Exponenial uiliy indifference valuaion in a general semimaringale model Chrisoph Frei and Marin Schweizer his version: 25.2.29. In: Delbaen, F., Rásonyi, M. and Sricker, C. eds., Opimaliy and Risk Modern rends in Mahemaical Finance. he Kabanov Fesschrif, Springer, Berlin 29, 49 86 his paper is dedicaed o Yuri Kabanov on he occasion of his 6h birhday. We hope he likes i even if i is no shor... Absrac We sudy he exponenial uiliy indifference valuaion of a coningen claim when asse prices are given by a general semimaringale S. Under mild assumpions on and S, we prove ha a no-arbirage ype condiion is fulfilled if and only if has a cerain represenaion. In his case, he indifference value can be wrien in erms of processes from ha represenaion, which is useful in wo ways. Firsly, i yields an inerpolaion expression for he indifference value which generalizes he explici formulas known for Brownian models. Secondly, we show ha he indifference value process is he firs componen of he unique soluion in a suiable class of processes of a backward sochasic differenial equaion. Under addiional assumpions, he oher componens of his soluion are BMO-maringales for he minimal enropy maringale measure. his generalizes recen resuls by Becherer 2 and Mania and Schweizer 19. Key words: exponenial uiliy, indifference valuaion, minimal enropy maringale measure, BSDE, BMO-maringales, fundamenal enropy represenaion FER MSC 2 subjec classificaion: 91B28, 6G48 JEL classificaion numbers: G13, C6 1 Inroducion One general approach o he problem of valuing coningen claims in incomplee markes is uiliy indifference valuaion. Is basic idea is ha he invesor valuing a Chrisoph Frei E Zurich, Deparmen of Mahemaics, 892 Zurich, Swizerland, e-mail: chrisoph.frei@mah.ehz.ch Marin Schweizer E Zurich, Deparmen of Mahemaics, 892 Zurich, Swizerland, e-mail: marin.schweizer@mah.ehz.ch 1
2 Chrisoph Frei and Marin Schweizer coningen claim should achieve he same expeced uiliy in he wo cases where 1 he does no have, or 2 he owns bu has his iniial capial reduced by he amoun of he indifference value of. Exponenial uiliy indifference valuaion means ha he uiliy funcion one uses is exponenial. Even in a concree model, i is difficul o obain a closed-form formula for he indifference value. he majoriy of exising explici resuls are for Brownian seings; see for insance Frei and Schweizer 1 and he references herein. In more general siuaions, Becherer 2 and Mania and Schweizer 19 derive a backward sochasic differenial equaion BSDE for he indifference value process. While 19 assumes a coninuous filraion, he framework in 2 has a coninuous price process driven by Brownian moions and a filraion generaed by hese and a random measure allowing he modeling of non-predicable evens. he main conribuion of his paper is o exend he above resuls o a seing where asse prices are given by a general semimaringale. We show ha he exponenial uiliy indifference value can sill be wrien in a closed-form expression similar o ha known for Brownian models, alhough he srucure of his formula is here much less explici. Independenly from ha, we esablish a BSDE formulaion for he dynamic indifference value process. Boh of hese resuls are based on a represenaion of he claim and on he relaionship beween a noion of no-arbirage, he form of he so-called minimal enropy maringale measure, and he indifference value. As our saring poin, we ake he work of Biagini and Frielli 3, 4. heir resuls yield a represenaion of he minimal enropy maringale measure which we can use o derive a decomposiion of a fixed payoff in a similar way as in Becherer 1. We call his decomposiion, which is closely relaed o he minimal enropy maringale measure, he fundamenal enropy represenaion of FER. I is cenral o all our resuls here, because we can express he indifference value for as a difference of erms from FER and FER. We derive from his a fairly explici formula for he indifference value by an inerpolaion argumen, and we also esablish a BSDE represenaion for he indifference value process. Is proof is based on he idea ha he wo represenaions FER and FER can be merged o yield a single BSDE. his direc procedure allows us o work wih a general semimaringale, whereas Becherer 2 as well as Mania and Schweizer 19 use more specific models because hey firs prove some resuls for more general classes of BSDEs and hen apply hese o derive he paricular BSDE for he indifference value. he price o pay for working in our general seing is ha we mus resric he class of soluions of he BSDE o ge uniqueness. Under addiional assumpions, he componens of he soluion o he BSDE for he indifference value are again BMO-maringales for he minimal enropy maringale measure; his applies in paricular o he value process of he indifference hedging sraegy. he paper is organized as follows. Secion 2 lays ou he model, moivaes, and inroduces he imporan noion of FER. In Secion 3, we prove ha he exisence of FER is essenially equivalen o an absence-of-arbirage condiion. Moreover, we develop a uniqueness resul for FER and is relaionship o he minimal enropy maringale measure. Secion 4 esablishes he link beween he exponenial
General exponenial uiliy indifference valuaion 3 indifference value of and he wo decomposiions FER and FER. By an inerpolaion argumen, we derive a fairly explici formula for he indifference value. In Secion 5, we exend o a general filraion he BSDE represenaion of he indifference value by Becherer 2 and Mania and Schweizer 19. We furher provide condiions under which he componens of he soluion o he BSDE are BMOmaringales for he minimal enropy maringale measure. Secion 6 rounds off wih an applicaion o a Brownian seing. 2 Moivaion and definiion of FER We sar wih a probabiliy space Ω,F,P, a finie ime inerval, for a fixed > and a filraion F = F saisfying he usual condiions of righ-coninuiy and compleeness. For simpliciy, we assume ha F is rivial and F = F. For a posiive process Z, we use he abbreviaion Z,s := Z s /Z, s. In our financial marke, here are d risky asses whose price process S = S is an R d -valued semimaringale. In addiion, here is a riskless asse, chosen as numeraire, whose price is consan a 1. Our invesor s risk preferences are given by an exponenial uiliy funcion Ux = exp γx, x R, for a fixed γ >. We always consider a fixed coningen claim which is a real-valued F -measurable random variable saisfying E P expγ <. Expressions depending on are inroduced wih an index so we can laer use hem also in he absence of he claim by seing =. owever, he dependence on γ is no explicily menioned. We define by := expγ/ E P expγ a probabiliy measure P on Ω,F equivalen o P. Noe ha P = P. We denoe by LS he se of all R d -valued predicable S-inegrable processes, so ha ϑ ds is a well-defined semimaringale for each ϑ in LS. We always impose wihou furher menion he following sanding assumpion, inroduced by Biagini and Frielli 3, 4 for =. We assume ha W / and W /, 1 where W is he se of loss variables W which saisfy he following wo condiions: 1 W 1 P-a.s., and for every i = 1,...,d, here exiss some β i LS i such ha P, s.. β i = = and β s i dss i W for all, ; 2 E P expcw < for all c >. Clearly, W = W if is bounded. Lemma 1 a he beginning of Secion 3 gives a less resricive condiion on for W = W. he sanding assumpion 1 is auomaically fulfilled if S is locally bounded since hen 1 W W by Proposiion 1 of Biagini and Frielli 3, using P P. Bu 1 is for example also saisfied if is bounded and S = S 1 is a scalar compound Poisson process wih Gaussian jumps. his follows from Secion 3.2 in Biagini and Frielli 3. So he model wih condiion 1 is a genuine generalizaion of he case of a locally bounded S.
