce he wider-sense 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 d-dimensional 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 lp-augmenaion 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 sub-σ-field G F and 1 p <, L p (G; le) denoes he space of all le-valued, G-measurable random variables ξ such ha E ξ p <. As usual, ξ L (G; le) means ha i is G-measurable and bounded. for 1 p <, L p (F, [, ]; le) denoes he space of all le-valued, F-progressively 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 well-known 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 expense-loading 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, non-negaive 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. Asmussen-Nielsen [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 non-cheap 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 non-degenerae: 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 i-h 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, F-adaped 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 F-predicive 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 D-financing (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., Karazas-Shreve ([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 so-called sae-price-densiy process is defined as H = γ Y Z. ρ s ds. (3.3) We now give wo lemmas concerning he Girsanov-Meyer 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 square-inegrable maringale. Furhermore, define dq Y = Y dp on F, hen (i) he process Z is a square-inegrable 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 square-inegrable 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 Q-Brownian moion, and N α is a Q-local maringale. Consequenly, N α W is a Q-local 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 Q-local maringale we mus noe ha he reinsurance policy α is assumed o be F-adaped, 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 Q-maringale. 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 sae-price-densiy process H. 13

14 4 Wider-sense 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 wider-sense sraegies. Definiion 4.1 We say ha a riple of F-adaped processes (π, α, D) a wider-sense 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 wider-sense sraegy a pseudo-reinsurance policy. he following lemma gives he exisence of he wider-sense sraegies. Lemma 4.2 Assume (H1) (H3). hen for any consumpion process D and any F - measurable non-negaive random variable B such ha E(B) > and { E H s D s ds + H B = x, (4.1) here exis a D-financing porfolio process π and a pseudo-reinsurance 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 (F-adaped) 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 pseudo-reinsurance 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 Q-Brownian moion and N α is a Q-local 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 Q-maringales 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 non-negaive 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 pseudo-reinsurance policy. In he res of he secion we will modify he wider-sense sraegy obained above o consruc an admissible sraegy. We shall follow he idea of he so-called 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 pseudo-reinsurance 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 wider-sense sraegies will be useful in our fuure discussion. Definiion 4.3 Le v D. A wider-sense sraegy (α, π, D) is called v-admissible 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 wider-sense v-admissible 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 v-admissible 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 Q-Brownian moion and N α is a Q-local 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 Doléans-Dade 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 D-financing 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 ill-behaved. 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/pseudo-reinsurance 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/pseudo-reinsurance 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 Cvianic-Karazas [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/pseudo-reinsurance 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,

21 where for any v D, H α,v { = γ α,v Y Z, γ α,v = exp [rs v + m(s, α v))]ds,. (5.6) hen he riple (π, α, D) A(x). Furher, he corresponding reserve X saisfies X = B, P -a.s. Proof. Firs noe ha wih he given u D and he associaed porfolio/pseudoreinsurance pair (π, α ), he BSDE (5.5) becomes y = B + {[rs u + m(s, α u )]ys D s ds ys π s, σ s dws α (s, z)f(s, z)ñ p (dsdz). (5.7) Following he same argumens as ha in Lemma 4.2 and using he assumpion of he heorem, we can show ha y = B, y >, for all [, ], and y = x. In oher words, we have shown ha (π, α, D) is a u -admissible wider-sense sraegy. We shall prove ha under he assumpions of he heorem, α is a rue reinsurance policy and y = X x,π,α,d. Bu his is, according o he remark (4.15), amouns o showing ha α (, z) 1, and δ(u (, z)) + α (, z)u (, z) =, d ν(dz) dp -a.e. In ligh of he argumens in [4], we inroduce he following noaions: for any λ D, and (, z) [, ], le δ u (λ) = δ(u ), λ = u, δ(λ), oherwise. (5.8) We hen define, for v D, x (v) = [ E H α,v B + L,v = m(s, δ u (v u ))ds; ] H α,v s D s ds ; M,v = m(s, α (v u ))ds. (5.9) In paricular, for any λ D, we denoe L λ = L,u +λ and M λ = M,u +λ. We hen define, for λ D, a sequence of sopping imes τ n = inf { : L λ M λ {[γ s u ys(l λ s +Ms λ ) σs πs ] 2 +1 ds n. (5.1) I is clear from definiion ha x (u ) = y = x and τ n, as n, P -a.s. for all λ D. Now for each λ D, ε (, 1), and n ln, we define a random field u,λ ε,n(, z) = u (, z) + ελ(, z)1 { τn, (, z) [, ]. (5.11) 21

