Differential Equations in Finance and Life Insurance

Transcription

1 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 of sreams of paymens beween he insurance company and he conrac holder. These paymen sreams may cover he life ime of he conrac holder. Therefore, ime valuaion of money is crucial for any measuremen of paymens due in he pas as well as in he fuure. Life insurance companies never pu heir money under he pillow, and accumulaion and disribuion of capial gains were always par of he insurance business. Wih respec o he fuure, appropriae discouning of conracual obligaions qualifies he esimaes of liabiliies. Financial conracs specify an exchange of sreams of paymens as well. However, while he life insurance paymen sream is parly linked o he sae of healh of he insured, he financial paymen sream is linked o he sae of healh of an enerprise. Tha could be he sream of dividends disribued o he owners of he enerprise or he sream of claims coningen on he price of he enerprise paid o he holder of a so-called derivaive. The discipline of personal finance is paricularly closely linked o life insurance. Decisions on e.g. consumpion, invesmen, reiremen, and insurance coverage belong o some of he mos subsanial life ime financial decisions of an individual. Valuaion of paymen sreams is probably he mos imporan discipline in he inersecion beween finance and life insurance. Various valuaion dogmas are in play here. The principle of no arbirage and he marke efficiency assumpion are aken as given in he majoriy of modern academic approaches o valuaion of financial conracs. Life insurance conrac valuaion ypically relies on independence, or a leas asympoic independence, beween insured lives. Then he law of large numbers ensures ha reasonable esimaes can be found if he porfolio of insurance conracs is sufficienly large. Boh dogmas reduce he valuaion problem o being primarily a maer of calculaion of condiional expeced values. Condiional expeced values can be approached by several differen echniques. E.g. Mone Carlo simulaion explois ha condiional expeced values can be approximaed by empirical means. Someimes, however, one can go a leas par of he way by explici calculaions. E.g. if a series of auxiliary models wih explici expeced values converges owards he real model in such a way ha he series of explici expeced values converges o he desired quaniy. A differen roue can be aken when he underlying sochasic sysem is Markovian, i.e. if given he presen sae, he fuure is independen of he pas. Then soluions o cerain sysems of deerminisic differenial equaions can ofen be proved o characerize he condiional expeced values. This is he roue aken o various valuaion problems and opimizaion problems in finance and life insurance in his exposiion. Here, we jus sae he differenial equaions and do no discuss possible numerical soluions o hese, hough. 1

2 Valuaion is performed by calculaion of condiional expeced values. However, he claim o be evaluaed may conain decision processes in case of which he valuaion problem is exended o a maer of calculaing exrema of condiional expeced values. The exrema are aken over he se of admissible decision processes. However, also exrema of condiional expeced values can be characerized by differenial equaions, albei more involved. Also decision problems ha are no par of a valuaion problem are relevan and are sudied here. We solve boh a problem of minimizing expeced quadraic disuiliy and a problem of maximizing expeced power uiliy. In boh cases we sae differenial equaions characerizing he soluions. Acually, from a echnical poin of view, valuaion under decision aking and uiliy opimizaion basically only differ by he firs measuring sreams of paymens and he second measuring sreams of uiliy of paymens. Even from a qualiaive poin of view he disciplines are closely relaed, e.g. in he valuaion approach called uiliy indifference pricing ha we shall no deal wih here, hough. The models used in his aricle combine he geomeric Brownian moion modelling of financial asses wih he finie sae Markov chain modelling of he sae of a life insurance policy. However, he finie sae Markov chain model appears in finance in oher connecions han life insurance. Therefore he saed differenial equaions apply o oher fields of finance. One example is reduced form modelling of credi risk where he sae of healh, or in his connecion crediworhiness, of an enerprise can be modelled by a Markov chain. Anoher example is valuaion of innovaive enerprise pipelines. Many ypes of innovaive projecs may be modelled by a finie sae Markov chain. In e.g. drug developmen, he drug candidae can be in differen saes (phases) and cerain milesone paymens are conneced o cerain saes of he drug candidae. The lis of discoverers in he field of Markov processes and sysems of parial differenial equaions is awe-inspiring: Feller, Kolmogorov and Dynkin are he fahers of he connecion beween Markov processes and mahemaical analysis. Afer hem conribuions by Feynman, Kac, Davis, Bensoussan and Lions among ohers are relevan in he conex of his aricle. However, we concenrae on a few references on more recen applicaions relaed o he maerial of his aricle and enclose a secionwise ouline. Secion 2: Thiele wroe down in 1875 an ordinary differenial equaion for he reserve of a life insurance conrac. His work was generalized by Hoem (1969) and furher by Norberg (1991). The Nobel prize awarded work by Black and Scholes (1973) and Meron (1973) iniiaed pricing of claims coningen on underlying financial processes. The heory of opion pricing has since hen urned ino one of he larger indusries of applied mahemaics worldwide. Shorly afer, applicaions o insurance producs wih coningen claims were suggesed by Brennan and Schwarz (1976). The firs hybrid beween Thiele s and Black and Scholes differenial equaions appeared in Aase and Persson (1994). Differenial equaions for he reserve ha connecs Hoem (1969) wih Aase and Persson (1994) appeared in Seffensen (2000). We sae and derive he differenial equaions of Thiele, Black and Scholes and a paricular hybrid equaion. Secion 3: Applicaions o more general life insurance producs are based on he noions of surplus and dividend disribuion. These were sudied by Norberg (1999,2001) who also evaluaed fuure dividends by sysems of ordinary differenial equaions. Seffensen (2006b) approached he dividend valuaion problem by solving sysems of parial differenial equaions, conforming wih a paricular specificaion of he underlying financial marke. We sae he parial differenial equaion sudied in Seffensen (2006b), including a paricular case wih a semi-explici soluion. Secion 4: Coningen claims wih early exercise opions are conneced o he heory of opimal sopping and variaional inequaliies. Grosen and Jørgensen (2000) realized he connecion o surrender opions in life insurance. In Seffensen (2002), he connecion was generalized o general inervenion opions and he Markov chain model for he insurance policy. We sae and prove he 2