4 Chrisoph Frei and Marin Schweizer o assign o a ime, a value based on our exponenial uiliy funcion, we firs fix an F -measurable random variable x, inerpreed as he invesor s saring capial a ime. hen we define V x := esssupe P exp ϑ A γx γ ϑ s ds s + γ F, 2 where he se A of -admissible sraegies on, consiss of all processes ϑi, wih ϑ LS and such ha ϑ ds is a Q-supermaringale for every Q P e, f f ; he se Pe, is defined in he paragraph afer he nex. We recall ha x + ϑ s ds s is he invesor s final wealh when saring wih x and invesing according o he self-financing sraegy ϑ over,. herefore, V x is he maximal condiional expeced uiliy he invesor can achieve from he ime- iniial capial x by rading during, and paying ou or receiving a he mauriy. he indifference seller value h x a ime for is implicily defined by V x = V x + h x. 3 his says ha he invesor is indifferen beween solely rading wih iniial capial x, versus rading wih iniial capial x +h x bu paying an addiional cash-flow a mauriy. o define our sraegies, we need he ses where P f := { Q P IQ P < and S is a Q-sigma-maringale }, P e, f := { Q P IQ P < and S is a Q-sigma-maringale }, { E IQ P := Q log dq if Q P + oherwise denoes he relaive enropy of Q wih respec o P. Since P is equivalen o P, he ses P f f and Pe, depend on only hrough he condiion IQ P <. By Proposiion 3 and Remark 3 of Biagini and Frielli 3, applied o P insead of P, here exiss a unique Q E P f ha minimizes IQ P over all Q P f, provided of course ha P f /. We call QE he minimal -enropy measure, or -MEM for shor. If P e, f /, hen QE is even equivalen o P, i.e., Q E Pe, f ; see Remark 2 of Biagini and Frielli 3. Noe ha he proper erminology would be minimal -enropy sigma-maringale measure or -MEσMM, bu his is oo long. We briefly recall he relaion beween Q E, QE and he indifference value h x a ime o moivae he definiion of FER, which we inroduce laer in his secion. Assume P e, f f / and Pe, /. he P -densiy of Q E and he P-densiy of have he form Q E
General exponenial uiliy indifference valuaion 5 dq E = c exp ζs ds s and dqe = c exp ζs ds s 4 for some posiive consans c, c and processes ζ, ζ in LS such ha ζ ds is a Q-maringale for every Q P f and ζ ds is a Q-maringale for every Q P f, whence ζ A and ζ A. his was firs shown by Kabanov and Sricker 16 in heir heorem 2.1 for a locally bounded S and =, and exended by Biagini and Frielli 4 in heir heorem 1.4 o a general S for = under he assumpion W /. By using his resul also under P insead of P, we immediaely obain 4. I is now sraighforward o calculae and also well known a leas for locally bounded S ha for x R, we can wrie V x = sup E P exp γx γ ϑ s ds s + γ and herefore ϑ A = exp γx E P expγ inf ϑ A = exp γx E P expγ inf ϑ A E P exp γ E Q E 1 c exp ϑ s ds s γϑs ζs dss = exp γx E P expγ c 5 h x = h = 1 γ log c E P expγ c. 6 In Secion 4, we sudy he relaion beween Q E, QE and V x, h for arbirary,. From his we can derive, on he one hand, an inerpolaion formula for each h in Secion 4 and, on he oher hand, a BSDE characerizaion of he process h in Secion 5. o generalize he saic relaions 5, 6 o dynamic ones, we inroduce a cerain represenaion of ha we call fundamenal enropy represenaion of FER. Is link o he minimal -enropy measure is elaboraed in he nex secion. We give wo differen versions of his represenaion. he idea is ha he firs definiion only requires some minimal condiions, whereas he second srenghens he condiions o guaranee uniqueness of he represenaion and ensure he idenificaion of he -MEM; see Proposiion 2. Definiion 1. We say ha FER exiss if here is a decomposiion where = 1 γ loge N + ηs ds s + k, 7 i N is a local P-maringale null a such ha E N is a posiive P- maringale and S is a P N -sigma-maringale, where P N is defined by N := E N ;
6 Chrisoph Frei and Marin Schweizer ii η is in LS and such ha η s ds s L 1 P N ; iii k R is consan. In his case, we say ha N,η,k is an FER. If moreover ηs ds s L 1 Q and E Q ηs ds s for all Q P f and η ds is a P N -maringale, 8 we say ha N,η,k is an FER. For any FER N,η,k, we se k := k + 1 γ loge N + ηs ds s for, 9 and call P N he probabiliy measure associaed wih N,η,k. Because E N is by i a posiive P-maringale, he local P-maringale N has no negaive jumps whose absolue value is 1 or more, and P N is a probabiliy measure equivalen o P. We consider wo FER N,η,k Ñ and, η, k as equal if Ñ and N are versions of each oher hence indisinguishable, since boh are RCLL, η ds is a version of η ds, and k = k. For fuure use, we noe ha 7 and 9 combine o give = k + 1 γ loge N, + ηs ds s for,. 1 he nex resul shows ha for coninuous asse prices, we can wrie FER in a differen and perhaps more familiar form. For is formulaion, we need he following definiion. We say ha S saisfies he srucure condiion SC if S i = S i + Mi + d j=1 λ j d M i,m j, i = 1,...,d, where M is a locally square-inegrable local P-maringale null a and λ is a predicable process such ha he final value of he mean-variance radeoff, K = d i, j=1 λ sλ i s j d M i,m j s = λ dm, is almos surely finie. Proposiion 1. Assume ha S is coninuous. hen a riple N,η,k is an FER if and only if S saisfies SC and Ñ = N + λ dm, η = η 1 γ λ, k = k saisfy and = 1 γ loge Ñ + η s ds s + 1 λ dm + k 11 2γ