22 hen clearly, u,v ε,n D. Furhermore, recalling (4.8) we see ha δ is a convex funcion such ha δ(cx) = cδ(x) for all c >, and x IR. hus i is easy o check ha δ(u,λ ε,n(, z)) δ(u (, z)) ε1 { τnδ u (λ(, z)), (, z) [, ]. (5.12) Now le λ D be fixed. For noaional simpliciy le us denoe H = H α,u and H ε,n,λ = H α,u,λ ε,n. hen, by he assumpion of he heorem we have, for any ε > and n ln, x (u ) x (u,λ ε,n) ε = 1 { ε E (H H ε,n,λ )B+ (Hs Hs ε,n,λ )D s ds = E{Θ ε,n,λ, (5.13) where Θ ε,n,λ = H ( 1 Hε,n,λ H ) (B ε ) ( + Hs 1 Hε,n,λ s Hs ) (Ds ε ) ds. (5.14) We claim ha, for each n ln, Θ ε,n,λ κ n {H B + H = s D s ds Θ n, ε >. (5.15) where κ n = 1 e 2εn sup<ε<1 ε. Indeed, recall from (5.6) and noe (5.12) we see ha H ε,n,λ H = γα,u,λ ε,n γ α,u { exp ε = exp { = exp τn { ε(l λ τ n + M τn ) ( ) m s, δ(u,λ ε,n) δ(u ) + α (u,λ ε,n u ) ds m(s, δ u (λ) + α λ)ds e 2εn, [, ]. (5.16) hus (5.15) follows easily from (5.14). Furhermore, noe ha EΘ n = κy = κ nx <, 1 H ε,n,λ and lim ε ε H (L λ τ n + M τ λ n ) for [, ], we can apply Faou s lemma o (5.13) o ge x(u) x(u v ε,n) lim ε ε ( { E H B lim ε 1 { E lim 1 Hε,n,λ ε { E H B(L λ τ n + Mτ λ n ) + H = E Q{ γ u B(L λ τ n + M λ τ n ) + Θ ε,n,λ ε ) + H 1 s D s lim ε ε ( 1 Hε,n,λ s Hs D s (L λ s τ n + Ms τ λ n )ds 22 H s γs u D s (L λ s τ n + Ms τ λ n )ds, ) ] ds

23 where Q is he probabiliy measure defined as before. Now leing n and applying he Dominaed Convergence heorem, we obain ha E Q{ γ u B(L λ + M λ ) + γs u D s (L λ s + Ms λ )ds. (5.17) On he oher hand, a simple applicaion of Iô s formula o γ u y (L λ + M λ ) from o τ n leads o ha τn τn γτ u n y n (L λ τ n + Mτ λ n ) + (L λ s + Ms λ )γs u D s ds = γs u ysm(s, δ(λ) + α λ)ds + M τn, (5.18) where M τn = τn τn (L λ s+m s λ )γs u ys π s, σ s dws + γs u (L λ s +Ms λ ) α (s, z)f(s, z)np (dsdz) is a Q-maringale. Firs aking expecaion on boh sides of (5.18), hen leing n and applying he Dominaed Convergence heorem again we obain ha E Q{ γ u B(L λ + M λ ) + hus (5.17) becomes (L λ s + Ms λ )γs u D s ds = E Q{ E Q{ Since λ D is arbirary, we claim ha his will lead o ha γs u ysm(s, δ(λ) + α λ)ds. γs u ysm(s, δ u (λ) + α λ)ds. (5.19) α (, z, ω)λ(, z, ω) + δ u (λ(, z, ω)), d dν dp -a.e. (, z, ω) (5.2) for all λ D. Indeed, if no, we define A = {(, z, ω) : α (, z, ω)λ(, z, ω) + δ u (λ(, z, ω)) <. hen i mus hold ha Q{ 1 A (, z)dν(dz) > >. Since γ u y > for all, we have I λ (A) = E Q{ γs u ysm(s, δ u (λ1 A ) + α λ1 A )ds <. Now for any η > consider λ η = λ1 A c + ηλ1 A D. Since η is arbirary, we can assume λ η u and hus δ u (λ η ) = δ(λ η ). Noe again ha δ(cx) = cδ(x) for all c > and x IR, i follows from (5.19) ha E Q{ γ u s y sm(s, δ u (λ η ) + α λ η )ds = I λ (A c ) + ηi λ (A). (5.21) Since I λ (A) <, we have a conradicion for η > sufficienly large. his proves (5.2). 23