3 variaional inequaliy for he price of a coningen claim and sae he corresponding sysem for an insurance conrac wih a surrender opion. Secion 5: Opimal arrangemen of paymen sreams in life insurance was firs based on he linear regulaor. We refer he reader o Fleming and Rishel (1975) for he linear regulaor and Cairns (2000) for an overview over is applicaions o life insurance. The linear regulaor was combined wih he Markov chain model of an insurance conrac in Seffensen (2006a). We sae and prove he Bellman equaion for he linear regulaor, and sae he Bellman equaion derived in Seffensen (2006a), including an indicaion of he soluion. Secion 6: The more convenional approach o decision making in finance is based on uiliy opimizaion, see Korn (1997) and Meron (1990). Meron (1990) approached decision problems in personal finance and inroduced uncerainy of life imes. A connecion o he Markov chain model of an insurance conrac was suggesed in Seffensen (2004). In Nielsen (2004) a relaed problem is solved. We sae he Bellman equaions for he decision problems solved by Meron (1990) and Seffensen (2004), including an indicaion of he soluion. 2 The Differenial Sysems of Thiele and Black-Scholes 2.1 Thiele s Differenial Equaion In his secion we sae and derive he differenial equaion for he so-called reserves conneced o a life insurance conrac wih deerminisic paymens. We give a proof for he differenial equaion ha corresponds o he proofs ha will appear in he res of he aricle. We end he secion by considering he sochasic differenial equaion for he reserve wih applicaion o uni-link life insurance. See Hoem (1969) and Norberg (1991) for differenial equaions for he reserve. We consider an insurance policy issued a ime 0 and erminaing a a fixed finie ime n. There is a finie se of saes of he policy, J = {0,..., J}. Le Z () denoe he sae of he policy a ime [0, n] and le Z be an RCLL process (righ-coninuous, lef limis). By convenion, 0 is he iniial sae, i.e. Z (0) = 0. Then also he associaed J-dimensional couning process N = ( N k) k J is an RCLL process, where N k couns he number of ransiions ino sae k, i.e. N k () = # {s s (0, ], Z (s ) k, Z (s) = k }. The hisory of he policy up o and including ime is represened by he sigma-algebra F Z () = σ {Z (s), s [0, ]}. The developmen of he policy is given by he filraion F Z = { F Z () } [0,n]. Le B () denoe he oal amoun of conracual benefis less premiums payable during he ime inerval [0, ]. We assume ha i develops in accordance wih he dynamics db () = db Z() () + k:k Z( ) bz( )k () dn k (). (1) Here, B j is a deerminisic and sufficienly regular funcion specifying paymens due during sojourns in sae j, and b jk is a deerminisic and sufficienly regular funcion specifying paymens due upon ransiion from sae j o sae k. We assume ha each B j decomposes ino an absoluely coninuous par and a discree par, i.e. db j () = b j () d + B j (). (2) Here, B j () = B j () B j ( ), when differen from 0, is a jump represening a lump sum payable a ime if he policy is hen in sae j. The se of ime poins wih jumps in ( B j) j J is D = { 0, 1,..., q } where 0 = 0 < 1 <... < q = n. 3

4 We assume ha Z is a ime-coninuous Markov process on he sae space J. Furhermore, we assume ha here exis deerminisic and sufficienly regular funcions µ jk () such ha N k admis he sochasic inensiy process { µ Z( )k () } [0,n], i.e. consiues an F Z -maringale. M k () = N k () 0 µ Z(s)k (s) ds 0 acive ( ) 1 disabled 2 dead Figure 1: Disabiliy model wih moraliy, disabiliy, and possibly recovery. Figure 1 illusraes he disabiliy model used o describe a policy on a single life, wih paymens depending on he sae of healh of he insured. We assume ha he invesmen porfolio earns reurn on invesmen by a consan ineres rae r. We use he noaion s = (s,] hroughou and inroduce he shor-hand noaion s r = s j r (τ) dτ = r (s ). Throughou we use subscrip for parial differeniaion, e.g. V () = V j (). The insurer needs an esimae of he fuure obligaions sipulaed in he conrac. The usual approach o such a quaniy is o hink of he insurer having issued a large number of similar conracs wih paymen sreams linked o independen lives. The law of large numbers hen leaves he insurer wih a liabiliy per insured ha ends o he expeced presen value of fuure paymens, given he pas hisory of he policy, as he number of policy holders ends o infiniy. We say ha he valuaion echnique is based on diversificaion of risk. The condiional expeced presen value is called he reserve and appears on he liabiliy side of he insurer s balance scheme. By he Markov assumpion he reserve is given by [ V Z() () = E ] r db (s) Z (). (3) We inroduce he differenial operaor A, he rae of paymens β, and he updaing sum R, AV j () = k:k j µjk () ( V k () V j () ), β j () = b j () + k:k j µjk () b jk (), R j () = B j () + V j () V j ( ). We can now presen he firs differenial equaion, in general spoken of as Thiele s differenial equaion. Proposiion 1 The saewise reserve defined in (3) is characerized by he following deerminisic 4

5 sysem of backward ordinary differenial equaions, 0 = V j () + AV j () + β j () rv j (), / D, (4a) 0 = R j (), D, (4b) 0 = V j (n). (4c) In mos exposiions on he subjec, (4a) is wrien as wih he so-called sum a risk R jk () defined by V j () = rv j () b j () k:k j µjk () R jk (), R jk () = b jk () + V k () V j (). In he succeeding secions, however, i urns ou o be convenien o work wih he differenial operaor abbreviaion. We choose o do his already a his sage in order o communicae he cross-secional similariies. There are several roads leading o (4). We presen a proof ha shows ha any funcion solving he differenial equaion (4) acually equals he reserve defined in (3). Such a resul shows ha (4) as a sufficien condiion on V in he sense ha he differenial equaion characerizes he reserve uniquely. Take an arbirary funcion H j () solving (4) and consider he process H Z() (). For his process he following line of equaliies holds, H Z() () = = = = d (e R ) s r H Z(s) (s) e R ( ) s r rh Z(s) (s) ds + dh Z(s) (s) r (db (s) k:k Z(s ) RZ(s )k R s s (,n] D e H ) dm k (s) ( ) r Hs Z(s) (s) + AH Z(s) (s) + β Z(s) (s) rh Z(s) (s) ds r R Z(s) H (s) r (db (s) k:k Z(s ) RZ(s )k H ) (s) dm k (s). Here R j H and Rjk H are defined as Rj and R jk wih V replaced by H. Now, aking condiional expecaion on boh sides and assuming sufficien inegrabiliy, he inegral wih respec o he maringale vanishes. This leaves us wih he conclusion ha any soluion o (4) equals he reserve, H Z() () = V Z() (). We end his secion by reviewing he dynamics of he reserve. Plugging (4) ino (??) leads o dv Z() () = rv Z() () d db Z() () k:k Z() µz()k () R Z()k () d (6) + ( ) V k () V Z( ) () dn k (), k:k Z( ) (5) 5