General exponenial uiliy indifference valuaion 7 i Ñ is a local P-maringale null a and srongly P-orhogonal o each componen of M, and E Ñ E λ dm is a posiive P-maringale; ii η is in LS and such ha η s + 1 γ λ s dss is P N -inegrable, where N := E Ñ E λ dm ; iii k R is consan. Proof. Le firs N,η,k be an FER. Is associaed measure P N is equivalen o P and S is a local P N -maringale since S is coninuous. By heorem 1 of Schweizer 23, S saisfies SC and we can wrie N = Ñ λ dm, where Ñ is a local P-maringale null a and srongly P-orhogonal o each componen of M, and E N = E Ñ E λ dm. he las equaliy uses ha Ñ, λ dm = due o he coninuiy of M. ence condiions i iii of FER imply i iii, and 7 is equivalen o 11 by SC and he coninuiy of S. Conversely, le Ñ, η, k be as in he proposiion. We claim ha he riple Ñ λ dm, η + 1 γ λ, k is an FER. Because M is a local P-maringale and E N = E Ñ E λ dm is he P-densiy process of P N, he process L defined by L := M N,M,, is a local P N -maringale by Girsanov s heorem; see for insance heorem III.4 of Proer 21 and observe ha E N,M = E N d N,M exiss since M is coninuous like S. Because Ñ is srongly P-orhogonal o each componen of M and M is coninuous, we have N,M i = Ñ λ dm,m i = d j=1 λ j d M j,m i, i = 1,...,d, and so SC shows ha S = L + S is also a local P N -maringale. he oher condiions of FER are easy o check. Remark 1. 1 Suppose ha S is coninuous and saisfies SC. If he sochasic exponenial E λ dm is a P-maringale, condiions i and ii in Proposiion 1 can be wrien under he probabiliy measure ˆP defined by d ˆP := E λ dm, which is called he minimal local maringale measure in he erminology of Föllmer and Schweizer 9. his means ha condiion i in Proposiion 1 is equivalen o i Ñ is a local ˆP-maringale null a and srongly ˆP-orhogonal o each componen of S, and E Ñ is a posiive ˆP-maringale, and P N can be defined by N := E Ñ. o prove he equivalence of d ˆP i and i, firs assume ha Ñ is a local P-maringale null a and srongly P- orhogonal o each M i. hen Ñ, λ dm = Ñ, λ dm =
8 Chrisoph Frei and Marin Schweizer by he coninuiy of M, and hence Ñ is also a local ˆP-maringale by Girsanov s heorem; see, for insance, heorem III.4 of Proer 21. he coninuiy of S, SC and he srong P-orhogonaliy of Ñ o M enail Ñ,S i Ñ =,M i =, i = 1,...,d, implying ha Ñ is srongly ˆP-orhogonal o each componen of S. he proof of i = i goes analogously. 2 Assume ha S is no necessarily coninuous bu locally bounded and saisfies SC wih λ i Lloc 2 M i, i = 1,...,d, and le N,η,k be an FER. hen we can sill wrie N = Ñ λ dm for a local P-maringale Ñ null a and srongly P-orhogonal o each componen of M, by using Girsanov s heorem, SC and he fac ha E N defines an equivalen local maringale measure. owever, we canno separae E Ñ λ dm ino wo facors. 3 No-arbirage and exisence of FER heorem 1 below says ha a cerain noion of no-arbirage is equivalen o he exisence of FER. I can be considered as an exponenial analogue o he L 2 -resul of heorem 3 in Bobrovnyska and Schweizer 5. For a locally bounded S, he implicaion = roughly corresponds o Proposiion 2.2 of Becherer 1, who makes use of he idea o consider known resuls under P insead of P. his echnique, which already appears in Delbaen e al. 6, will also be cenral for he proofs of our heorem 1 and Proposiion 2. We sar wih a resul ha gives sufficien condiions for W W and P e, f P e, f as well as for W = W and P e, f = P e, f f. he relaion beween Pe, and P e, f will be used laer, while W = W is helpful in applicaions o verify he condiion 1. Lemma 1. If saisfies hen W W, P f P f E P exp ε < for some ε >, 12 and Pe, f P e, f. If saisfies E P exp γ + ε < and EP exp ε < for some ε >, 13 hen W = W, P f = P f and Pe, f = P e, f. Proof. We firs show W W under 12. For c >, ölder s inequaliy yields
General exponenial uiliy indifference valuaion 9 E P expcw = EP exp cw + εγ ε + γ exp εγ ε + γ ε ε + γ ε+γ E P exp cw + γ E γ ε+γ P exp ε 14 ε ε + γ = E P exp cw E ε+γ ε P expγ E γ ε+γ P exp ε. ε Because of E P expγ < and 12, his is finie if W W, and hen W W. o prove W = W under 13, we only need o show W W. For c > and W W, we obain similarly o 14 ha γ ε EP exp ε + γ ε+γ ε + γ ε+γ E P expcw E P exp cw < E P expγ ε by 13, and hence W W. he remainder of he second par follows from Lemma A.1 in Becherer 1. he proof of he res of he firs par is very similar. Indeed, 12 and he sanding assumpion ha E P expγ < imply EP exp ε <, where ε := minε,γ. Lemma 3.5 of Delbaen e al. 6 yields 1 E Q ε IQ P + e E P exp ε for Q P. 15 If Q P f, he righ-hand side is finie, hus E Q <, and we have IQ P = E Q log dq log = IQ P + loge P expγ γeq, which is finie. his shows Q P f f, and Pe, P e, f follows analogously. heorem 1. We have ha P e, f / FER exiss FER exiss. In paricular, if P e, f / and saisfies 12, hen FER exiss. Proof. We firs show ha P e, f / yields he exisence of FER. As already menioned, P e, f / and he sanding assumpion W / imply by Proposiion 3 and Remarks 2, 3 of Biagini and Frielli 3, applied o P insead of P, exisence and uniqueness of he -MEM Q E Pe, f. Using QE P P, we can wrie dq E = E N 16 for some local P-maringale N null a such ha E N is a posiive P-maringale and S is a Q E -sigma-maringale. Moreover, by heorem 1.4 of Biagini and Fri-
1 Chrisoph Frei and Marin Schweizer elli 4, applied o P insead of P, we have as in 4 dq E = c exp ζs ds s 17 for a consan c > and some ζ in LS such ha ζ ds is a Q-maringale for every Q P f. Since = expγ/ E P expγ, comparing 17 wih 16 gives E N = c 1 exp ds s + γ, where c 1 := c/ E P expγ is a posiive consan. We hus obain = 1 γ loge N 1 γ ζ s ζ s ds s + c 2 wih c 2 := 1 γ logc 1, and hence N, 1 γ ζ,c 2 is an FER. Noe ha ζ ds is a P N -maringale because he -MEM Q E equals he probabiliy measure P N associaed wih N, 1 γ ζ,c 2 by consrucion; compare 16. o esablish he equivalences of heorem 1, i remains o show ha he exisence of FER implies P e, f /, because every FER is obviously an FER. So le N,η,k be an FER and recall ha is associaed measure P N is defined by N := E N. We prove ha P N P e, f. By condiion i on FER, P N is a probabiliy measure equivalen o P and S is a P N -sigmamaringale. o show ha P N has finie relaive enropy wih respec o P, we wrie N = N = E N exp γe P expγ = exp γk EP expγ exp γ ηs ds s, 18 where he las equaliy is due o he decomposiion 7 in FER. his yields by ii of FER ha I P N P = E PN log N = γk + loge P expγ γepn ηs ds s <. Finally, he las asserion follows direcly from he firs par of Lemma 1.