24 o complee he proof we noe ha he se D conains all consans. hus (5.2) implies ha for any r IR, α (, z, ω)r + δ(r), d dν dp -a.e. Using he coninuiy of δ we can deduce easily ha δ(r) + α (, z, ω)r, r IR, d dν dp -a.e. Consequenly we ge from he definiion of δ ha α, r, (, z, ω)r δ(r) = r, r <, d dν dp -a.e. ha is, α (, z, ω) [, 1], d dν dp -a.e. Furhermore, noe ha (5.2) and definiion of δ u ( ) imply ha for boh λ = ±u i holds ha α (, z)λ(, z)) + δ u (λ(, z)), d ν(dz) dp -a.s. ha is, α (, z)u (, z)) + δ(u (, z)) =, d ν(dz) dp -a.e. he proof is complee. 6 Uiliy Opimizaion In his secion we sudy a general uiliy opimizaion problem under our reserve model, by combining he resuls esablished in he previous secions, and he resuls of [4, 5]. We begin by inroducing some necessary noaions: Definiion 6.1 A funcion U : [, ) [, ] is called a uiliy funcion if i enjoys he following properies: (i) U C 1 ((, )), and U (x) >, for all x IR; (ii) U ( ) is sricly decreasing; (iii) U ( ) = lim x U (x) =. We denoe dom(u) = {x [, ); U(x) >. We should noe ha our definiion of a uiliy funcion is slighly differen from hose in [4, 5] or [11] in ha he dom(u) [, ) insead of he whole real line. Recall (from [11], for example) ha for any uiliy funcion, if we define x = inf{x : U(x) >, and U ( x+) = lim x x U (x). hen U ( x+) (, + ]. Furhermore, for each x [, ), le I : (, U ( x+)) ( x, ) be he inverse of U. hen I is coninuous and is sricly decreasing, and can be exended o (, ] by seing I(y) = x for y U ( x+). Furhermore, one has U y, < y < U ( x+) (I(y)) = U ( x+), U ( x+) y ; I(U (x)) = x, x < x <. (6.1) In our opimizaion problem we consider he following runcaed version of a uiliy funcion. 24

25 Definiion 6.2 A funcion U is a runcaed uiliy funcion if for some K >, U is a uiliy funcion on [, K] bu U(x) = U(K) for all x K. We call he inerval [, K] he effecive domain of U. Furhermore, we say ha a runcaed uiliy funcion is good if i saisfies U ( x+) <. We noe ha for any uiliy funcion U, here exiss a sequence of good runcaed uiliy funcions {U n such ha U n U, as n. Indeed, le U be a uiliy funcion. For any n >, we define x n x + n x x n ξ n (x) = U (x) x n x x n 1 n x > x n, (6.2) where x n and x n are such ha U (x n ) = n and U ( x n ) = 1 n. hen i is fairly easy o check ha he funcion defined by U n (x) = U(x n ) 1 x x n 2 (xn ) 2 nx n + ξ n (y)dy, x is a good runcaed uiliy funcion wih x =, U n (+) = U n () = U(x n ) 1 2 (xn ) 2 nx n, and K = x n. Clearly, U n (x) U(x), as n, for all x IR. We remark ha if U is a good runcaed uiliy funcion, hen we can assume wihou loss of generaliy ha x =, and U () <. Hence U : [, K] [U (K ), U ()]. If we define he inverse of U by I(y) = inf{x : U (x) y. (6.3) hen I : [U (K ), U ()] [, K] is coninuous and sricly decreasing. We can also exend I o [, ) by defining I(y) = for y U () and I(y) = K for y [, U (K )]. I is worh noing ha such a funcion I is hen bounded over [, )(!). Now le U be a good runcaed uiliy funcion wih effecive domain [, K], and le us define he convex dual (or Legendre-Fenchel ransform) of he funcion U as follows Ũ(y) = max {U(x) xy, < y <. (6.4) <x K he following lemma shows ha uiliy funcions. Ũ can be expressed in erms of I, jus as he sandard Lemma 6.3 Suppose ha U is a modified uiliy funcion, and le Ũ be is convex dual. hen i holds ha Ũ(y) = U(I(y)) yi(y), y >. (6.5) 25

