Mean-Variance Portfolio Selection for a Non-life Insurance Company

Transcription

1 Mean-Variance Porfolio Selecion for a Non-life Insurance Company Łukas Delong 1,2,, Russell Gerrard 2 1 Insiue of Economerics, Division of Probabilisic Mehods, Warsaw School of Economics, Niepodległości 162, Warsaw, Poland 2 Faculy of Acuarial Science and Insurance, Cass Business School, 16 Bunhill Row, London EC1Y 8TZ, Unied Kingdom Absrac We consider a collecive insurance risk model wih a compound Cox claim process, in which he evoluion of a claim inensiy is described by a sochasic differenial equaion driven by a Brownian moion. The insurer operaes in a financial marke consising of a risk-free asse wih a consan force of ineres and a risky asse which price is driven by a Lévy noise. We invesigae wo opimiaion problems. The firs one is he classical mean-variance porfolio selecion. In his case he efficien fronier is derived. The second opimiaion problem, excep he mean-variance erminal objecive, includes also a running cos penaliing deviaions of he insurer s wealh from a specified profi-solvency arge which is a random process. In order o find opimal sraegies we apply echniques from he sochasic conrol heory. Keywords: Lévy diffusion financial marke, compound Cox claim process, Hamilon-Jacobi-Bellman equaion, Feynman-Kac represenaion, efficien fronier. corresponding auhor, lukas.delong@sgh.waw.pl, el/fax:+48)

2 1 Inroducion The idea of he classical mean-variance porfolio selecion is o consruc he bes allocaion of wealh among available asses in order o achieve he opimal rade-off beween he expeced reurn on an invesmen and is risk, measured by he variance. The mean-variance opimiaion problem was firs inroduced in Markowi 1952) and his work has laid down foundaions of he modern financial heory. In his paper we exend he exising resuls concerning he mean-variance porfolio selecion. The new feaures are a financial marke wih an asse driven by an infinie acive Lévy process, a sochasic claim inensiy leading o a compound Cox claim process and a mean-variance erminal objecive wih a running cos. To bes our knowledge he opimiaion in his framework is aken for he firs ime. The goal is o derive, in our new general model, an invesmen sraegy, verify is opimaliy and idenify he efficien fronier. Anoher conribuion is o find a classical soluion of he corresponding Hamilon-Jacobi-Bellman equaion. We arrive a a quadraic soluion wih coefficiens depending on he ime variable and he level of a claim inensiy. The exisence of a classical soluion, and is form, is no surprising, however, i seems o appear for he firs ime in he conrol lieraure. Opimal invesmen problems for non-life insurers have recenly gained a lo of aenion, see among ohers, Browne 1995), Schmidli 22), Yang, Zhang 25), Schäl 25), Taksar 2), Højgaard, Taksar 24). The objecives adoped in hese papers are he ulimae ruin probabiliy or he expeced value of dividends paid unil ruin, which are he mos common ones in he acuarial lieraure. This is in conras wih he objecive applied widely in he financial lieraure, which is he expeced uiliy of a erminal wealh. We are aware of only hree papers, Browne 1995), Yang, Zhang 25) and Wang 26), in which he expeced value of an exponenial uiliy of a erminal reserve is maximied in order o arrive a opimal invesmen sraegies. The applicaion of a quadraic disuiliy in he field of a non-life insurance business seems o be quie new. A he same ime we mus noice ha he arge-based approach o decision making under uncerainy and he mean square objecive is very common in pensions, see for example Gerrard e al 24), Delong e al 27) and references herein. In Delong 25) a quadraic problem, wihou a erminal consrain bu wih a running cos penaliing deviaions from a deerminisic arge, is considered for a general insurer operaing in a Black-Scholes marke and covering claims generaed by a compound Poisson process. Very recenly, he Markowi s porfolio selecion problem for a non-life insurer has been solved, and he efficien fronier has been derived in Wang e al 26). We would like o poin ou ha here are significan differences beween our paper and heir. In Wang e al 26) maringale mehods and Backward Sochasic Differenial Equaions echniques are applied in order o find he opimal invesmen sraegy for an insurer which risk process is modelled as a Lévy process and he surplus is invesed in a sandard Black-Scholes 2

