Optimal Reinsurance/Investment Problems for General Insurance Models


 Clementine Norman
 1 years ago
 Views:
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 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,
Optimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 57 85 Barcelona Absrac We see ha he price of an european call opion
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationVerification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing
MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp. 62 635 issn 364765X eissn 526547 364 62 hp://dx.doi.org/.287/moor..57 2 INFORMS Verificaion Theorems for Models of Opimal Consumpion
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION
QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS BY MOGENS STEFFENSEN ABSTRACT Quadraic opimizaion is he classical approach o opimal conrol of pension funds. Usually he paymen sream is approximaed
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationModeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling
Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se
More informationDynamic Information. Albina Danilova Department of Mathematical Sciences Carnegie Mellon University. September 16, 2008. Abstract
Sock Marke Insider Trading in Coninuous Time wih Imperfec Dynamic Informaion Albina Danilova Deparmen of Mahemaical Sciences Carnegie Mellon Universiy Sepember 6, 28 Absrac This paper sudies he equilibrium
More informationOn the Role of the Growth Optimal Portfolio in Finance
QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 144 January 2005 On he Role of he Growh Opimal Porfolio in Finance Eckhard Plaen ISSN 14418010 www.qfrc.us.edu.au
More informationTime Consistency in Portfolio Management
1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland Time Consisency in Porfolio Managemen
More informationA UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS
A UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS RICHARD A. TAPIA Appendix E: Differeniaion in Absrac Spaces I should be no surprise ha he differeniaion
More informationA martingale approach applied to the management of life insurances.
A maringale approach applied o he managemen of life insurances. Donaien Hainau Pierre Devolder 19h June 2007 Insiu des sciences acuarielles. Universié Caholique de Louvain UCL. 1348 LouvainLaNeuve, Belgium.
More informationJumpDiffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach
umpdiffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach By Ioan Mihai Oancea and Sylianos Perrakis This version February 00 Naional Bank of Canada, 30 King Sree Wes, Torono,
More informationA MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES.
A MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES. DONATIEN HAINAUT, PIERRE DEVOLDER. Universié Caholique de Louvain. Insiue of acuarial sciences. Rue des Wallons, 6 B1348, LouvainLaNeuve
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationTechnical Appendix to Risk, Return, and Dividends
Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,
More informationAn Optimal Selling Strategy for Stock Trading Based on Predicting the Maximum Price
An Opimal Selling Sraegy for Sock Trading Based on Predicing he Maximum Price Jesper Lund Pedersen Universiy of Copenhagen An opimal selling sraegy for sock rading is presened in his paper. An invesor
More informationOptimal Life Insurance Purchase, Consumption and Investment
Opimal Life Insurance Purchase, Consumpion and Invesmen Jinchun Ye a, Sanley R. Pliska b, a Dep. of Mahemaics, Saisics and Compuer Science, Universiy of Illinois a Chicago, Chicago, IL 667, USA b Dep.
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationOptimal Life Insurance, Consumption and Portfolio: A Dynamic Programming Approach
28 American Conrol Conference Wesin Seale Hoel, Seale, Washingon, USA June 1113, 28 WeA1.5 Opimal Life Insurance, Consumpion and Porfolio: A Dynamic Programming Approach Jinchun Ye (Pin: 584) Absrac A
More informationCredit Index Options: the noarmageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 2324, 2008 Credi Index Opions: he noarmageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo  Join work
More informationAND BACKWARD SDE. Nizar Touzi nizar.touzi@polytechnique.edu. Ecole Polytechnique Paris Département de Mathématiques Appliquées
OPIMAL SOCHASIC CONROL, SOCHASIC ARGE PROBLEMS, AND BACKWARD SDE Nizar ouzi nizar.ouzi@polyechnique.edu Ecole Polyechnique Paris Déparemen de Mahémaiques Appliquées Chaper 12 by Agnès OURIN May 21 2 Conens
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationOn Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations
On Galerkin Approximaions for he Zakai Equaion wih Diffusive and Poin Process Observaions An der Fakulä für Mahemaik und Informaik der Universiä Leipzig angenommene DISSERTATION zur Erlangung des akademischen
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationnonlocal conditions.