6 ha is a backward sochasic differenial equaion. The erm backward refers o he fac ha he soluion is fixed by he erminal condiion (4c), i.e. V Z(n) (n) = 0. Usually his erminal condiion is rewrien by (4b) ino V j (n ) := B j (n) where B j (n) is a fixed erminal paymen. However, one can urn hings upside down by aking his erminal condiion o be he defining relaion of B Z(n) (n) in erms of V Z(n) (n ), i.e. B Z(n) (n) := V Z(n) (n ) wih V Z(n) (n ) given by (6). Then he erminal condiion V Z(n) (n) = 0 is fulfilled by consrucion. We hen jus need an iniial condiion on V o consider i as a forward sochasic differenial equaion. Here, one should ake he so-called equivalence relaion V 0 (0 ) as iniial condiion. Hereafer, V k () can be aken o be anyhing and plays he role as iniial condiion a ime on V, given ha he policy jumps ino sae k. The ype of life insurance where erminal paymens are linked o he developmen of he policy is, generally speaking, known as uni-link life insurance. The consrucion described above is indeed a kind of uni-link life insurance wih no guaranee in he sense ha here are no predefined bounds on B Z(n) (n). The simples implemenaion urns ou by puing V k () = V Z( ) () so ha dv Z() () = rv Z() () d db Z() () k:k Z() µz()k () b Z()k () d. (7) This means ha he reserve is mainained upon ransiion and he risk sum R jk () reduces o he ransiion paymen b jk (). Then he reserve is really nohing bu an accoun from ha he infiniesimal benefis less premiums db Z() () are paid and from ha he so-called naural risk premium k:k Z() µz()k () b Z()k () is wihdrawn o cover he benefis b Z()k (), k Z (). 2.2 Black-Scholes Differenial Equaion In his secion we sae and prove he differenial equaion for he value of a financial conrac wih paymens linked o a sock index. See Black and Scholes (1973) and Meron (1973) for he original conribuions. We consider a financial conrac issued a ime 0 and erminaing a a fixed finie ime n. The payoff from he financial conrac is linked o he value of a sock index. Le X () denoe he sock index a ime [0, n]. The hisory of he sock index up o and including ime is represened by he sigma-algebra F X () = σ {X (s), s [0, ]}. The developmen of he sock index is formalized by he filraion F X = { F X () } [0,n]. Le B () denoe he oal amoun of conracual paymens during he ime inerval [0, ]. We assume ha i develops in accordance wih he dynamics db () = b (, X ()) d + B (, X ()), (8) where b (, x) and B (, x) are deerminisic and sufficienly regular funcions specifying paymens if he sock value is x a ime. The decomposiion of B ino an absoluely coninuous par and a discree par conforms wih (2). Again, we denoe he se of ime poins wih jumps in B by D = { 0, 1,..., q } where 0 = 0 < 1 <... < q = n. The mos classical example of a conracual paymen funcion is he European call opion given by he following specificaion of paymen coefficiens, for some consan K. b (, x) = 0, B (, x) = 0, < n, (9) B (n, x) = max (x K, 0), 6

7 We assume ha X is a ime-coninuous Markov process on R + wih coninuous pahs. Furhermore, we assume ha he dynamics of X are given by he sochasic differenial equaion, dx () = αx () d + σx () dw (), X (0) = x 0, where W is a Wiener-process, and α and σ are consans. We assume ha one may inves in X bu, a he same ime, a riskfree invesmen opporuniy is available. The riskfree invesmen opporuniy earns reurn on invesmen by a consan ineres rae r, corresponding o he invesmen porfolio underlying he insurance porfolio in he previous secion. The issuer of he financial conrac wishes o calculae he value of he fuure paymens in he conrac. The idea of so-called derivaive pricing is ha he conrac value should preven he conrac from imposing arbirage possibiliies, i.e. riskfree capial gains beyond he reurn rae r. The enrepreneurs of modern financial mahemaics realized ha, in cerain financial markes like he one given here, his idea is sufficien o produce he unique value of he financial conrac. This conrac value equals he condiional expeced value, where V (, X ()) = E Q [ dx () = rx () d + σx () dw Q (), ] r db (s) X (), (10) wih W Q being a Wiener-process under he measure Q. The measure Q is called a maringale measure because he discouned sock index e r X () is a maringale under his measure. This consrucion ensures ha he price prevening arbirage possibiliies can be represened in he form (10). Thus, i is acually jus a probabiliy heoreical ool for represenaion. We inroduce he differenial operaor A, he rae of paymens β, and he updaing sum R, AV (, x) = V x (, x) rx V xx (, x) σ 2 x 2, β (, x) = b (, x), R (, x) = B (, x) + V (, x) V (, x). We can now presen he second differenial equaion. Proposiion 2 The conrac value given by (10) is characerized by he following deerminisic backward parial differenial equaion, 0 = V (, x) + AV (, x) + β (, x) rv (, x), / D, (11a) 0 = R (, x), D, (11b) 0 = V (n, x). (11c) The usual siuaion in financial exposiions is ha here are no paymens unil erminaion, in he case of which (12) reduces o 0 = V (, x) + AV (, x) rv (, x), V (n, x) = B (n, x), 7

8 in general spoken of as he Black-Scholes equaion. For he European call opion given by (9), he erminal condiion is given by V (n, x) = max (x K, 0). In his case, he sysem has an explici soluion ha is known as he Black-Scholes formula. This can be found in almos any exbook on derivaive pricing. As in he previous secion we prove ha he differenial equaion is a sufficien condiion on he conrac value in he sense ha any funcion solving (12) indeed equals he conrac value given by (10). Take an arbirary funcion H solving (12) and consider he process H (, X ()). For his process he following line of equaliies holds, H (, X ()) = = = = d (e R ) s r H (s, X (s)) r ( rh (s, X (s)) + dh (s, X (s))) r ( db (s) H x (s, X (s)) σx (s) dw Q (s) ) r (H s (s, X (s)) + AH (s, X (s)) + β (s, X (s)) rh (s, X (s))) ds R s s (,n] D e r R H (s, X (s)) r ( db (s) H x (s, X (s)) σx (s) dw Q (s) ). Now, aking condiional expecaion on boh sides and assuming sufficien inegrabiliy, he inegral wih respec o he maringale vanishes. This leaves us wih H (, X ()) = V (, X ()). Thus, any funcion solving (12) equals he conrac value, and he differenial equaion is hen a sufficien condiion o characerize he conrac value. 2.3 A Hybrid Equaion In his secion we sae he differenial equaion for he reserves conneced o a life insurance conrac wih paymens linked o a sock index. We end he secion by considering a sochasic differenial equaion for he reserve wih applicaions o uni-link life insurance. See Brennan and Schwarz (1976), Aase and Persson (1994), and Seffensen (2000) for he original ideas and he general hybrid equaions, respecively. As in Secion 2.1, we consider an insurance policy issued a ime 0 and erminaing a a fixed finie ime n wih a paymen sream given by (1). However, insead of leing each B j and each b jk be deerminisic funcions of ime, we inroduce dependence on he sock index as formalized in Secion 2.2. We assume ha he accumulaed paymen process develops in accordance wih he dynamics db () = db Z() (, X ()) + k:k Z( ) bz( )k (, X ()) dn k (), (13) where db j (, x) = b j (, x) d + B j (, x), wih sufficienly regular funcions b jk (, x), b j (, x), and B j (, x). As in he previous secions, we are ineresed in valuaion of he fuure paymens in he paymen process. The quesion is now how we should inegrae he wo approaches o risk pricing presened 8