3 financial marke. The erminal mean-variance objecive is only invesigaed. In his paper, we arrive a he opimal sraegy by solving he Hamilon-Jacobi-Bellman equaion in he case when an insurer s risk process is modelled as a compound Cox process a non-lévy process) wih a sochasic, diffusion-ype, claim inensiy and he surplus is invesed in a financial marke wih an asse which price is driven by a Lévy process possibly infinie acive). We addiionally consider a running cos, wih a random arge process o be reached by he insurer, and his leads o a socalled wealh-pah dependen opimiaion problem, see Bouchard, Pham 24). Moreover, he sraegy derived in Wang e al 26) is expressed in he erms of an "arificial" process, resuling from solving he backward sochasic differenial equaion, which would be difficul o applied in he real-life seing. Our sraegy is expressed direcly in he erms of he insurer s wealh process and can easily be applied. As far as he mean-variance opimiaion objecives are concerned, i is also worh menioning he paper of Bäuerle 25), where he opimal proporional reinsurance sraegy is derived in a Cramér-Lundberg model. In he financial lieraure he mean-variance objecive is usually applied o solve porfolio selecion problems for self-financing wealh processes. In Zhou, Li 2) he original mean-variance problem is embedded ino an auxiliary problem, which is hen solved by he sochasic maximum principle. Techniques of Backward Sochasic Differenial Equaions are applied in Bielecki e al 25) and Lim 24), in he presence of random marke coefficiens. We refer he reader o Guo, Xu 24) where he Markowi s porfolio selecion is solved in a financial marke consising of asses which prices are driven by a jump-diffusion process, as well as o discree ime model in Cakmak, Öekici 26), where random reurns of asses depend on he sae of an observable Markov chain. We would like o poin ou ha in all he above menioned works here is no running cos included, excep in Gerrard e al 24) and Delong 25), an insurance risk process is modelled as a Brownian moion or as/wih a compound Poisson process, excep in Wang 26) and Wang e al 26), where a pure jump process, respecively a Lévy process is considered, a financial marke is of Black-Scholes ype, excep in Guo, Xu 24), and a arge is always deerminisic. This paper is srucured as follows. In secion 2 we inroduce a financial marke and an insurance risk process. Our wo opimiaion problems are formulaed in secion 3 and hen solved in secion 4. In secion 5, in he case of he Markowi s classical mean-variance porfolio selecion for he company, he efficien fronier is derived. Some numerical example is also invesigaed. We end wih summariing commens in secion 6. The proofs are posponed o he appendix. 3

4 2 The model Le us consider a probabiliy space Ω, F, P) wih a filraion F = F ) T for some finie T which denoes he invesmen ime horion. The filraion saisfies he usual hypoheses of compleeness F conains all ses of P-measure ero) and righ coninuiy F = F + ). The filraion F consiss of hree subfilraions: we se F = F F F C F I, where F F conains informaion abou a financial marke, F C conains informaion abou a claim process and F I conains informaion abou a claim inensiy. We assume ha he subfilraions F F and F C, F I ) are independen. The measure P is he real-world, objecive probabiliy measure. All expeced values are aken wih respec o measure P and he condiional expeced value E P [ X) = x, λ) = λ] is denoed as E,x,λ [ ]. The class of funcions C 1,2,2 [, T ) R, )) C[, T ] R, )), which consiss of funcions coninuous on [, T ] R, ) and once coninuously differeniable wih respec o ime variable and wice coninuously differeniable wih respec o space variables on [, T ) R, ), is denoed simply by C. In he following subsecions we inroduce a financial marke, a claim process and a claim inensiy process. 2.1 The financial marke We consider a Lévy diffusion version of a Black-Scholes financial marke. The price of a risk-free asse S := S ), T ) is described by he ordinary differenial equaion ds ) S ) = rd, S ) = 1, 2.1) where r denoes a rae of ineres. The second radeable financial insrumen in he marke is a risky sock and he dynamics of is price S := S), T ) is given by he sochasic differenial equaion ds) S ) = µd + ξdl), S) = 1, 2.2) where µ and ξ denoe a drif and a volailiy, and L := L), T ) denoes a ero-mean Lévy process a process wih independen and saionary incremens), F F -adaped wih càdlàg sample pahs pahs which are coninuous on he righ and have limis on he lef). I is argued ha Lévy processes can capure price movemens in a much beer way, see Con, Tankov 24) and Kyprianou e al 25). The ero-mean Lévy process L is assumed o saisfy he following Lévy-Iô decomposiion L) = σw ) + Mds d) νd)ds ), 2.3),] R 4

5 where W := W ), T ) is a Brownian moion and Ms, ] A) = #{s < u : Lu) Lu )) A} is a Poisson random measure, independen of W, wih a deerminisic, ime-homogeneous inensiy measure νd)d. This Poisson random measure couns he number of jumps of a paricular sie in a given ime inerval. Le us recall ha M, ] A) = M, ] A) νa) is a maringale-valued measure, ha is M := M, ] A), T ) is a F F -maringale for all Borel ses A BR {}). We refer he reader o Applebaum 24) for mahemaical deails concerning Lévy processes and Poisson random measures. We make he following assumpions concerning he coefficiens and he inensiy measure: A1) r, µ, σ are non-negaive consans and r < µ, A2) we se ξ 1, his is no loss of generaliy as he process ξl has also independen and saionary incremens and saisfies he Lévy-Iô decomposiion, A3) ν is a Lévy measure on 1, ), such ha ν{}) = and 1 4 νd) <. We recall ha a Lévy measure is a measure which verifies R 2 1)νd) <. The momen condiion in A3) ensures ha sup [,T ] E[ L) 4 ] <. The sochasic differenial equaion 2.2) has a unique, posiive and almos surely finie soluion, given explicily by Doléans-Dade exponenial S) = exp { µ 1 2 σ2 + log1 + ) )νd) ) > 1 +σw ) + log1 + ) Mds d) },] > 1 = exp { µ E + σw ) + M E ds d) } 2.4) which is an exponenial Lévy process wih he measure ν E A) = ν{ : log1 + ) A}), see Con, Tankov 25), proposiions 8.21 and The measure ν E saisfies he equivalen condiion A3 ): A3 ) ν E is a Lévy measure on R, such ha ν E {}) = and 1 e4 ν E d) <. I is well known ha here is one o one correspondence beween he measures and he sock price models 2.2) and 2.4). We would like o poin ou ha in chaper 3 in Kyprianou e al 25) he inensiy measures ν E for Variance Gamma and CGMY processes are esimaed for five world index markes and in each case he esimaed measure saisfies A3 ). We refer he reader o Con, Tankov 25) and Kyprianou e al 25) in which differen aspecs of financial modelling wih Lévy diffusion processes are invesigaed.,] R 5