26 Proof. For each y >, consider he funcion F (x) = U(x) xy. Differeniaing wih respec o x we ge F (x) = U (x) y. If y [U (K ), U ()] = Dom(I), hen we have Ũ(y) = max x F (x) = U(I(y)) yi(y). If y > U (), hen we have F (x) = U (x) y < for all x, since U is decreasing. hus F (x) is decreasing and Ũ(y) = max F (x) = F () = U(). Since I(y) = for all y > U (), (6.3) sill holds. Similarly, if y [, U (K )), hen F (x) = U (x) y > for all x. hus Ũ(y) = max F (x) = F (K) = F (I(y)), for y [, U (K )]. hus (6.5) holds for all y >, proving he lemma. o formulae our opimizaion problem, we now consider a pair of funcions U 1 : [, ] (, ) [, ) and U 2 : [,, ) [, ). As usual we denoe he (parial) derivaive of U 1 and U 2 wih respec o x by U 1 and U 2 respecively. definiion ha is based on he one in [11]. We inroduce he following Definiion 6.4 A pair of funcions U 1 : [, ] (, ) [, ) and U 2 : [,, ) [, ) is called a (von Neumann-Morgensern) preference srucure if (i) for each [, ], U 1 (, ) is a uiliy funcion, such ha he subsisence consumpion defined by x 1 () = inf{x IR; U 1 (, x) > is coninuous on [, ], and ha boh U 1 and U 1 are coninuous on he se D(U 1) = {(, x) : x > x 1 (), [, ]; (ii) U 2 is a uiliy funcion, wih subsisence erminal wealh defined by x 2 = inf{x : U 2 (x) >. Moreover, he pair (U 1, U 2 ) is called a modified preference srucure if U 2 is a good runcaed uiliy funcion. Our opimizaion problem is formulaed as follows. Recall firs he risk neural measure Q defined by (3.4). Le (U 1, U 2 ), x IR be a modified preference srucure. We assume ha he effecive domain of U 2 is [, K]. For any (π, α, D) A(x), we define he oal expeced uiliy by J(x; π, α, D) = { ( E U 1 (, D )d + U 2 X x,α,π,d ). (6.6) he goal is o maximize J(x; π, α, D) over all (π, α, D) A(x), and we denoe he value funcion by V (x) = sup J(x; π, α, D). (6.7) (π,α,d) A(x) We shall proceed along he lines of he dualiy mehod of [4, 5]. o be more precise, we shall firs consider he ficiious marke defined in 4 for find a candidae (wider-sense) opimal sraegy, and hen o verify ha i is acually an opimal sraegy in he sricly sense, using heorem