9 here. In Secion 2.1, we assumed insured risk o obey he law of large numbers and based he risk valuaion on diversificaion. This lef us wih a condiional expeced presen value under he objecive probabiliy measure. In Secion 2.2, we based he risk valuaion on he no arbirage paradigm of derivaive pricing. This lef us wih a condiional expeced presen value under an arificial measure Q called he maringale measure. Which measure should we now use for valuaion of inegraed insurance and financial risk in he paymen process (13)? The prevenion of arbirage possibiliies is no sufficien o ge a unique maringale measure. Insead, his idea leaves us wih an infinie se of maringale measures. From hese measures, some can be said o play more imporan roles han ohers. Probably he mos imporan role is played by he produc measure ha combines he objecive measure of insurance risk wih he maringale measure of financial risk. We denoe, wih a sligh misuse of noaion, also his produc measure by Q. This paricular maringale measure appears boh in several so-called quadraic hedging approaches and in he heory of asympoic arbirage. Typically, his measure is applied for valuaion of inegraed financial and insurance risk. Here, we simply ake his measure for given and proceed. I should be menioned ha he differenial equaion below holds for a much larger class of maringale measures in he following sense: Insead of valuaing insurance risk under he objecive measure one could change his measure and sill have a maringale measure. However changing he measure of insurance risk is jus a maer of changing he ransiion inensiies for Z. So changing he inensiies in he formulas below corresponds o picking ou an alernaive maringale measure o he produc measure described in he previous paragraph. We can now define he reserve by V Z() (, X ()) = E Q [ ] r db (s) Z (), X (). (14) Noe here ha we choose he erm reserve for he hybrid (14) of he reserve given in (3) and he conrac value given in (??). This reflecs ha he reserve (14) ypically appears on he liabiliy side of an insurance company s balance scheme. We inroduce he differenial operaor A, he paymen rae β and he updaing sum R, AV j (, x) = k:k j µjk () ( V k (, x) V j (, x) ) (15a) +V j x (, x) rx V j xx (, x) σ 2 x 2, β j (, x) = b j (, x) + k:k j µjk () b jk (, x), (15b) R j (, x) = B j (, x) + V j (, x) V j (, x). (15c) We can now presen he hird differenial equaion. Proposiion 3 The reserve given by (14) is characerized by he following deerminisic sysem of backward parial differenial equaions, 0 = V j (, x) + AV j (, x) + β j (, x) rv j (, x), / D, (16a) 0 = R j (, x), D, (16b) 0 = V j (n, x). (16c) 9

10 We shall no go hrough he derivaion of he differenial equaion sufficien condiion for characerizing he reserve. The recipe and he calculaions can be copied from he previous secion bu hey become more messy as he valuaion problem expands. Bu i is worhwhile o realize ha he differenial equaion (16) is a rue generalizaion of boh (4) and (12). The specializaion of (16) ino (4) comes from erasing all sock index dependence. The specializaion ino (12) comes from erasing all sae dependences and all paymens riggered by ransiions of Z. We end his secion by sudying he special insurance conrac inroduced a he end of Secion 2.1 in he presence of sock index dependence. The backward sochasic differenial equaion corresponding o (6) describing he dynamics of he reserve urns ino wih dv Z() (, X ()) = ( rv Z() (, X ()) + (α r) V Z() x ) (, X ()) X () d (17) +Vx Z() (, X ()) σx () dw () db Z() (, X ()) k:k Z( ) µz()k () R Z()k (, X ()) d + ( ) V k (, X ()) V Z( ) (, X ()) dn k () k:k Z( ) R jk (, x) = b jk (, x) + V k (, x) V j (, x). As in Secion 2.1 we le B Z(n) (n) := V Z(n) (n ) be he defining relaion implying ha he erminal condiion V Z(n) (n) = 0 is fulfilled by consrucion. Furhermore, we assume ha from he reserve a proporion π () is invesed in he sock index a ime. Then, leing h denoe he number of sock indices held a ime and noing ha π () V Z() (, X ()) = h () X (), we hen have ha Vx Z() (, X ()) = h () π () = V Z() (, X ()). X () Plugging his relaion ino (17) gives us a general version of (6). We wrie down here he special case coming from V k () = V Z( ) (), corresponding o (7), dv Z() (, X ()) = (r + π () (α r)) V Z() (, X ()) d +σπ () V Z() (, X ()) dw () db Z() (, X ()) k:k Z() µz()k () b Z()k (, X ()) d. Now, his is an invesmen accoun wih he proporion π invesed in he sock index and wih a flow of paymens corresponding o (7), excep for he possibiliy of sock index dependence in all paymens. 3 Surplus and Dividends 3.1 The Dynamics of he Surplus In his secion we inroduce he noion of surplus ha measures he excess of asses over liabiliies. Also he noion of dividends ha allows he insured o paricipae in he performance of he insurance conrac, is inroduced. For he succeeding secions, only he process of dividends and 10