ISSN 17493889 prin, 17493897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.39 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed
More informationOptimal Time to Sell in Real Estate Portfolio Management
Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and JeanLuc Prigen hema, Universiy of CergyPonoise, CergyPonoise, France Emails: fabricebarhelemy@ucergyfr; jeanlucprigen@ucergyfr
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationOptimal Life Insurance Purchase and Consumption/Investment under Uncertain Lifetime
Opimal Life Insurance Purchase and Consumpion/Invesmen under Uncerain Lifeime Sanley R. Pliska a,, a Dep. of Finance, Universiy of Illinois a Chicago, Chicago, IL 667, USA Jinchun Ye b b Dep. of Mahemaics,
More informationLongevity 11 Lyon 79 September 2015
Longeviy 11 Lyon 79 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univlyon1.fr
More informationValuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate
Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More information= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting OrnsteinUhlenbeck or Vasicek process,
Chaper 19 The BlackScholesVasicek Model The BlackScholesVasicek model is given by a sandard imedependen BlackScholes model for he sock price process S, wih imedependen bu deerminisic volailiy σ
More informationT ϕ t ds t + ψ t db t,
16 PRICING II: MARTINGALE PRICING 2. Lecure II: Pricing European Derivaives 2.1. The fundamenal pricing formula for European derivaives. We coninue working wihin he Black and Scholes model inroduced in
More informationPATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM
PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu
More informationMultiprocessor SystemsonChips
Par of: Muliprocessor SysemsonChips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,
More informationOptimalCompensationwithHiddenAction and LumpSum Payment in a ContinuousTime Model
Appl Mah Opim (9) 59: 99 46 DOI.7/s45895 OpimalCompensaionwihHiddenAcion and LumpSum Paymen in a ConinuousTime Model Jakša Cvianić Xuhu Wan Jianfeng Zhang Published online: 6 June 8 Springer Science+Business
More informationKeldysh Formalism: Nonequilibrium Green s Function
Keldysh Formalism: Nonequilibrium Green s Funcion Jinshan Wu Deparmen of Physics & Asronomy, Universiy of Briish Columbia, Vancouver, B.C. Canada, V6T 1Z1 (Daed: November 28, 2005) A review of Nonequilibrium
More informationEfficient Risk Sharing with Limited Commitment and Hidden Storage
Efficien Risk Sharing wih Limied Commimen and Hidden Sorage Árpád Ábrahám and Sarola Laczó March 30, 2012 Absrac We exend he model of risk sharing wih limied commimen e.g. Kocherlakoa, 1996) by inroducing
More informationWorking Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits
Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 20080530 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More informationSEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION. Chavchavadze Ave. 17 a, Tbilisi, Georgia, Email: toronj333@yahoo.
SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION N. LAZRIEVA, 2, T. SHARIA 3, 2 AND T. TORONJADZE Georgian American Universiy, Business School, 3, Alleyway II, Chavchavadze Ave.
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationDifferential Equations in Finance and Life Insurance
Differenial Equaions in Finance and Life Insurance Mogens Seffensen 1 Inroducion The mahemaics of finance and he mahemaics of life insurance were always inersecing. Life insurance conracs specify an exchange
More informationOptimal Consumption and Insurance: A ContinuousTime Markov Chain Approach
Opimal Consumpion and Insurance: A ConinuousTime Markov Chain Approach Holger Kraf and Mogens Seffensen Absrac Personal financial decision making plays an imporan role in modern finance. Decision problems
More informationForecasting and Information Sharing in Supply Chains Under QuasiARMA Demand
Forecasing and Informaion Sharing in Supply Chains Under QuasiARMA Demand Avi Giloni, Clifford Hurvich, Sridhar Seshadri July 9, 2009 Absrac In his paper, we revisi he problem of demand propagaion in
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationDistributing Human Resources among Software Development Projects 1
Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources
More informationThe Uncertain Mortality Intensity Framework: Pricing and Hedging UnitLinked Life Insurance Contracts
The Uncerain Moraliy Inensiy Framework: Pricing and Hedging UniLinked Life Insurance Conracs Jing Li Alexander Szimayer Bonn Graduae School of Economics School of Economics Universiy of Bonn Universiy
More informationDETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUNSHAN WU
Yugoslav Journal of Operaions Research 2 (22), Number, 67 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUNSHAN WU Deparmen of Bussines Adminisraion
More informationOn Valuing EquityLinked Insurance and Reinsurance Contracts
On Valuing EquiyLinked Insurance and Reinsurance Conracs Sebasian Jaimungal a and Suhas Nayak b a Deparmen of Saisics, Universiy of Torono, 100 S. George Sree, Torono, Canada M5S 3G3 b Deparmen of Mahemaics,
More informationTimeinhomogeneous Lévy Processes in CrossCurrency Market Models
Timeinhomogeneous Lévy Processes in CrossCurrency Marke Models Disseraion zur Erlangung des Dokorgrades der Mahemaischen Fakulä der AlberLudwigsUniversiä Freiburg i. Brsg. vorgeleg von Naaliya Koval
More informationINDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES
Journal of Applied Analysis 1, 1 (1995), pp. 39 45 INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES A. LUCZAK Absrac. We find exponens of independen marginals
More informationOPTIMAL LIFE INSURANCE PURCHASE, CONSUMPTION AND INVESTMENT ON A FINANCIAL MARKET WITH MULTIDIMENSIONAL DIFFUSIVE TERMS
OPTIMAL LIFE INSURANCE PURCHASE, CONSUMPTION AND INVESTMENT ON A FINANCIAL MARKET WITH MULTIDIMENSIONAL DIFFUSIVE TERMS I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA Absrac. We inroduce an exension
More informationLIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b
LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.