6 2.2 The insurance risk process We consider a collecive insurance risk model. Le C) denoe an aggregae claim amoun paid up o ime. We assume ha he process C := C), T ) is a compound Cox process which means ha i has he following represenaion N) C) = Y i, 2.5) i=1 where {Y i, i N} is a sequence of posiive, independen and idenically disribued random variables wih a disribuion F y) = PY i y) and N := N), T ) is a Cox process wih a sochasic inensiy process Λ := λ), T ) and a law given by PN) = k F I ) = e H) H)) k, k =, 1, 2..., 2.6) k! where H denoes a cumulaive haard funcion H) = λs)ds. 2.7) We assume ha he aggregae claim process C is F C -adaped wih càdlàg sample pahs. Clearly, Y 1, Y 2,... denoe he amouns of successive claims and N couns he number of claims. A Cox process is a common alernaive o a Poisson process in he risk heory and in insurance modelling of claim processes, see Klüppelberg, Mikosch 1995), Rolski e al 1999), Dassios, Jang 26) and references herein. Noice ha a Cox process does no have independen and saionary incremens and is of finie variaion on [, T ]. We would like o poin ou ha he compound Cox process can also be defined hrough a measure Nd, d) as C) = Nds d), 2.8),] where Ns, ] A) = #{s < u : Cu) Cu )) A} is a finie random measure wih a random compensaor df )λ)d. We find i convenien o use he represenaion 2.8) in he proof of he verificaion heorem. We assume ha he dynamics of he claim inensiy Λ is given by he sochasic differenial equaion dλ) = θ, λ))d + η, λ))d W ), λ) = λ, 2.9) where W := W ), T ) is an F I -adaped Brownian moion, independen of W and M. We make he following assumpions concerning he insurance risk process: 6

7 B1) he disribuion F has a finie fourh momen, 4 df ) <, B2) θ : [, T ], ) R, η : [, T ], ), ) are coninuous funcions, locally Lipschi coninuous in λ, uniformly in, B3) here exiss a sequence E n ) n N of bounded domains wih Ēn, ) and n 1 E n =, ), such ha he funcions θ, λ) and η, λ) are uniformly Lipschi coninuous on [, T ] Ēn, B4) P s [,T ] λs), ) λ) = λ) = 1 for all saring poins, λ) [, T ], ), B5) sup s [,T ] E,λ [ λs) 4 ] < for all saring poins, λ) [, T ], ). Under he assumpions B2) and B4), for each saring poin, λ) [, T ], ), he inensiy process is nonexplosive on [, T ] and here exiss a unique srong soluion o he sochasic differenial equaion 2.9), such ha he mapping, λ, s) λ,λ s) is P-a.s. coninuous, see Heah, Schweier 2), Becherer, Schweier 25). The assumpions B1), B3), B5) ogeher wih A3)/A3 ), are required in he verificaion resul and ensure ha a classical soluion o our Hamilon-Jacobi-Bellman equaion exiss. We remark ha from he poin of he company he assumpion concerning he measurabiliy of he claim inensiy and is diffusion ype is very reasonable. In he lieraure one can find ha, for example, a disconinuous sho noise process is quie useful in modelling claim inensiies, see Klüppelberg, Mikosch 1995), Dassios, Jang 25) for commens. However, i is proved in Dassios, Jang 25), and inuiively i is clear, ha a sho noise process converges o a diffusion process in he case of high frequency evens, such as accidens from a large collecive porfolio. Moreover, he company should also have enough daa o esimae he inensiy correcly. 3 Problem formulaion In his paper we invesigae wo opimiaion problems which are quie similar in mahemaical formulaion bu he moivaion for saing and solving hem is a lile differen. Firsly, le us deal wih a porfolio selecion for a general insurance company. Le us consider a wealh process of he insurer X π := X π ), T ). Is dynamics are given by he sochasic differenial equaion dx π ) = π) µd + σdw ) + Md d) ) > 1 + X π ) π) ) rd + c)d dc), X) = x, 3.1) where π denoes he amoun of he wealh invesed in he risky asse, c denoes a premium rae and x denoes an iniial capial. We assume, as in he classical risk 7