27 o begin wih, for given v D recall he se A v (x), he v-admissible sraegies, defined by Definiion 4.3. For any (π, α, D) A v (x), he perurbed risk reserve, denoed by X v for simpliciy, saisfies he following SDE under Q (recall (4.12)). X v () = x + Xs v + [ ] r s + m(s, δ(v) + αv) ds + α(s, z)f(s, z)ñ p (dsdz) X v s π s, σ s dw s D s ds, [, ]. (6.8) Le γ α,v be he discouning facor defined by (4.16). hen as in Lemma 4.2 we can easily derive he following ficiious budge consrain: Nex, define x E Q{ γ α,v Xv + Xv α (y) = { E H α,v I 2(yH α,v ) + { γs α,v D s ds = E H α,v Xv + H α,v Hs α,v D s ds. (6.9) I 1 (, yh α,v )d, < y <, (6.1) where I 1 (, ) is he inverse of U 1 (, ) and I 2 he inverse of U 2. Since I 1(, ) is coninuous, sricly decreasing, and I 1 (, +) =, by he Monoone Convergence heorem we see ha X α v (+) = and X α v ( ) is also coninuous. On he oher hand, noe ha lim x I 1 (, x) = x 1 () and lim x I 2 (x) = x =, applying Dominaed Convergence heorem we have Xv α ( ) = { lim X α v (y) = E y H α,v x 1 ()d. (6.11) Furhermore, le y = sup{y > ; Xv (y) > X v ( ), hen y (, ] and one can show (as in [11]) ha Xv α ( ) is decreasing on he inerval (, y ). We can define he inverse of Xv α (, y ) by Yv α (x) = inf{y : Xv α (y) < x. hen, Yv α (x) (, y ), x (Xv α ( ), ). Noe ha he definiion of x 1 ( ) ells us ha J(x; π, α, D) > would imply Bu for all (α, π, D) saisfying (6.12), i holds ha { E H α,v Xv + D x 1 (), [, ], P -a.s. (6.12) { H α,v D d E on H α,v D d Xv α ( ). (6.13) herefore, he ficiious budge consrain (6.9) ells us ha if x < X α v ( ), hen (6.13) (whence (6.12)) canno hold. Consequenly one has V (x) =. We now sudy he case when X α v ( ) < x holds. We noe ha his condiion is raher unusual since i couples he iniial sae and he conrol. We now inroduce a subse of A v (x): A v(x) = {(π, α, D) A v (x) : x > X α v ( ) (6.14) 27

28 Clearly, we need only consider he problem of maximizing J(D, B) = { E U 1 (, D())d + U 2 (B) (6.15) over all pairs (D, B), where D is a consumpion process and B L F (Ω), subjec o he budge consrain { E H α,v D d + H α,v B x. (6.16) We again use he usual Lagrange muliplier mehod. For all y >, v D, and α F 2 p, le us ry o maximize he following funcional of (D, B): J α v (D, B; x, y) { = E = xy + E ( { U 1 (, D())d + U 2 (B) + y x E [U 1 (, D()) yh α,v ) H α,v D d + H α,v B D ]d + E[U 2 (B) yh α,v B]. (6.17) Bu recalling he definiion of he convex duals of U 1 and U 2 we see ha { Jv α (D, B; x, y) xy + E wih equaliy holds if and only if D α,v P -a.s. Ũ 1 (, yh α,v α,v )d + Ũ2(yH, ) (6.18) = I 1 (, yh α,v ),, and B α,v = I 2 (yh α,v ), We noe ha unlike he usual siuaions in finance (see, e.g., [4, 5]), he maximizer D α,v and B α,v depends on he reinsurance policy α as well. Since B α,v is in a place of being he erminal reserve, our soluion o he opimizaion problem herefore has a novel srucure, which we now describe. For any y and v D, consider he following so-called forward-backward SDEs : for [, ], H = 1 + H s [r s + m(s, δ(v) + αv))]ds H s θ s, dw s + H s ρ s Ñ p (dsdz); { X = I 2 (yh ) X s [r s +m(s, δ(v)+αv)+ π s, σ s θ s ]+(1+ρ s )m(s, α) ds X s π s, σ s dw s + α(s, z)f(s, z)n p(dsdz)+ I 1 (s, yh s )ds, (6.19) Denoe he adaped soluion o (6.19), if i exiss, by (H y,v, X y,v, π y,v, α y,v ). Comparing he reserve equaion (2.16) o he backward SDE in (6.19) we see ha for a fixed x and D y,v = I 1 (, yh y.v ) for, he riple (π y,v, α y,v, D y,v ) A v (x) if and only if X y,v = x. Bu his is by no means clear from he FBSDE alone. In fac, we can only hope ha for each x, here exiss a y = y(x), so ha X y(x),v = x. he following heorem is new. 28