General exponenial uiliy indifference valuaion 11 While he exisence of FER and of FER is equivalen by heorem 1, he wo represenaions are obviously differen since FER imposes more sringen condiions. he nex resul serves o clarify his difference. Proposiion 2. Assume P e, f / and le N,η,k be an FER wih associaed measure P N. hen he following are equivalen: a N,η,k is an FER, i.e., N,η,k saisfies 8; b P N equals he -MEM Q E, and η ds is a P N -maringale; c η ds is a Q E -maringale and E PN η s ds s = ; d η ds is a Q-maringale for every Q P f. Moreover, he class of FER consiss of a singleon. Proof. Clearly, d implies a, and also c since Q E exiss by Proposiion 3 of Biagini and Frielli 3, using P e, f / and he sanding assumpion W /. We prove a = b, c = b and finally b = d. he firs implicaion goes as in he proof of heorem 2.3 of Frielli 11, because we have by 18 ha N = c 3 exp γ ηs ds s wih c 3 := exp γk EP expγ. 19 he implicaion c = b follows from he firs par of he proof of Proposiion 3.2 of Grandis and Rheinländer 12, which does no use he assumpion ha S is locally bounded. o show b = d, noe ha b, 17 and 19 yield c 3 exp γ ηs ds s = c exp ζs ds s P-a.s., 2 where ζ in LS is such ha ζ ds is a Q-maringale for every Q P f. aking logarihms and P N -expecaions in 2, we obain c 3 = c by using ha P N P e, f by he proof of heorem 1. hus η s ds s = 1 γ ζ s ds s P-a.s. and hence η ds = 1 γ ζ ds since boh η ds and ζ ds are P N -maringales. herefore, η ds = 1 γ ζ ds is a Q-maringale for every Q P f. heorem 1 implies he exisence of FER because P e, f /. o show uniqueness, le N,η,k Ñ and, η, k be wo FER. Since he minimal - enropy measure is unique by Proposiion 3 of Biagini and Frielli 3, we have from a = b ha E N = dqe = E Ñ. So E Ñ is a version of E N since boh are P-maringales, and aking sochasic logarihms implies ha Ñ is a version of N. Similarly, 19 and c yield γk + log E P expγ = E Q log dqe = γ k E + log E P expγ,
12 Chrisoph Frei and Marin Schweizer hus k = k, and herefore again from 19 ha η s ds s = 1 γ log 1 c 3 dq E = η s ds s. Bu boh η ds and η ds are Q E -maringales due o d, and so η ds is a version of η ds. Remark 2. Exploiing Proposiion 3.4 of Grandis and Rheinländer 12, applied o P insead of P, gives a sufficien condiion for FER by using our Proposiion 2. Indeed, assume ha S is locally bounded and P e, f /. If for an FER N,η,k, η ds is a BMO P N N -maringale and E ε P < for some ε >, hen N,η,k is he FER. Anoher sufficien crierion is obained from Proposiion 3.2 of Rheinländer 22 in view of our Proposiion 2. Namely, if S is locally bounded and for an FER N,η,k here exiss ε > such ha EP exp ε η ds <, hen N,η,k is he FER. While here is always a mos one FER by Proposiion 2, he nex example shows ha here may be several FER. his also illusraes ha he uniqueness for FER is closely relaed o inegrabiliy properies. Example 1. ake wo independen P-Brownian moions W and W, denoe by F heir P-augmened filraion and choose d = 1, S = W and. he MEM Q E hen equals P since S is a P-maringale, and,, is he unique FER. o consruc anoher FER, choose N := W. hen E N = E W is clearly a posiive P-maringale srongly P-orhogonal o S = W so ha condiion i in FER holds. Define P N as usual by N := E N = E W. By Girsanov s heorem, W and W := W,, are hen P N -Brownian moions and we can explicily compue E P loge N I P N P = E PN = E P W /2 = /2, loge N = E PN W + /2 = /2. 21 his shows ha P N P e, f. Since S = W is a P-Brownian moion, Proposiion 1 of Emery e al. 8 now yields for every c R a process η c in LS such ha 1 γ loge W c = ηs cds s P-a.s. 22 Because I P N P <, using he inequaliy x logx xlogx + 2e 1 shows ha η s cds s is in L 1 P N so ha ii of FER is also saisfied. ence N,η c,c is an FER, bu does no coincide wih,, which is he
General exponenial uiliy indifference valuaion 13 FER. o check ha propery 8 indeed fails, we can easily see from 21 and 22 ha η cds canno be a P N -maringale if c 1 2γ. If c = 1 2γ, we can simply compue, for P P f, ha E P ηs cds s = 1 γ E P loge N + 1 2γ = 1 γ >. We have jus consruced an FER differen from FER. Ye anoher FER can be obained by choosing for k R\{} a process β k in LS such ha /2 βs kds s = k and βs kds s = k P-a.s., which is possible by Proposiion 1 of Emery e al. 8. Clearly, β s kds s = P-a.s. and,β k, is an FER wih associaed measure P, which even saisfies E Q β s kds s = for all Q P f ; bu β kds is no a P-maringale. his ends he example. Example 1 shows ha we should focus on FER if we wan o obain good resuls. If S is coninuous and we impose addiional assumpions, he nex resul gives BMO-properies for he componens of FER. his will be used laer when we give a BSDE descripion for he exponenial uiliy indifference value process. We firs recall some definiions. Le Q be a