8 heory, see Rolski e al 1999), ha c)d = 1 + θ)µ 1 λ )d, µ 1 = df ), 3.2) which means ha he premium colleced over an infiniesimal inerval d is equal o he expeced value of claims paid over his inerval wih an addiional safey loading θ. According o he classical porfolio heory due o Markowi an invesmen sraegy should be chosen in he following way { inf π Var[X π T )], 3.3) E[X π T )] = P where P is a specified arge. The second opimiaion problem which we consider, can be applied o an individual policy. In his case he value process of he issued policy X π evolves according o he sochasic differenial equaion dx π ) = π) µd + σdw ) + Md d) ) > 1 + X π ) π) ) rd dc), X) = x, 3.4) where, his ime, x denoes a single premium for he conrac. The opimiaion funcional can include, apar from he erminal mean-variance objecive, a running cos penaliing deviaions of he value process from a arge process R), T ). An invesmen sraegy can be chosen in he following way { inf π E [ X π ) R) ) 2 ] d + αvar[x π T ) RT )] E [ X π T ) RT, 3.5) )] = where α > aaches a weigh o he erminal cos. The arge can combine wo elemens R) = P ) + R, λ)), [, T ] 3.6) where P ) denoes a profi which should accumulae unil ime, for example P ) = xe ρ, ρ >, and R, λ) denoes a reserve which should be kep a ime, when he claim inensiy equals λ, in order o mee fuure conracual obligaions. We need he following echnical assumpion concerning he profi funcion P : C) P : [, T ] [, ) is Lipschi coninuous. We believe ha he inclusion of he running cos in he opimiaion problem is very reasonable as i means ha he opimally conrolled policy should generae values saisfying he desired profi as well as he required solvency consrains during he whole erm of he conrac. We assume ha he reserve for ousanding liabiliies R is calculaed as he 8

9 expeced presen value of fuure undiscouned paymens, condiioned on he given level of he claim inensiy. I is sraighforward o calculae he value of he reserve R, λ) = µ 1 E,λ [ We can prove he following lemma. λs)ds]. 3.7) Lemma 3.1. The funcion R : [, T ], ) [, ), defined in 3.7), is in he class C and saisfies he following parial differenial equaion R, λ) + θ, λ) R λ, λ) η2, λ) 2 R λ, λ) + µ 1λ =, RT, λ) =. 3.8) 2 Moreover, he equaion 3.8) has he unique soluion in he class C. 4 Soluion of opimiaion problems In his par of he paper we solve boh problems saed in he previous secion. We deal wih he following consrained opimiaion problem { inf π αe [ X π ) R) ) 2 ] d + 1 α)var[x π T ) RT )] E [ X π T ) RT, 4.1) )] = where α [, 1) and he dynamics of he wealh process X π is given by he sochasic differenial equaion dx π ) = π) µd + σdw ) + Md d) ) > 1 + X π ) π) ) rd θ)µ 1 λ)d dc), X) = x,4.2) Clearly, by seing α = and RT ) = P we recover he opimiaion problem 3.3), while seing θ = 1 leads o he opimiaion problem 3.5). The variance crierion can be handled by incorporaing he equaliy consrain on he expeced value ino he objecive funcion by using a Lagrange muliplier. Firs we can solve he sochasic conrol problem inf E,x,λ[ α π A X π ) P ) R, λ)) ) 2 d +1 α) X π T ) P T ) ) 2 β X π T ) P T ) )], 4.3) and hen choose a Lagrange muliplier β such ha he consrain on he expeced value of he erminal wealh is saisfied E,x,λ [X ˆπ,β T )] = P T ), 4.4) where ˆπ is he opimal sraegy for 4.3). Le us inroduce he se of admissible sraegies and hree operaors. 9

10 Definiion 4.1. A sraegy π), < T ) is admissible, π A, if i saisfies he following condiions: 1. π :, T ] Ω R is a predicable mapping wih respec o filraion F, 2. E,x,λ [ π2 )d] <, 3. he sochasic differenial equaion 4.2) has a unique soluion X π on [, T ]. I is well-known ha i is sufficien o consider only Markov sraegies, see Øksendal, Sulem 25), chaper 3. We poin ou ha for any π A he process X π, which saisfies 4.2), is a square inegrable semimaringale wih cádlág sample pahs, see Applebaum 24), chaper Moreover, for all [, T ] he following inequaliy holds R, λ)) 2 T µ 2 1E[ λ 2 s)ds F ], P a.s., 4.5) due o he Markov propery of he claim inensiy process, Jensen s inequaliy for condiional expecaions and he Cauchy-Schwar inequaliy. The esimae 4.5) allows us o prove he square inegrabiliy of he reserve process R, λ)), T ). For all [, T ] we have E,λ [ R, λ)) 2 ] T µ 2 1E,λ [ λ 2 s)ds] T µ 2 1E,λ [ λ 2 s)ds] <, 4.6) where he law of ieraed expecaions has been applied. We can conclude ha he objecive funcion 4.3) is well-defined. Definiion 4.2. The inegro-differenial operaor L F is given by L π F φ, x, λ) = πµ r) + xr θ)µ 1 λ ) φ x, x, λ) π2 σ 2 2 φ, x, λ) x2 φ + φ, x + π, λ) φ, x, λ) π x, x, λ)) νd), 4.7) > 1 he inegral operaor L C is given by L C φ, x, λ) = λ and he differenial operaor L I is given by φ, x, λ) φ, x, λ) ) df ), 4.8) L I φ, x, λ) = θ, λ) φ λ, x, λ) η2, λ) 2 φ, x, λ). 4.9) λ2 This operaors are defined for all funcions φ such ha he parial derivaives and he inegrals in 4.7), 4.8) and 4.9) exis poinwise. 1