More informationA TwoAccount Life Insurance Model for ScenarioBased Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul
universiy of copenhagen Universiy of Copenhagen A TwoAccoun Life Insurance Model for ScenarioBased Valuaion Including Even Risk Jensen, Ninna Reizel; Schomacker, Krisian Juul Published in: Risks DOI:
More informationLectures # 5 and 6: The Prime Number Theorem.
Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζfuncion o skech an argumen which would give an acual formula for π( and sugges
More informationUNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment.
UNIVERSITY OF CALGARY Modeling of Currency Trading Markes and Pricing Their Derivaives in a Markov Modulaed Environmen by Maksym Terychnyi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL
More informationA ProductionInventory System with Markovian Capacity and Outsourcing Option
OPERATIONS RESEARCH Vol. 53, No. 2, March April 2005, pp. 328 349 issn 0030364X eissn 15265463 05 5302 0328 informs doi 10.1287/opre.1040.0165 2005 INFORMS A ProducionInvenory Sysem wih Markovian Capaciy
More informationCredit risk. T. Bielecki, M. Jeanblanc and M. Rutkowski. Lecture of M. Jeanblanc. Preliminary Version LISBONN JUNE 2006
i Credi risk T. Bielecki, M. Jeanblanc and M. Rukowski Lecure of M. Jeanblanc Preliminary Version LISBONN JUNE 26 ii Conens Noaion vii 1 Srucural Approach 3 1.1 Basic Assumpions.....................................
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationMASAHIKO EGAMI AND HIDEKI IWAKI
AN OPTIMAL LIFE INSURANCE PURCHASE IN THE INVESTMENTCONSUMPTION PROBLEM IN AN INCOMPLETE MARKET MASAHIKO EGAMI AND HIDEKI IWAKI Absrac. This paper considers an opimal life insurance purchase for a household
More informationStochastic Calculus and Option Pricing
Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus 15.450, Fall 2010 1 / 74 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 BlackScholes
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationI. Basic Concepts (Ch. 14)
(Ch. 14) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing
More informationLIFE INSURANCE MATHEMATICS 2002
LIFE INSURANCE MATHEMATICS 22 Ragnar Norberg London School of Economics Absrac Since he pioneering days of Black, Meron and Scholes financial mahemaics has developed rapidly ino a flourishing area of science.
More informationWhen to Cross the Spread?  Trading in TwoSided Limit Order Books 
When o Cross he Spread?  rading in wosided Limi Order Books  Ulrich Hors and Felix Naujoka Insiu für Mahemaik HumboldUniversiä zu Berlin Uner den Linden 6, 0099 Berlin Germany email: {hors,naujoka}@mah.huberlin.de
More informationA Probability Density Function for Google s stocks
A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural
More informationDifferential Equations and Linear Superposition
Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y
More informationNetwork Effects, Pricing Strategies, and Optimal Upgrade Time in Software Provision.
Nework Effecs, Pricing Sraegies, and Opimal Upgrade Time in Sofware Provision. YiNung Yang* Deparmen of Economics Uah Sae Universiy Logan, UT 84322353 April 3, 995 (curren version Feb, 996) JEL codes:
More informationMarkov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension
Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical
More informationNiche Market or Mass Market?
Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.
More information