11 he derived dynamics of he surplus are imporan. See Norberg (1999,2001) and Seffensen (2004) for deailed sudies of he noions of surplus and dividends. Life insurance conracs are ypically long-erm conracs wih ime horizons up o half a cenury or more. Calculaion of reserves is based on assumpions on ineres raes and ransiion inensiies unil erminaion. Two difficulies arise in his connecion. Firsly, hese are quaniies ha are difficul o predic even on a shorer-erm basis. Secondly, he policy holder may be ineresed in paricipaing in reurns on risky asses raher han riskfree asses. A he end of Secion 2.3 we gave one approach o he second difficuly: Le he erminal lump sum paymen be defined by he erminal value of he reserve. Then he prospecive expeced value given by (14) can be calculaed rerospecively. The uni-linked insurance wihou a guaranee is hereby consruced. For various reasons, however, only few life insurance conracs were consruced like ha in he pas. Insead he insurer makes a firs pruden guess on he fuure ineres raes and ransiion inensiies in order o be able o pu up a reserve, knowing quie well ha realized reurns and ransiions differ. This firs guess on ineres raes and ransiion inensiies, here denoed by (r, µ ), is called he firs order basis, and gives rise o he firs order reserve, V. The se of paymens B seled under he firs order basis is called he firs order paymens or he guaraneed paymens. However, he insurer and he policy holder agree ha he realized reurns and ransiions should be refleced in he realized paymen sream. For his reason he insurer adds o he firs order paymens a dividend paymen sream. We denoe his paymen sream by D and assume ha is srucure corresponds o he srucure of B, i.e. dd () = dd Z() () + k:k Z( ) δz( )k () dn k (), (18) dd j () = δ j () d + D j (). Here, however, he coefficiens of D, δ jk (), δ j (), and D j (), are no assumed o be deerminisic. In conras, he dividends should reflec realized reurns and ransiions relaive o he firs order basis assumpions. One can now caegorize basically all ypes of life and pension insurance by heir specificaion of D. Such a specificaion includes possible consrains on D, he way D is seled, and he way in ha D maerializes ino paymens for he policy holder or ohers. We shall no give a horough exposiion of he various ypes of life insurance exising bu jus give a few hins o wha we mean by caegorizaion. When dividends are consrained o be o he benefi of he policy holder, i.e. D is posiive and increasing, one speaks of paricipaing or wih-profi life insurance. In so-called pension funding here is no such consrain. There, however, ofen he insured himself is no affeced by dividends. In reurn, an employer pays or receives dividends. No maer wheher dividends affec he insured or his employer, he dividends do no necessarily maerialize ino cash paymens. The insurer may conver hem ino adjusmens o firs order paymens. Such a conversion is hen agreed upon in he conrac. In paricipaing life insurance his adjusmen of firs order paymens is called bonus. We could coninue he caegorizaion of life insurance conracs bu we sop here. For all ypes of conracs, however, remains he quesion: How should dividends reflec he realized reurns and ransiions? A naural measure of realized performance is he surplus given by excess of asses over liabiliies. Assuming ha paymens are invesed in a porfolio wih value process Y and ha liabiliies are measured by he firs order reserve, we ge he surplus X () = 0 Y () Y (s) d ( (B + D) (s)) V Z() (), 11

12 where he firs par is he oal paymens in he pas accumulaed wih capial gains from invesing in Y. Noe ha X in his secion is defined as he surplus, in conras o he previous secion where X was he sock index. We now assume ha a proporion of Y given by π (, X ()) X () / ( X () + V Z() () ) is invesed in a risky asse modelled as in Secion 2.2. Then he dynamics of Y are given by π (, X ()) X () dy () = ry () d + σ X () + V Z() () Y () dw Q (). Noe ha we choose o specify he dynamics of Y direcly in erms of W Q, he Wiener process under he valuaion measure. Deriving he dynamics of X, using hese dynamics for Y, one arrives, afer a number of rearrangemens and abbreviaions, a dx () = rx () d + π (, X ()) σx () dw () + d (C D) (), (19) X (0) = x 0, where C is a surplus conribuion process wih a srucure corresponding o he srucure of B and D, i.e. dc () = dc Z() () + k:k Z( ) cz( )k () dn k (), (20) dc j () = c j () d + C j (). The dynamics of X show ha π is acually he proporion of he surplus invesed in he risky asse. This is he reason for saring ou wih he proporion π (, X ()) X () / ( X () + V Z() () ). The elemens c jk, c j, and C j of C are deerminisic funcions. They are, of course, imporan for a closer sudy on he elemens of he surplus. However, hey are no crucial for derivaion and comprehension of he formulas in wha follows. Having inroduced he surplus above as a performance measure, a naural nex sep is o link he dividend paymens direcly o he surplus, i.e. δ j () = δ j (, X ()), δ jk () = δ jk (, X ()), D j () = D j (, X ()), where we, wih a sligh misuse of noaion, use he same noaion for he dividend paymens and heir funcional dependence on (, X ()). This formalizaion of dividends would cerainly be a way of geing realized reurns (in Y ) and ransiions (in N) refleced in he dividend paymens. We could have inroduced oher performance measures han he surplus defined above. However, oher well-founded performance measures would ypically also follow he dynamics given by (19) wih appropriae definiion of he coefficiens in C. The formulas derived below would hold rue. Thus, in his respec, he sory abou firs order quaniies and surplus can be seen as jus one example of he sae process X underlying he dividend paymens. 3.2 The Differenial Equaion for he Marke Reserve In his secion we sae he differenial equaion for he reserves conneced o a life insurance conrac wih dividend paymens linked o he surplus. This formalizes mos pracical life insurance conracs where dividends are linked o he performance of he insurance conrac. Furhermore, for he special case of dividends ha are linear in he surplus, we separae variables of he reserves. Thereby one sysem of parial differenial equaions is reduced o wo sysems of ordinary 12

13 differenial equaions. See Seffensen (2006b) for furher sudies on parial differenial equaions for evaluaion of surplus-linked dividends. The insurer is ineresed in valuaion of he oal fuure liabiliies. We inroduce as reserve he expeced presen value of fuure oal paymens given he pas hisory of he policy. The expecaion is aken under he produc measure Q inroduced in Secion 2.2. Since fuure paymens depend on (Z (), X ()) only and (Z (), X ()) is a Markov process, he reserve is given by [ ] V Z() (, X ()) = E Q r d (B + D) (s) Z (), X (). (21) We inroduce he differenial operaor A, he paymen rae β and he updaing sum R, AV j (, x) = ( ( ) ) k:k j µjk () V k, x + c jk () δ jk (, x) V j (, x) +V j x (, x) ( rx + c j () δ j (, x) ) V j β j (, x) = b j () + δ j (, x) + k:k j µjk () xx (, x) π2 (, x) σ 2 x 2, (22a) ), (22b) ( b jk () + δ jk (, x) R j (, x) = B j () + D j (, x) (22c) +V j (, x + C j () D j (, x) ) V j (, x). We are now ready o presen he fourh differenial equaion. Proposiion 4 The reserve given by (14) is characerized by he following deerminisic sysem of backward parial differenial equaions, 0 = V j (, x) + AV j (, x) + β j (, x) rv j (, x), / D, (23a) 0 = R j (, x), D, (23b) 0 = V j (n, x). (23c) As in Secion 2.3 we shall no go hrough he derivaion of he differenial equaion. The calculaions are even more messy han hose leading o he sysem (16), bu he basic ingrediens remain he same. However, we explain how (23) generalizes (16) in several respecs. Firsly, compare he differenial operaors (15a) and (22a). In (15a), he change in he reserve corresponding o a ransiion from j o k is refleced in he difference V k (, x) V j (, x). In his secion a sae ransiion also affecs he variable X such ha afer a jump from j o k a ime, X () = X ( ) + c jk () δ jk (, X ( )). In (22a), his is seen in he change in he reserve by an updaing of he variable x accordingly. A similar difference appears beween (15c) and (22c). In (15c) he sae process X is no affeced by a lump sum paymen a a deerminisic poin in ime. This leads o a change in he reserve of V j (, x) V j (, x). In his secion, a lump sum paymen a ime yields X () = X ( ) + C j () D j (, X ( )). This is hen seen in (22c) by an updaing of he variable x accordingly. Secondly, in (15a) he coefficien on Vx j (, x), rx, sems from he sysemaic reurn rae on invesmen rx (). In his secion, he sysemaic rae of incremens of X, given sojourn in sae j, equals rx () + c j () δ j (, X ()). This is hen refleced in he coefficien on Vx j (, x), rx + 13