11 Le us inroduce he opimal value funcion for he problem 4.3) V, x, λ) = inf π A E,x,λ[ α X π ) P ) R, λ)) ) 2 d +1 α) X π T ) P T ) ) 2 β X π T ) P T ) )], 4.1) In he appendix we prove he following classical verificaion heorem. Theorem 4.1. Le v C saisfy for all π A α x P ) R, λ) ) 2 + v, x, λ) + Lπ F v, x, λ) +L I v, x, λ) + L C v, x, λ), 4.11) for all, x, λ) [, T ) R, ), wih vt, x, λ) = 1 α) x P T ) ) 2 β x P T ) ), x, λ) R, ). 4.12) Assume ha for all π A E,x,λ[ v, X π ) + π), λ)) v, X π ), λ)) 2 νd)d ] <, and E,x,λ[ > 1 > 1 v, X π ) + π), λ)) v, X π ), λ)) 4.13) π) v x, Xπ ), λ)) νd)d ] <, 4.14) E,x,λ[ v, X π ), λ)) v, X π ), λ)) 2 λ)df )d ] <, 4.15) {v + τ, X π τ), λτ))} <τ T is uniformly inegrable for all F-sopping imes τ.4.16) Then v, x, λ) V, x, λ),, x, λ) [, T ] R, ). 4.17) Moreover, if here exiss an admissible conrol ˆπ such ha = α x P ) R, λ) ) 2 + v, x, λ) + Lˆπ F v, x, λ) +L I v, x, λ) + L C v, x, λ), 4.18) for all, x, λ) [, T ) R, ), and hen {vτ, X ˆπ τ), λτ))} <τ T is uniformly inegrable for all F-sopping imes τ,4.19) v, x, λ) = V, x, λ),, x, λ) [, T ] R, ). 4.2) and ˆπ is he opimal sraegy. 11

12 In he above heorem v + := maxv, ) denoes a posiive par of a funcion v. As our opimiaion problem 4.3) is quadraic i is naural o ry o find a soluion in he form v, x, λ) = A, λ)x 2 + B, λ)x + C, λ). Wih his choice of he value funcion he opimal sraegy ˆπ which minimies he righ hand side of 4.11) is given by ˆπ, x, λ) = δ x + ) B, λ) µ r, δ = 2A, λ) σ 2 + > 1 2 νd). 4.21) Subsiuing 4.21) ino 4.18) and collecing he erms we arrive a hree parial differenial equaions { A, λ) + L IA, λ) + 2r δ)a, λ) + α =, 4.22) AT, λ) = 1 α, B, λ) + L IB, λ) + r δ)b, λ) 2α P ) + R, λ) ) +2θµ 1 λa, λ) =, BT, λ) = 21 α)p T ) β, C, λ) + L IC, λ) + λµ 2 A, λ) + θµ 1 λb, λ) + α P ) + R, λ)) ) 2 B2,λ) 4A,λ) δ =, CT, λ) = 1 α)p 2 T ) + βp T ), 4.23) 4.24) where δ = δµ r) and µ 2 = 2 df ). A soluion o 4.22) can be saed explicily by noicing ha he ime-dependen funcion of he form A) = 1 α)e 2r δ)t ) + α 2r δ e2r δ)t ) 1) 4.25) saisfies 4.22). I is easy o check ha he funcion A) is sricly posiive and is inverse is bounded uniformly on [, T ]. As far as he nex wo parial differenial equaions are concerned we can prove he following lemma. Lemma 4.1. There exis unique soluions B, C C o he parial differenial equaions 4.23) and 4.24). These soluions saisfy B, λ) K 1 + E,λ[ λs)ds ]), 4.26) C, λ) K 1 + E,λ[ λ 2 s)ds ]), 4.27) for all, λ) [, T ], ) and some finie consan K >. In paricular, one can sae he Feynman-Kac represenaion of he unique soluion o 4.23) B, λ) = 21 α)p T ) + β ) e r δ)t ) 2α P s)e r δ)s ) ds +2µ 1 E,λ[ ) θas)e r δ)s ) α er δ)s ) 1 λs)ds ].4.28) r δ 12