probabiliy measure on Ω,F equivalen o P and p > 1. An adaped posiive RCLL sochasic process Z is said o saisfy he reverse ölder inequaliy R p Q if here exiss a posiive consan C such ha Z p ess sup E Q = ess sup E Q Zτ, p Fτ C. τ sopping ime Z τ F τ /2 τ sopping ime Recall ha Z τ, = Z /Z τ for a posiive process Z. We say ha Z saisfies he reverse ölder inequaliy R LlogL Q if here exiss a posiive consan C such ha ess sup E Q Z τ, log + Z τ, F τ C. τ sopping ime Z saisfies condiion J if here exiss a posiive consan C such ha 1 C Z Z CZ. heorem 2. Assume ha S is coninuous, is bounded and here exiss Q P e, f whose P-densiy process saisfies R LlogL P. Le N,η,k be an FER. hen he following are equivalen: a N,η,k is he FER ;
14 Chrisoph Frei and Marin Schweizer b N is a BMOP-maringale, E N saisfies condiion J, and η ds is a P N -maringale; c N is a BMOP-maringale, E N saisfies condiion J, and η ds is a BMO P N -maringale; d η dm is a BMOP-maringale, where M is he P-local maringale par of S; e here exiss ε > such ha E P exp ε η ds <. he hypoheses of heorem 2 are for insance fulfilled if is bounded, S is coninuous and saisfies SC, and λ dm is a BMOP-maringale. o see his, noe ha E λ dm hen saisfies he reverse ölder inequaliy R p P for some p > 1 by heorem 3.4 of Kazamaki 18. he fac ha here exiss k < such ha xlogx k + x p for all x > now implies ha E λ dm also saisfies R LlogL P. ence he minimal local maringale measure ˆP given by d ˆP := E λ dm is in P e, f and is P-densiy process saisfies R LlogL P. = Pe, f Proof of heorem 2. By Lemma 1, P e, f / so ha here exiss an FER N,η,k by heorem 1. Before we show ha a e are equivalen, we need some preparaion. Le Q be a probabiliy measure equivalen o P. Denoing by Z he P-densiy process of Q and by Y he P -densiy process of Q, we prove ha Z saisfies R LlogL P if and only if Y saisfies R LlogL P, 23 Z saisfies condiion J if and only if Y saisfies condiion J. 24 o ha end, observe firs ha because is bounded, here exiss a posiive consan k wih 1 k k, which yields 1 Z Y kz. 25 k For any sopping ime τ, 25 implies E P Y τ, log + Y τ, F τ E P Z τ, log + Z τ, k 2 Fτ, and so he inequaliy log + ab log + a + logb for a > and b 1 yields E P Z τ, log + Z τ, k 2 Fτ E P Z τ, log + Z τ, F τ + 2logk, which is bounded independenly of τ if Z saisfies R LlogL P. If Z saisfies condiion J wih consan C, hen 25 gives Y kz kcz k 2 CY and Y 1 k Z 1 kc Z 1 k 2 C Y. So he only if par of boh 23 and 24 is clear, and he if par is proved symmerically.
General exponenial uiliy indifference valuaion 15 By assumpion, here exiss Q P e, f whose P-densiy process saisfies R LlogL P, and so he P -densiy process of Q saisfies R LlogL P by 23. Because P e, f = Pe, f is nonempy, he unique minimal -enropy measure Q E exiss, and is P -densiy process also saisfies R LlogL P by Lemma 3.1 of Delbaen e al. 6, used for P insead of P. Since S is coninuous, he P -densiy process of Q E also saisfies condiion J by Lemma 4.6 of Grandis and Rheinländer 12. I follows from 23, 24 and Lemma 2.2 of Grandis and Rheinländer 12 ha he P-densiy process Z QE,P of Q E saisfies R LlogL P and condiion J, and he sochasic logarihm of Z QE,P is a BMOP-maringale. 26 a = b. Since N,η,k is he FER, Proposiion 2 implies ha he P-densiy process Z QE,P of Q E is given by E N and ha η ds is a P N - maringale. We deduce b from 26. b = c. We have o show ha η ds is in BMO P N. By condiioning 7 under P N on F τ for a sopping ime τ, we obain by b τ τ ηs ds s = 1 γ E PN loge N F τ + E PN F τ k, and hence ηs ds s = 1 γ loge N + 1 γ E PN loge N F τ + E PN F τ. By Proposiion 6 of Doléans-Dade and Meyer 7, here is a BMO P N - maringale ˆN wih E N 1 = E ˆN. his uses ha Z QE,P = E N saisfies condiion J and N is a BMOP-maringale by 26. Since is bounded, we ge E PN ηs ds s F τ τ 2 L P + 1 γ E PN loge N E PN loge N F τ F τ = 2 L P + 1 γ E PN loge ˆN E PN loge ˆN F τ F τ, 27 and now we proceed like on page 131 in Grandis and Rheinländer 12 o show ha 27 is bounded uniformly in τ. his proves he asserion since S is coninuous. c = d. Due o 26, Proposiion 7 of Doléans-Dade and Meyer 7 implies ha η ds + η ds,n is a BMOP-maringale. By Proposiion 1, S saisfies SC and N = Ñ λ dm for a local P-maringale Ñ null a and srongly P-orhogonal o each componen of M. Since S is coninuous and saisfies SC,