14 c j () δ j (, x). Finally, we have in his secion allowed for a cerain proporional invesmen of he surplus in he risky asse. The volailiy π (, X ()) σx () dw () leads o a differen coefficien on V j xx (, x) in (22a) han in (15a). Apar from he difference beween he differenial operaors, he sysems (23) and (16) are almos idenical. In his secion we have added he wo paymen sreams B and D, of which only D is linked o X. In Secion 2.3 he paymen sream B was linked o wha X presened here. This is refleced in he according replacemen of paymens in (15b) and (15c), such ha (22b) and (22c) appear. So far we have jus presened he differenial equaion characerizing he reserve. We have no discussed which funcional dependence of dividends on X ha migh be relevan. For such a discussion we need o know he insurer s and he policy holder s agreemen on reflecion of performance in dividends. In pracice, dividends are always increasing in X. Then a good performance is shared beween he wo paries by he insurer paying back par of he surplus as posiive dividends. A bad performance is shared beween he wo paries by he insurer collecing par of he defici as negaive dividends. Since here may be consrains on D, e.g. D increasing, hese qualiaive esimaes are no necessarily sric, hough. There are only few examples of a funcional dependence ha allow for more explici calculaions. However, luckily he mos imporan one allows us o ake an imporan sep furher. We end his secion by specifying a paricular funcional dependence of dividends on X ha allows for more explici calculaions of he reserve. We inroduce dividends ha are linear in he surplus in he sense ha δ j () = p j () + q j () X (), δ jk () = p jk () + q jk () X (), D j () = P j () + Q j () X (), where p j, p jk, P j, q j, q jk, and Q j are posiive deerminisic funcions. I is an easy exercise o plug hese dividends ino he sysem (23). The nex sep is hen o sugges a useful separaion of variables in V. Lineariy of dividends inspires a guess on he form V j (, x) = f j () + g j () x. Plugging his guess and is derivaives ino (23) and collecing all erms including and excluding x, respecively, gives us sysems of ordinary differenial equaions for f and g. We leave i o he reader o verify ha he differenial equaions covering f and g are similar in srucure o (4). This makes furher sudies, inerpreaions, and represenaions possible. In his exposiion we jus noify he separaion of variables of he reserve funcion for linear dividends. This separaion reduces he sysem (23) of parial differenial equaions o wo sysems of ordinary differenial equaions characerizing f and g. 4 Inervenion 4.1 Opimal Sopping and Early Exercise Opions In his secion we sae and prove he differenial equaion for he value of a financial conrac wih paymens linked o a sock index and wih an early exercise opion. The proof shows ha he differenial equaion is sufficien for a characerizaion of he conrac value. In Secion 2.2 we sudied he price of a financial conrac where he paymen raes and lump sum paymens a deerminisic poins in ime were linked o a sock index. Typically, here is he 14

15 addiional feaure o such a conrac ha he conrac holder can, a any poin in ime unil erminaion, close he conrac. He hen receives a payoff ha depends on he sock value upon closure. This feaure is known as he premaure or early exercise opion, since i gives he conrac holder he opporuniy o conver fuure paymens ino an immediae premaure paymen. Recall he paymen sream (8) in Secion 2.2. Now assume ha, given exercise a ime, all fuure paymens are convered ino one exercise paymen, due a ime, and denoed by Φ () = Φ (, X ()), where we, wih a sligh misuse of noaion, use Φ for boh he process and is sufficienly regular funcional dependence on (, X ()). We are now ineresed in calculaing he value of he conrac. I is possible o give an arbirage argumen for he unique conrac value, V (, X ()) = [ τ sup E Q τ [,n] r db (s) + e R τ ] r Φ (τ) X (). (24) The decision no o exercise premaurely is included in he supremum in (24) by specifying Φ (n) = 0 (25) and leing he decision no o exercise premaurely be presened by τ = n. Assume ha X is modelled as in Secion 2.2 and ha he marke available is as in Secion 2.2. One canno from he resuls in he previous secions immediaely see how he differenial equaion from here can be generalized o he siuaion in his secion. For a fixed τ he valuaion problem is he same as in Secion 2.2 wih n replaced by τ bu how does he supremum affec he resuls? Does here sill exis a deerminisic differenial equaion characerizing he conrac value? We define he differenial operaor A, he rae of paymens β, and he sum R as in Secion 2.2, and inroduce furhermore he sum ϱ by ϱ (, x) = B (, x) + Φ (, x) V (, x). We can now presen he fifh differenial equaion. Proposiion 5 The conrac value given by (24) is characerized by he following deerminisic backward parial variaional inequaliy, 0 V (, x) + AV (, x) + β (, x) rv (, x), / D, (26a) 0 Φ (, x) V (, x), / D, (26b) 0 = (V (, x) + AV (, x) + β (, x) rv (, x)) (V (, x) Φ (, x)), / D, (26c) 0 R (, x), D, (26d) 0 ϱ (, x), D, (26e) 0 = R (, x) ϱ (, x), D, (26f) 0 = V (n, x). (26g) This sysem should be compared wih (12). Firsly, (11a) is replaced by (26a)-(26c). The equaion in (11a) urns ino an inequaliy in (26a). An addiional inequaliy (26b) saes ha he conrac value always exceeds he exercise payoff. This is reasonable, since one of he possible 15