13 Noice ha afer specifying he claim inensiy process and calculaing is expeced value, he soluion o 4.23) migh be given explicily hrough 4.28), afer inegraing some deerminisic funcions. Le us now invesigae he wealh process X ˆπ under he opimal sraegy. Is dynamics is given by he sochasic differenial equaion dx ˆπ ) = { ) δ X ˆπ B, λ)) ) + + X ˆπ )r θ)µ 1 λ) } d dc) δ X ˆπ ) + 2A) B, λ)) 2A) ) σdw ) + > 1 ) Md d), 4.29) wih he iniial condiion X) = x. We can arrive a he following resul. Lemma 4.2. The sochasic differenial equaion 4.29), given he iniial condiion X) = x R, has a unique soluion on [, T ] in he space of semimaringale processes wih càdlàg sample pahs. This soluion has a finie fourh momen, sup [,T ] E,x,λ [ X ˆπ ) 4 ] <. We proceed now o find he value of he Lagrange muliplier. The Iô differenial 4.29) can be rewrien in he inegral form ) { X ˆπ ) = x + δ X ˆπ Bs, λs)) s ) + 2As) +X ˆπ s )r θ)µ 1 λs) } ds C) δ X ˆπ s ) + Bs, λs)) 2As) ) σdw s) + > 1 ) Mds d) 4.3) Taking he expeced value on boh sides of 4.3) and applying Fubini s heorem we arrive a ϕ) = x + where we define for s T and { Bs) ) δ ϕs ) + + rϕs ) + θµ1 ms)}ds, 4.31) 2As) s ϕs) = E,x,λ [X ˆπ s)], 4.32) ms) = E,λ [λs)], 4.33) Bs) = 21 α)p T ) + β ) e r δ)t s) 2α P u)e r δ)u s) du s ) +2µ 1 θau)e r δ)u s) α er δ)u s) 1 mu)du. 4.34) r δ We poin ou ha he expeced values of he sochasic inegrals in 4.3) are indeed equal o ero due o he square inegrabiliy of he processes X ˆπ and B, λ)), 13

14 T ). In he view of lemma 4.2 he firs saemen is clear, and in order o esablish he second one, noice ha for all [, T ] he following inequaliies hold P-a.s. B, λ)) 2 K 1 + E[ λs)ds F ] ) 2 T K1 1 + E[ λ 2 s)ds F ] ), 4.35) due o he Markov propery of he inensiy process, he esimae 4.26), Jensen s inequaliy for condiional expecaions and he Cauchy-Schwar inequaliy. Then, aking he expeced value on boh sides of 4.35) and applying he law of ieraed expecaions we arrive a E,λ [ B, λ) 2 ] K E,λ [ λ 2 s)ds] ) T K E,λ [ λ 2 s)ds] ) <, 4.36) for all [, T ]. The funcion 4.34) arises due o he Markov propery of he inensiy process and he law of ieraed expecaions. I is easy o show ha he funcion ϕ saisfying 4.31) mus be coninuous and differeniable. The inegral equaion 4.31) can be ransformed back ino he ordinary differenial equaion dϕ d ) = r δ ) ϕ) + θµ 1 m) δ B), ϕ) = x, 4.37) 2A) which can be solved resuling in ϕt ) = xe r δ)t + θµ 1 m)e r δ)t ) d δ B) 2A) er δ)t ) d. 4.38) I is lef o find he value of β such ha he consrain ϕt ) = P T ) is saisfied. Wih a lile algebra we can arrive a he value of he Lagrange muliplier ) 1 δ1 α)β2 P T ) xe r δ)t β 1 δβ 3 β = 2, 4.39) δβ 2 where and β 3 = µ 1 +α β 1 = θµ 1 m)e r δ)t ) d, 4.4) β 2 = r δ)t ) e T A) r δ)t ) e A) 2r δ)t ) e A) α er δ)s ) 1 r δ We conclude wih he heorem summariing our resuls. d, 4.41) θas)e r δ)s ) ) ms)dsd P s)e r δ)s ) dsd. 4.42) 14

15 Theorem 4.2. The invesmen sraegy given by ˆπ) = δ X ˆπ ) + B, λ)) 2A) ), δ = µ r σ 2 + > 1 2 νd), 4.43) is he opimal invesmen sraegy for he consrained quadraic opimiaion problem 4.1), and he minimum cos is equal o A)x 2 + B, λ)x + C, λ). The funcions A, B, C and he consan β are given by 4.22)-4.24), 4,39), whereas X ˆπ is he wealh process under he opimal sraegy, evolving according o 4.29). 5 Efficien fronier and efficien porfolios In his secion we derive he efficien fronier and he efficien porfolios of he meanvariance porfolio selecion problem 3.3) for he general insurance company. Le us recall ha an efficien porfolio is one for which here does no exis anoher porfolio which has higher mean and no higher variance, and/or has less variance and no less mean, see Bielecki e al 25). As saed in Bielecki e al 25), he efficien fronier can be derived from he variance minimiing fronier. In our case he variance minimiing fronier is jus he opimal value funcion v, x, λ), which can be recovered from he resuls of he previous secion by seing α =. Afer edious calculaions, which we omi, one can show ha he Lagrange muliplier is equal o β = 2 P ert x + θµ 1 E,λ [ e r λ)d]), 5.1) e δt 1 and he variance minimiing fronier, as he funcion of P = E[X ˆπ T )], is equal o where Var[X ˆπ T )] = µ 2 e 2r δ)t E,λ[ e 2r δ) λ)d ] + θ 2 µ 2 1e 2r δ)t W + P e rt x + θµ 1 E,λ [ e r λ)d] )) 2 e δt 1 [ ] W = 2E,λ e δ e r λ) e rs λs)dsd [ E,λ δe δ E [ ] ) ] 2 e rs λs)ds F d, 5.2) E,λ[ e r λ)d ]) ) Le us deal wih he consan W. Due o Jensen s inequaliy for condiional expecaions, Fubini s heorem and he law of ieraed expecaions, which are applied o 15