16 Chrisoph Frei and Marin Schweizer η ds,n = η dm,n = η dm, λ dm = d i, j=1 η i λ j d M i,m j. ence η ds + η ds,n = η dm is a BMOP-maringale. d = e. We se 1 ε := 2 η dm 2 BMO 2 P and L := ε η dm. Clearly, L is like η dm a coninuous BMOP-maringale and we have ha L BMO2 P = 1 / 2 < 1. Since S is coninuous, he John-Nirenberg inequaliy see heorem 2.2 of Kazamaki 18 yields E P exp ε η ds = E P exp 1 L 1 L 2 <. BMO 2 P e = a. his is based on he same idea as he proof of Proposiion 3.2 of Rheinländer 22. Lemma 3.5 of Delbaen e al. 6 yields ε E Q η ds IQ P + 1e E P exp ε η ds < for any Q P f because is bounded and e holds. So η ds is Q-inegrable and hus he local Q-maringale η ds is a square-inegrable Q-maringale for any Q P f. his concludes he proof in view of Proposiion 2. 4 Relaing FER and FER o he indifference value In his secion, we esablish he connecion beween FER, FER and he indifference value process h. We hen derive and sudy an inerpolaion formula for h. hroughou his secion, we assume ha P e, f f / and Pe, /, and we denoe by N,η,k and N,η,k he unique FER and FER wih associaed measures P N = Q E and P N = Q E, respecively. Our firs resul expresses he maximal expeced uiliy and he indifference value in erms of he given FER and FER. For a locally bounded S, his is very similar o Becherer 1; see in paricular here Proposiions 2.2 and 3.5 and he discussion on page 12 a he end of Secion 3. Indeed, he main differences are ha
General exponenial uiliy indifference valuaion 17 he represenaion in 1 is given in erms of cerainy equivalens insead of maximal condiional expeced uiliies and S is locally bounded; bu he resuls are he same. heorem 3. V, V and h are well defined and, for any, and any F - measurable random variable x, we have V x = exp γx + γk 28 and h x = h = k k, 29 where k and k, wih he obvious adapaions are defined in 9. Proof. Le us firs wrie 2 as V wih he abbreviaion ϕ ϑ := E P exp γ x = exp γx essinf ϕ ϑ 3 ϑ A ϑ s ds s + γ F. Because N,η,k is he FER, ϕ ϑ can be wrien by 1 as ϕ ϑ = exp γk EP E N γ, exp η s ϑ s dss F = exp γk exp γ η s ϑ s dss F, EPN 31 using Bayes formula. Since P N = Q E Pe, f and η ds is a Q-maringale for every Q P e, f E PN η s ϑ s dss F and ϑ ds is a Q-supermaringale, we have which implies ϕ ϑ exp γk by Jensen s inequaliy and 31. On he oher hand, he choice ϑs := ηs, s,, 32 gives ϕ ϑ = exp γk by 31. Because ϑ ds = η ds is a Q-maringale for every Q P e, f, ϑ is in A, and 28 now follows from 3. By he same reasoning as for 28, we obain V x = exp γx + γk. Solving he implici equaion 3 for h x hen immediaely leads o 29.
18 Chrisoph Frei and Marin Schweizer he proof of heorem 3, especially 32, gives an inerpreaion for he FER. An invesor who mus pay ou he claim a ime uses, under exponenial uiliy preferences, he decomposiion 7. he porion of ha he hedges by rading in S is η s ds s, whereas 1 γ loge N remains unhedged. Moreover, he proof of heorem 3 shows ha for, and an F -measurable x, he value of V x is no affeced if we resric he se A o hose ϑ A such ha ϑ ds is no only a Q-supermaringale, bu a Q-maringale for every Q P e, f. Proposiion 3. Assume ha saisfies 12. hen for any Q P f and,, h = E Q F 1 γ E Q log E N, E N F. 33, In paricular, h = E Q + 1 γ I Q Q E I Q Q E. 34 he decomposiion 34 of he indifference value h can be described as follows. he firs erm, E Q, is he expeced payoff under a measure Q P f. his is linear in he number of claims. he second erm is a nonlinear correcion erm or safey loading. I can be inerpreed as he difference of he disances from Q E and QE o Q alhough I is no a meric. his correcion erm is no based on all of, bu only on he processes N and N from he FER and FER, i.e., on he unhedged pars of and, respecively. A similar decomposiion also appears for indifference pricing under quadraic preferences; see Schweizer 24. If saisfies 12, hen he indifference value process h is a Q E -supermaringale. In fac, Jensen s inequaliy and 33 wih Q = Q E yield h E Q E F P-a.s. for, and so h L 1 Q E since is Q E -inegrable due o 12; compare 15. Moreover, Z := E N / E N is a Q E -maringale as i is he QE -densiy process of Q E. hus logz has he QE -supermaringale propery by Jensen s inequaliy, and so has h since h = E Q E F 1 γ E Q E logz F + 1 γ logz for, by 33. Now E Q E h h < shows ha h is Q E -inegrable for every,. Proof of Proposiion 3. Since Q P f P f by Lemma 1, η ds is a Q-maringale by Proposiion 2. Moreover, is Q-inegrable due o 12; compare 15. From 1, we hus obain for, ha k = E Q 1 γ loge N, F. 35 Plugging 35 and he analogous expression for k ino 29 leads o 33. o prove 34, we firs show ha I Q Q E is finie. We can wrie I Q Q E = EQ log dq + log dq E = IQ P E Q loge N < 36