16 exercise sraegies is o exercise immediaely and his would give an immediae exercise payoff. The equaliy (26c) is he mahemaical version of he following saemen: A any poin in he sae space (, x) a leas one of he inequaliies in (26a) and (26b) mus be an equaliy. Secondly, (11b) is replaced by (26d)-(26f). The equaion in (11b) urns ino an inequaliy in (26d). An addiional inequaliy (26e) saes ha he conrac value on he ime se D exceeds he lump sum plus he exercise payoff falling due. The equaliy (26f) saes ha a leas one of he inequaliies in (26d) and (26e) mus be an equaliy. Noe ha (26d)-(26f) easily can be wrien as V (, x) = B (, x) + max (V (, x), Φ (, x)), D, (27) while here is no such abbreviaion available for (26a)-(26c). However, we choose he version (26d)-(26f) o illusrae he symmery wih (26a)-(26c). The usual siuaion in financial exposiions is ha here are no paymens unil exercise or erminaion whaever comes firs. In ha case β (, x) disappears from (26a)-(26c) and (26d)-(26g) reduce o V (n, x) = B (n, x) since boh V (n, x) and Φ (n, x) are zero. Wih his specificaion, (26) is he variaional inequaliy characerizing he value of a so-called American opion. By he variaional inequaliy (26) one can divide he sae space ino wo regions, possibly inersecing. In he firs region, (26a) and (26d) are equaliies. This region consiss of he saes where he opimal sopping sraegy for he conrac holder is no o sop. In his region he conrac value follows a differenial equaion as if here were no exercise opion. In he second region (26b) and (26e) are equaliies. This region consiss of he saes where he opimal sopping sraegy for he conrac holder is o sop. Thus, in his region he value of he conrac equals he exercise payoff. I is possible o show ha (26) is a necessary condiion on he conrac value. However, insead we go direcly o verifying ha (26) is also a sufficien condiion. The proof sars ou in he same way as he verificaion argumen in Secion 2.2. Take an arbirary funcion H solving (26) and consider he process H (, X ()). Then we can wrie, by replacing n by τ in (5), H (, X ()) = e R τ + τ τ r H (τ, X (τ)) r ( db (s) H x (s, X (s)) σx (s) dw Q (s) ) r (H s (s, X (s)) + AH (s, X (s)) + β (s, X (s)) rh (s, X (s))) ds s (,τ] D e R s r R H (s, X (s)). Now consider an arbirary sopping ime τ. For his sopping ime we know from (26a), (26b) and (26d) ha H (, X ()) τ τ r db (s) + e R τ r Φ (τ) r H x (s, X (s)) σx (s) dw Q (s). Taking, firsly, condiional expecaion given X () on boh sides and hen aking supremum over τ gives ha [ τ ] H (, X ()) sup E Q r db (s) + e R τ r Φ (τ) X (). (28) τ [,n] Now consider insead he sopping ime defined by τ = inf {H (s, X (s)) = Φ (s, X (s))}. s [,n] 16

17 This sopping ime is indeed well-defined since, from (25) and (26g), H (n, X (n)) = Φ (n, X (n)) = 0, so ha τ occurs no laer han n. We now know from (26c) and (26f) ha such ha 0 = H s (s, X (s)) + AH (s, X (s)) + β (s, X (s)) rh (s, X (s)), s [, τ ], 0 = R H (s, X (s)), s [, τ ] D, H (, X ()) = τ τ r db (s) + e R τ r Φ (τ ) r H x (s, X (s)) σx (s) dw Q (s). Taking, firsly, condiional expecaion given X () on boh sides and hen esimaing over all possible sopping imes yields he inequaliy [ τ ] H (, X ()) sup E Q r db (s) + e R τ r Φ (τ) X (). (29) τ [,n] By (28) and (29), we conclude ha H (, X ()) = V (, X ()). Thus, any funcion solving (26) characerizes he conrac value. Noe ha he proof also produces he opimal exercise sraegy. The conrac holder should exercise according o he sopping ime τ. However, in order o know when o exercise, one mus be able o calculae he value. Only rarely, he variaional inequaliy (26) has an explici soluion. However, here are several numerical procedures developed for his purpose. One may e.g. use Mone Carlo echniques, general parial differenial equaion approximaions, or cerain specific approximaions developed for specific funcions Φ. 4.2 Inervenion Opions in Life and Pension Insurance In his secion we sae he differenial equaion for he reserve of a life insurance conrac wih dividends linked o he surplus and wih a surrender opion. Furhermore we commen on he generalizaion o general inervenion opions. See Grosen and Jørgensen (2000) and Seffensen (2002) for resuls on he surrender opions and general inervenion opions. In correspondence wih he previous secion, also he holder of a life insurance conrac can, ypically, erminae his policy premaurely. The ac of erminaing a life insurance policy is called surrender, and he exercise opion is in his conex called a surrender opion. We consider he insurance conrac described in Secion 3, i.e. a conrac wih he oal accumulaed paymens given by B + D. Assume now ha he conrac holder can erminae his policy a any poin in ime. Given ha he does so a ime, he receives he surrender value Φ () = Φ Z() (, X ()), for a sufficienly regular funcion Φ j (, x). Here, we ake X o be he surplus process inroduced in Secion 3. We are now ineresed in calculaing he value of fuure paymens specified in he policy. We consider he reserve, V Z() (, X ()) = [ τ sup E Q τ [,n] r d (B + D) (s) + e R τ ] r Φ (τ) Z (), X (), (30) where Q is he produc measure described in Secion 2.3. As in he previous secion, one canno immediaely see how he differenial equaion (23) generalizes o his siuaion. The resuls in he 17