16 he second erm in 5.2), we can arrive a W 2E,λ [ e δ e r λ) ] e rs λs)dsd ] [ 2 E,λ δe δ e λs)ds) rs d E,λ[ e r λ)d ]) ) Le us consider for a momen a coninuous funcion f. I is easy o show, by inegraing by pars, ha he following equaliy holds 2 e δ f) fs)dsd = e δ d T fs)ds ) 2 d d = fs)ds ) 2 + δe δ fs)ds ) 2 ds. 5.5) Le us now pu fs) = e rs λs) and apply 5.5) o he firs wo erms in 5.4) under he expeced value. We can arrive a W E,λ[ e r λ)d ) 2] E,λ[ e r λ)d ]) ) This proves a raher obvious fac ha here exiss no "risk-free asse" in our economy. Even by choosing he expeced reurn arge P = e rt x + θµ 1 E,λ [ e r λ)d] ), which corresponds o invesing a ime = all our wealh in he bank accoun, we are sill lef wih sricly posiive variance of he surplus due o possible claims. We can sae he following sraighforward lemma. Lemma 5.1. Le P = e rt x + θµ 1 E,λ [ e r λ)d] ). The variance minimiing fronier 5.1) is sricly decreasing for, P ) and sricly increasing for P, ). The efficien fronier is he subse of he variance minimiing fronier corresponding o P [P, ). I is inuiively clear, ha he insurer should se he arge such ha P P holds, as i should require he profi which would be no less han he expeced value of he fuure cash inflows/ouflows invesed in he bank accoun. The efficien porfolios are hose porfolios whose expeced reurns and corresponding variances lie on he efficien fronier. We poin ou ha he capial marke line, drawn in a mean-sandard deviaion plane, is no longer a sraigh line. Finally, le us recall he well-known facs, see Bielecki e al 25), ha in he case wihou claims, he variance of he erminal wealh could be represened as a complee square, here would be a "risk free asse" - a bank accoun, and he capial marke line would be a sraigh line. 16

17 Example 5.1. Le us assume ha he sock price follows an exponenial Variance Gamma process of he form S) = e,28+l), L) =, 2h) +, 2W h)), 5.7) where h) is a Gamma disribued random variable wih he densiy funcion g h) y) = 1 Γ/, 3), 3) /,3 y,3 1 e y,3. 5.8) For he subordinaed Brownian moion represenaion of a Variance Gamma process we refer he reader o Con, Tankov 24) or Kyprianou e al 25). We remark ha his choice of parameers corresponds o µ =, 1. We assume ha he individual claims are exponenially disribued, wih he expeced value equals o µ 1 = 1, and ha he claim inensiy process follows an exponenial maringale of he form λ) = 1e,2 W ),2. 5.9) In able 1 we give some quaniies of he empirical disribuion of he erminal wealh XT ), based on our simulaion resuls, in hree cases: a) when he wealh is conrolled in order o minimie he mean square error and he sochasic naure of he claim inensiy is no aken ino accoun in he derivaion of he opimal sraegies and in he seing of he premiums, β =, E,λ [ λs)ds] = 1T ); b) when he wealh is conrolled in order o minimie he mean square error bu he sochasic naure of he claim inensiy is aken ino accoun, β = ; c) when he wealh is conrolled according o he sraegy 4.43) in order o minimie he variance and when he sochasic naure of he inensiy is aken ino accoun as well. We assume ha T = 2, r =, 5, θ =, 5, P = 75, X) = 5. Table 1: Disribuion of he wealh X2) Case "a" Case "b" Case "c" Mean value 66, ,462 75, Sandard deviaion 385,482 25,81 325,141 1s percenile -369,2 12,497-27,462 5h percenile,14 322, ,517 1h percenile 168, , ,588 9h percenile 1135, , ,875 95h percenile 1252,99 989, ,262 99h percenile 1434, , ,399 17

18 SD Expeced reurn Figure 1: Capial marke line One should noice ha he mean values of he wealh in he case "a" and he case "b" are he same, which can also be concluded based on 4.38). However, he sandard deviaion in he case "b" is lower han in he case "a", and he disribuion of he erminal wealh is hicker-ailed in he case "a", compared wih he case "b". This is he resul of aking ino accoun he changes in he claim inensiy over ime, as his decreases he variabiliy of he cash flows. The erminal consrain and posiive value of β lead o he sraegy of invesing higher amouns in he sock, see 4.43) and 4.28), and his explains he increase in he mean value, as well as in he sandard deviaion for he case "c", compared wih he case "b". The Lagrange muliplier in our example is equal o β = I is also clear ha he disribuion of he erminal wealh in he case "c" has hicker ails han in he case "b". Finally, i is worh noicing ha he sandard deviaion in he case "c" is lower han in he case "a", and ha he disribuion of he wealh has hicker ails in he case "a", compared wih he case "c". A he end, le us consider he efficien fronier and he efficien porfolios. In our example, he minimum expeced reurn, which should be required by he insurer, is equal o P = 665, 75. The capial marke line can be found in figure 1. 6 Conclusions In his paper we have deal wih a mean-variance porfolio selecion for a non-life insurance company. We have assumed ha insurance claims are generaed according 18