29 heorem 6.5 Assume (H1) (H3). Le (U 1, U 2 ) be a modified preference srucure. he following wo saemens are equivalen: (i) For any x IR, B = I 2 (Y(x)H ) and D where Y(x) is such ha = I 1 (, Y(x)H ),, saisfy { V (x) = E U 1 (, D )d + U 2 (B ) = sup J(x; π, α, D), (6.2) (π,α,d) A(x) (ii) here exiss a u (H, X, π, α ), wih y saisfying { x = E I 1 (, Y(x)H )d + I 2 (Y(x)H ) ; (6.21) D, such ha he FBSDE (6.19) has an adaped soluion { x = E I 1 (, yh )d + I 2 (yh ). (6.22) In paricular, if (i) or (ii) holds, hen (π, α, D ) A(x) is an opimal sraegy for he uiliy maximizaion insurance/invesmen problem. Proof. We firs assume (i) and prove (ii). By assumpion here exiss a porfolio and reinsurance pair (π, α ) such ha (π, α, D ) A(x), X π,α,d = B, and ha { J(x; π, α, D ) = V (x) = E U 1 (, D )d + U 2 (B ). Since α (, z) [, 1], we can define a random field u by 1 α (, z) = ; u (, z) = 1 {α (,z)= 1 {α (,z)=1 = 1 α (, z) = 1; oherwise so ha, by virue of (4.13), δ(u ) + α u = u {α 1 {u + (1 α )1 {u <. Consequenly, we mus have m(, δ(u ) + α u ) =, γ α,u = γ, and H α,u = H. Noe ha he process H saisfies he SDE H = 1 + H s r s ds H s θ s, dw s + and he reserve X = X π,α,d X = x + { saisfies he SDE Xs [r s + πs, σ s θ s ] + (1 + ρ s )m(s, α ) ds + α (s, z)f(s, z)n p (dsdz) 29 H s ρ s Ñ p (dsdz), (6.23) X s π s, σ s dw s D sds, (6.24)

30 and he erminal condiion X = B = I 2 (Y(x)H ). Combining (6.23) and (6.24) we see ha (H, X, π, α ) acually saisfies he FBSDE (6.19) wih y = Y(x) and v = u. Noe ha when H = H he equaions (6.21) and (6.22) coincide, we proved he saemen (ii). We now assume (ii) holds and ry o prove (i). Suppose ha for some u D, FBSDE (6.19) has an adaped soluion (H, X, π, α ) wih y = Y α u (x) = Y (x) (ha is, y saisfies (6.22)). We define D = I 1 (, Y (x)h ),, B = I 2 (Y (x)h ). Since we have already seen ha (D, B ) is a maximizer of he Lagrange muliplier problem for J α u (x, Y (x); D, B), defined by (6.17), we mus have { x = E H B + H D d, (6.25) and { V (x) = sup Ju α (D, B; x, Yα u (x)) = E U 1 (, D )d + U 2 (B ). (6.26) (D,B) Since I 2 is bounded by K >, and H saisfy an SDE, we have P {B > = P {Y (x)h U 2 () >. Namely, B is bounded and E(B ) >. Furhermore, for any oher v D, he budge consrain (6.16) ells us ha { E H α,v D d + H α,v B { x = E H D d + H B. hus, applying heorem 5.2 we can conclude ha (α, π, D ) A(x), o wi, α (, z) 1; δ(u (, z)) + α (, z)u (, z) =, (, z) [, ], P -a.s., and X = x. hus, we mus again have H = H, Y (x) = Y(x), and X = X x,π,α,d. Consequenly, we see ha (D, B ) become he same as ha defined in (i), and V (x) = V (x) = E{ U 1(, D )d + U 2 (B ), proving (i). he las claim is clear from he proof. his complees he proof. We noe ha he saemen (ii) in heorem 6.5 is indeed more han he solvabiliy of he FBSDE (6.19), due o he special choice of y. In fac in a sense y iself becomes a par of he soluion. his, along wih he solvabiliy of FBSDE (6.19), forms a class of new problems in he heory of BSDEs, and i is currenly under invesigaion. We hope o be able o address his issue in our fuure publicaions. 3

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