18 previous secion indicae, however, ha he differenial equaion can be replaced by a variaional inequaliy. We define he differenial operaor A, he paymen rae β, and he updaing sum R as in Secion 3, and inroduce furhermore he sum ϱ j by ϱ j (, x) = B j (, x) + Φ j (, x) V j (, x). We can now presen he sixh differenial equaion. Proposiion 6 The reserve given by (30) is characerized by he following deerminisic sysem of backward parial variaional inequaliies, 0 V j (, x) + AV j (, x) + β j (, x) rv j (, x), / D, 0 Φ j (, x) V j (, x), / D, ( (V 0 = V j (, x) + AV j (, x) β j (, x) rv j (, x)) j (, x) Φ j (, x) ), / D, 0 R j (, x), D, 0 ϱ j (, x), D, 0 = R j (, x) ϱ j (, x), D, 0 = V j (n, x). This differenial equaion can be compared wih (23) in he same way as (26) was compared wih (12). Is verificaion goes in he same way as he verificaion of (??) alhough i becomes somewha more involved. We shall no go hrough his here. As in he previous secion, one can now divide he sae space ino wo regions, possibly inersecing. In he firs region, he reserve follows a differenial equaion as if surrender were no possible. This region consiss of saes from where immediae surrender is subopimal. In he second region, he reserve equals he surrender value, This region consiss of he saes where immediae surrender is opimal. The surrender value is ofen in pracice given by he firs order reserve defined in Secion 3, in he sense ha Φ j (, x) = V j (), and is, hus, no surplus dependen. The ile of his secion is Inervenion. So far we have only deal wih sopping, firs of a financial conrac in he previous subsecion, and second of an insurance conrac in his subsecion. In pracice he insurance policy holder ypically holds oher opions ha in some respecs are similar in naure o he surrender opion bu in oher respecs no. The mos imporan one is he free policy opion ha allows he policy holder o sop all premium paymens bu coninue he conrac in a so-called free policy sae. Exercising a free policy opion leads o a reducion of he firs order benefis ha were seled under he assumpion of full premium paymen. Thus, exercising a free policy opion does no sop he insurance policy ha coninues under free policy condiions, bu sops only he premium paymens. Therefore, one should raher speak of inervenion in han sopping of he insurance policy. Of course, sopping is a special example of inervenion. For a sopping or surrender opion, here is always only one conrol ac, namely he ac of sopping since hereafer he conrac has expired. Given ha he policy has been convered ino a free policy, he policy holder may sill hold a surrender opion. Thus, inroducing inervenions, he policy holder may choose beween differen series of inervenions. This feaure produces echnical challenges in he verificaion of a variaional inequaliy characerizing he reserve. However, he basic srucure of he resuling variaional inequaliy remains he same. 18

19 5 Quadraic Opimizaion 5.1 Porfolio Quadraic Opimizaion of Dividends In his secion we sae and prove he differenial equaion for a value funcion of an opimizaion problem where preferences over surplus and dividends are specified by a quadraic disuiliy funcion. We speak of he value process as a disuiliy reserve. The surplus inroduced in Secion 3 is here approximaed by a considerably simpler process. We also indicae he soluion o he differenial equaion and he opimal dividend sraegy. The conrol problem sudied in his secion is known as he linear regulaor. See Fleming and Rishel (1975) for he linear regulaor in general and Cairns (2000) for is applicaions o life insurance. In Secion 3, we inroduced he noion of surplus. The surplus accumulaes a sochasic process of surplus conribuions C and capial gains from invesmen in a Black-Scholes marke. From he surplus is wihdrawn redisribuions o he policy holders in erms of dividends. We modelled he process of dividends similarly o he underlying paymen process B (and he process of surplus conribuions C). In (22) a deerminisic differenial equaion for he reserve was presened where he coefficiens in he dividend process are linked o he surplus. We concluded Secion 3 by proposing dividends o be affine in he surplus. This led o a reserve ha is affine in he surplus. Thus, Secion 3 deal wih valuaion of cerain dividend plans. The quesion ha we did no address was wheher, or raher when, surplus linked dividends, or dividends affine in he surplus for ha maer, are paricularly aracive. Quesions of ha kind appear in he discipline of opimizaion raher han valuaion. We approximae he surplus by a diffusion process on he basis of he following lis of adapaions: where We assume ha he surplus is invesed in he riskfree asse exclusively. We approximae he process of surplus conribuions by a Brownian moion wih volailiy ρ and drif c. We assume ha accumulaed dividends are absoluely coninuous and paid ou by he rae δ. These adapaions give us he following surplus dynamics, dx () = rx () d + d (C D) (), X (0) = x 0, dc () = c () d + ρ () dw (), dd () = δ () d. We are now ineresed in deciding on a dividend rae δ ha we prefer over oher dividend raes according o some preference crierion. For his purpose we inroduce a process of accumulaed disuiliies U, ha is absoluely coninuous wih disuiliy rae u (, δ (), X ()), i.e. du () = u (, δ (), X ()) d. We now inroduce a cerain quadraic disuiliy crierion, u (, δ, x) = p () (δ a ()) 2 + q () x 2. (31) 19

20 This crierion punishes quadraic deviaions of he presen dividend rae from a dividend arge rae a and deviaions of he surplus from 0. Such a disuiliy crierion reflecs a rade-off beween policy holders preferring sabiliy of dividends, relaive o a, over non-sabiliy, and he insurance company preferring sabiliy of he surplus relaive o 0. The preference over he surplus could be driven by regulaory rules saing ha earned surplus conribuions should be redisribued upon earning in some sense. The deerminisic funcions p and q give weighs o hese preference formalizaions. A ime he fuure disuiliies are measured by heir condiional expecaion. We define he disuiliy reserve as he infimum of all such condiional expecaions over all admissible dividend paymen sreams, i.e. [ ] V (, X ()) = inf E du (s) D X (). (32) Excep for he infimum over D, noe he similariy wih e.g. (14). The primary difference is ha, insead of measuring an expeced (presen) value of paymen raes δ, we now measure an expeced disuiliy funcion of paymen raes, p () (δ () a ()) 2. Hereo we add an expeced disuiliy funcion of he posiion of he surplus, q () X () 2. Now, we inroduce he differenial operaor A and he rae of disuiliies β, AV (, x) = V x (, x) (rx + c () δ) V xx (, x) ρ 2, β (, x) = u (, δ, x). We are now ready o presen he sevenh differenial equaion ha is a so-called Bellman equaion. Proposiion 7 The disuiliy reserve given by (32) is characerized by he following Bellman equaion, 0 = V (, x) + inf [AV (, x) + β (, x)], (33a) δ 0 = V (n, x). (33b) An appendix o his differenial equaion is he specificaion of he opimal dividend sream, i.e. he dividend sream ha acually minimizes he disuiliy reserve (32). This opimal dividend sream, specified by he opimal rae δ, is simply he argumen of he supremum in (33a), i.e. δ = arg inf δ [AV (, x) + β (, x)]. (34) I is worhwhile o commen on he connecion beween (33) and e.g. he variaional inequaliy (26). In (26a)-(26b) and in (26d)-(26e), we had wo inequaliies, corresponding o wo differen acions, sopping and no sopping. From (26c) and (26f) one of he inequaliies mus be an equaliy. The srucure of (33a) is he same in he sense ha (33a) represens an infinie se of inequaliies, corresponding o each possible dividend rae. However, one of he inequaliies mus hold wih equaliy. Since for each dividend rae, he disuiliy reserve is described by he same parial differenial equaion, we can wrie his in he very compac way (33a). This compac way acually corresponds o he compac wriing of (26d)-(26f) in (27). 20