19 o a compound Cox process and ha an asse reurn is driven by a Lévy process. We believe ha his are very imporan exensions as far as inegraed risk managemen is concerned. We have arrived a he smooh opimal value funcion, he opimal sraegy and derived he efficien fronier in he Markowi s case. We have only considered diffusion ype claim inensiy processes. This assumpion is reasonable when dealing wih large collecive porfolios of independen risks. However, for an individual porfolio of a small sie, a disconinuous claim inensiy process seems o fi beer. We would like o poin ou ha one can choose a specific disconinuous claim inensiy process, like a sho noise process or a more general Ornsein-Uhlenbeck process driven by a subordinaor, apply he Feynman-Kac formula informally and hen ry o calculae he inegrals from he represenaion explicily by subsiuing he expeced value funcion of he inensiy. This migh be edious bu if he resul is a smooh funcion hen all lemmas and heorems saed in his paper will also be saisfied. Finally, we would like o refer he ineresed reader o he paper of Delong e al 26), where mean-variance opimiaion problems for he accumulaion phase in a defined benefi plan are considered, wih he aim of hedging an annuiy paymen for a reiree in he presence of a sochasic moraliy inensiy. 7 Appendix In order o prove he exisence of a unique soluion o a parial differenial equaion and is smoohness we use he heorem 1 from Heah, Schweier 2). Noice ha in our case he coefficiens are unbounded and he heorem canno be applied direcly. However, he problem lies only in aking limis under he expecaion so i is sufficien o esablished uniform inegrabiliy, as poined ou in Heah, Schweier 2). In he sequel, K denoes a consan whose value may change from one occurrence o he nex. Proof of lemma 3.1: Fix, λ) [, T ], ). Define he sequence { λ,y s)ds},y) U where U is a compac se around, λ). Due o boundness of he coninuous mapping, y, s) λ,y s) on compac ses, Lebesgue s dominaed convergence heorem yields ha lim,y),λ) λ,y s)ds = Applying he Cauchy-Schwar inequaliy we can arrive a E[ λ,y s)ds 2 ] KE[ λ,λ s)ds, P a.s. 7.1) λ,y s)) 2 ds] <,, y) U. 7.2) This implies ha he sequence { λ,y s)ds},y) U is uniformly inegrable, which ogeher wih a.s. convergence 7.1), esablishes he convergence in L 1 P). We can 19

20 now conclude ha he mapping, λ) R, λ) is coninuous. The res of he lemma follows from Heah, Schweier 2). Proof of lemma 4.1: Define he funcion B, λ) = 21 α)p T ) + β ) r δ)t ) e + E,λ[ { ) 2α P s) + Rs, λs)) + 2θµ1 As)λs) } e r δ)s ) ds ], 7.3) for all, λ) [, T ], ). Due o he Markov propery of he inensiy process and he law of ieraed expecaions we have E,λ [Rs, λs))] = E,λ[ µ 1 E[ s λu)du F s ] ] = µ 1 E,λ [ s λu)du], 7.4) for all s [, T ]. By applying Fubini s heorem, subsiuing 7.4) ino 7.3), and changing he order of inegraion under he expeced value in E,λ[ e r δ)s ) λu)duds ] = E,λ[ λu) s u e r δ)s ) dsdu ], 7.5) we can arrive a 4.28). From he proof of lemma 3.1, i is clear ha he sequence { ) } θas)e r δ)s ) α er δ)s ) 1 λ,y s)ds 7.6) r δ,y) U converges P-a.s., as, y), λ), and is uniformly inegrable, hence converges in L 1 P). We can now conclude ha he mapping, λ) B, λ) is coninuous. In order o prove ha 7.3) is he probabilisic represenaion of a unique classical soluion o he parial differenial equaion 4.23) we follow he proof of proposiion 2.3 in Becherer, Schweier 25). Choose ɛ > and consider he equaion 4.23) on he ime inerval [, T ɛ]. Noice ha he funcion R is uniformly Hölder coninuous on compac subses of [, T ɛ] Ēn. Based on heorem 1 in Heah, Schweier 2) we can conclude ha he coninuous mapping B : [, T ɛ], ) R B, λ) = E,λ r δ)t ɛ ) [BT ɛ, λt ɛ))e ɛ { ) + 2α P s) + Rs, λs)) + 2θµ1 As)λs) } e r δ)s ) ds ], 7.7) is he unique classical soluion o 4.23) on [, T ɛ]. As ɛ is arbirary, he exisence of a classical soluion o he parial differenial equaion 4.23) and is probabilisic represenaion 7.3) follow. The esimae 4.26) can be derived immediaely from he represenaion 4.28). Define he funcion C, λ) = 1 α)p 2 T ) + βp T ) + E,λ[ { λs)µ2 As) + θµ 1 λs)bs, λs)) +α P s) + Rs, λs)) ) 2 B 2 s, λs)) δ } ds ], 7.8) 4As) 2