General exponenial uiliy indifference valuaion 19 because Q P f and E Q loge N = γk by 35 for = and =. Moreover, Q P Q E dq gives > Q-a.s. and hus from ha dq dq E = dq loge N dq E = dq and analogously for insead of. ence E Q log E N E N 1 E N dq = log dq E log dq Q-a.s. Q-a.s., = E Q log dq dq E log dq dq E = I Q Q E I Q Q E, where we have used 36 for he las equaliy. Now 34 follows from 33. We nex come o he announced inerpolaion formula for he indifference value. heorem 4. Le Q P e, f and ϕ in LS be such ha ϕ ds is a Q- and Q E - maringale. Fix,, denoe by Z he P-densiy process of Q, se Ψ := exp γ + ϕ s ds s Z, 37 and assume ha Ψ and logψ are Q-inegrable. hen here exiss an F -measurable random variable δ : Ω 1, such ha for almos all ω Ω, where k ω = 1 γ log E Q Ψ 1/δ F ω δ δ=δ ω, 38 Ψ ω log E Q 1/δ δ Ψ ω F := lim log E Q 1/δ δ F 39 δ= δ = E Q logψ F ω for almos all ω Ω. In view of h = k k by heorem 3, 38 gives us a quasi-explici formula for he exponenial uiliy indifference value if is bounded and if we can find a measure Q P e, f such ha he corresponding Ψ given in 37 and logψ are Q- inegrable for some predicable ϕ such ha ϕ ds is a Q-, Q E - and QE -maringale. For =, one possible choice is he minimal -enropy measure Q E which is by 19 and Proposiion 2 of he form dqe = c 3 exp ζ s ds s for a consan c 3 and a predicable process ζ such ha ζ ds is a Q-maringale for every Q P f. One disadvanage of his choice is ha Q E is in general unknown; a second is ha we sill
2 Chrisoph Frei and Marin Schweizer need o find some ϕ, and we know almos nohing abou he poenial candidae ζ. In Corollary 1, we give condiions under which he explicily known minimal local maringale measure ˆP saisfies he assumpions of heorem 4. Proof of heorem 4. From 1 and 37, we obain via dqe = E N and Bayes formula ha exp γk E N EQ Ψ F, = EQ exp ϕs + γηs dss F Z, = E Q E exp ϕs + γη s dss F exp E Q E ϕs + γη s dss F = 1 by Jensen s inequaliy and because ϕ ds and η ds are Q E -maringales. ence 4 k 1 γ loge Q Ψ F. 41 On he oher hand, 35, 37 and Jensen s inequaliy yield γk = E Q γ loge N, F = E Q logψ log E N, Z, F E Q logψ F. 42 Consider he sochasic process f, : 1, Ω R defined by Ψ 1 ω f δ,ω := log E Q δ F δ, δ,ω 1, Ω. Because Ψ 1/δ 1 +Ψ L 1 Q for all δ 1,, Lebesgue s dominaed convergence heorem and Jensen s inequaliy for condiional expecaions allow us o choose a version of f which is coninuous and nonincreasing in δ for all fixed ω Ω, so ha by monooniciy, he limi f,ω := lim δ f δ,ω exiss for all ω Ω. We nex show ha f,ω = E Q logψ F ω for almos all ω Ω. 43 o ease he noaion, we define g, : 1, Ω R by gδ,ω := exp 1δ f δ,ω = E Q Ψ 1 δ F ω, δ,ω 1, Ω
General exponenial uiliy indifference valuaion 21 so ha f δ,ω = δ loggδ,ω. Again since Ψ 1/δ 1 +Ψ δ 1,, dominaed convergence gives L 1 Q for all lim gn,ω = 1 for almos all ω Ω. 44 n For x > 1/2 we have x 1 logx x 1 x 1 2, from which we obain by 44 ha for almos all ω Ω, here exiss n ω N such ha n gn,ω 1 f n,ω n gn,ω 1 n gn,ω 1 2, n n ω. 45 In view of 44 and 45, we ge 43 if we show ha lim n gn,ω 1 = E Q logψ F n ω for almos all ω Ω. 46 Bu 46 follows from Lebesgue s convergence heorem and Ψ lim n 1 n n 1 1 = lim n exp n n logψ 1 = logψ P-a.s. if we show ha n Ψ 1/n, 1 n N, is dominaed by a Q-inegrable random variable. Due o e x 1 x for x R and d dx x a 1 x 1 = a 1 x 1 1 x loga 1 a 1 x exp 1 x loga 1 = for a > and x >, i follows for a = Ψ logψ 1 n exp n logψ ha 1 Ψ 1, n N. his gives n Ψ 1/n 1 logψ +Ψ L 1 Q, n N, and proves 43. Combining 41, 42 and 43 yields f,ω γk ω f 1,ω for almos all ω Ω. By he inermediae value heorem, he se ω := { δ 1, f δ,ω = γk ω } is hus nonempy for almos all ω Ω. Define δ : Ω 1, by δ ω := sup ω, ω Ω, 47 seing δ := 1 on he P-null se {ω Ω ω = /}. By coninuiy of f in δ, ω is closed in R {+ } for all ω Ω, and we ge for almos all ω Ω ha f δ ω,ω = γk ω. 48
22 Chrisoph Frei and Marin Schweizer I remains o prove ha he mapping ω δ ω is F -measurable. Because f is nonincreasing and due o 47 and 48, we have for any a 1, ha { ω Ω δ ω < a } = { ω Ω f δ ω,ω > f a,ω } = { ω Ω γk ω > f a,ω } = {ω Ω γk ω > q } { ω Ω q > f a,ω } q Q up o a P-null se. he las se is in F because k and f a, for fixed a 1, are F -measurable random variables. Since F is complee, { ω Ω δ ω < a } is in F for every a R {+ }, and so δ is F -measurable. he nex resul provides a simplified version of heorem 4 based on he use of he minimal local maringale measure ˆP. Corollary 1. Fix, and assume ha is bounded and S saisfies SC. Suppose furher ha ˆP given by d ˆP := E λdm f is in Pe,, ha λds is a ˆP-, Q E - and Q E -maringale, and ha he random variable exp λ dm + 2 1 c e λ s M s λ dm, 1 λ s M s <s and is logarihm are ˆP-inegrable. hen here exis F -measurable random variables δ, δ : Ω 1, such ha for almos all ω Ω, h ω = 1 Ψ E γ log ω ˆP 1/δ δ F 1 Ψ E γ log ˆP 1/δ ω δ F where we use he convenion 39 and he definiion Ψ δ =δ ω δ =δ ω, := exp γ λ s ds s E λdm = eγ exp λ ds, E λdm. 49,, Proof. We only need o check ha Ψ, Ψ given by 49 and logψ, logψ are ˆPinegrable as he resul hen follows from heorems 3 and 4 wih he choice Q := ˆP and ϕ := λ. Using he formula for he sochasic exponenial and SC, we ge Ψ = exp λ dm + 1 c λ dm 2, <s e λ s M s 1 λ s M s, and hus Ψ, logψ L 1 ˆP by assumpion. he same is rue for Ψ because is bounded by assumpion.
General exponenial uiliy indifference valuaion 23 o he bes of our knowledge, resuls like heorem 4 and Corollary 1 have no been available in he lieraure so far. A closed-form expression for he exponenial uiliy indifference value has been known only in specific cases when he asse prices are modeled by coninuous semimaringales; see for example 1 for explici expressions of he indifference value in wo Brownian seings. here he adaped process δ, called he disorion power, is closely relaed o he insananeous correlaion beween he driving Brownian moions. he model in 1 consiss of a risk-free bank accoun and a sock S = S 1 driven by a Brownian moion W. he claim depends on anoher Brownian moion Y which has a ime-dependen and fairly general insananeous sochasic correlaion ρ wih W, wih ρ uniformly bounded away from 1. heorem 2 of 1 proves ha he indifference value is of he form of Corollary 1 above, wih δ and δ aking values beween 1 δ := inf s, 1 ρ s 2 and δ := sup 1 L P 1 ρ s 2 L P. s, For small ρ uniformly in s, in he L -norm, he claim is almos unhedgeable and 1/δ is nearly 1, whereas for ρ close o 1, he claim is well hedgeable and 1/δ is nearly. So in ha Brownian model, 1/δ is closely relaed o some kind of disance of from being aainable or hedgeable. In he subsequen discussion, we exend his idea o a more general seing, while we come back o he Brownian model in Secion 6. Consider he seing of Corollary 1 where S is in addiion coninuous and saisfies SC, and is bounded. hen he P-maringale par M of S is also coninuous and he mean-variance radeoff process K = λ dm = λ ds is P-a.s. finie by SC. he quaniy Ψ from 49 hen reduces o Ψ = exp γ 2 1K K, and he assumpions of Corollary 1 are saisfied if K is bounded, because λ dm is hen a BMOP-maringale. If we now even suppose ha K is deerminisic, he indifference value a ime simplifies o h = 1 γ log E ˆP exp γ/δ δ δ=δ by Corollary 1. If δ <, we can wrie h = Ũ 1 E ˆPŨ, where Ũ x := exp γx/δ, x R, which means ha h is a cerainy equivalen of. Noe, however, ha his is done under ˆP, no P, and wih respec o he uiliy funcion Ũ, no U, where Ũ depends iself on he claim. If δ = 1, hen Ũ and U coincide and is valued by he U-cerainy equivalen under ˆP. Moreover, 38 shows ha we hen mus have equaliy in 4 for =, which implies ha γη s λ s dss is deerminisic, hence γη λ ds =. In oher words, he equivalen formulaion 11 of FER in Proposiion 1 simplifies in his case o 5
24 Chrisoph Frei and Marin Schweizer = 1 γ loge Ñ + 1 2γ K + k, which means ha consiss only of a consan plus an unhedged erm. his may be inerpreed as saying ha has maximal disance o aainabiliy. On he opposie exreme, he case δ = leads by 5 and 39 and sill under he same assumpions o h = E ˆP. ence for δ =, we ge a familiar no-arbirage value for. In his case, 38 and 39 show ha we mus have equaliy in 42 for = ; hence E N = E λ dm and hus 11 simplifies o = η s ds + 1 2γ K + k, showing ha is aainable. Summing up, we can inerpre 1/δ as he disance of from being aainable; for 1/δ = convenion: 1/ =, he disance is minimal, whereas for 1/δ = 1, i is maximal. he following remark shows how his idea can be made mahemaically more precise. Remark 3. Assume ha S is coninuous, saisfies SC and ha K = λ dm is bounded, bu no necessarily deerminisic. By heorem 4 and Corollary 1, we can aribue o any L P a number δ := δ in 1, uniquely defined via 47 wih Q = ˆP and ϕ = λ. Defining for G, L P G : δ G + 1 2γ K = δ + 1 2γ K gives an equivalence relaion on L P. We denoe by D := L P / he se of is equivalence classes and associae o each equivalence class a represenaive. We furher define he mapping d : D D,1 for G, D by 1 dg, := δ 1 G + 2γ 1 K δ + 2γ 1 K. Clearly, d is a meric on D. A claim G L P is called ˆP- aainable if i can be wrien as G = E ˆP G + β s ds s for a predicable process β such ha β ds is a ˆPmaringale, which is hen even a BMO ˆP -maringale. If G is aainable, he FER of G+ 1 2γ K equals λ dm,β + 1 γ λ,e ˆP G, and so he erm log E N E λ dm vanishes idenically. his implies δ G + 2γ 1 K = by he proof of heorem 4, hence G. herefore, 1 d, = δ + 2γ 1 K is a disance of L P from aainabiliy. he maximal value of d, depends on he diversiy of he filraion F. If S has he predicable represenaion propery in F in he sense ha any L P is aainable as above, hen has only one equivalence class and d. On he oher
General exponenial uiliy indifference valuaion 25 hand, suppose ha here exiss a nondeerminisic local ˆP-maringale N null a and srongly ˆP-orhogonal o each componen of S such ha E N is a ˆP-maringale bounded away from zero and infiniy. he maximal disance o aainabiliy is hen aained by 1 γ loge N since d, 1 γ loge N = 1. 5 A BSDE characerizaion of he indifference value process In his secion, we prove ha he indifference value process h is he firs componen of he unique soluion, in a suiable class of processes, of a backward sochasic differenial equaion BSDE. his resul is similar o Becherer 2 and Mania and Schweizer 19, bu obained here in a general no even locally bounded semimaringale model. We assume hroughou his secion ha P e, f / and denoe by Q E he minimal -enropy measure. Le us consider he BSDE wih he boundary condiion Γ = Γ + 1 γ loge L + ψ s ds s,, 51 Γ =. 52 We inroduce hree differen noions of soluions o 51, 52. Definiion 2. We say ha he riple Γ,ψ,L is a soluion of 51, 52 if Si Γ is a real-valued semimaringale; Sii ψ is in LS; Siii L is a local Q E -maringale null a such ha E L is a posiive QE - maringale and S is a QL-sigma-maringale, where QL is defined by := E L. dql dq E We call Γ,ψ,L a special soluion of 51, 52 if furhermore Siv ψ ds is a Q-maringale for every Q P e, f ; dq Sv E P E L E log dq E L E <, i.e., he probabiliy measure QL defined by dql dq E := E L has finie relaive enropy wih respec o P. If S is locally bounded, we say ha Γ,ψ,L is an orhogonal soluion of 51, 52 if i saisfies 51, 52, Si, Sii and Siii L is a local Q E -maringale null a and srongly QE -orhogonal o every componen of S and such ha E L is posiive.
26 Chrisoph Frei and Marin Schweizer Under he assumpion ha S is locally bounded, a riple Γ,ψ,L is a soluion of 51, 52 if and only if i is an orhogonal soluion and E L is a Q E -maringale. 53 o see his, noe firs ha a locally bounded S is a QL-sigma-maringale if and only if E LS is a local Q E -maringale, under he assumpion ha QL is a probabiliy measure. If Γ,ψ,L is a soluion, hen Siii holds and all of E LS, E L and S are local Q E -maringales. ence E L is srongly QE -orhogonal o every componen of S, and herefore so is L. Conversely, if Siii holds, hen E L is like L srongly Q E - orhogonal o every componen of he local Q E -maringale S. ence E LS is a local Q E -maringale and hus S is a QL-sigma-maringale if E L is a QE -maringale. Our main resul in his secion is hen heorem 5. Assume ha saisfies 13. hen he indifference value process h is he firs componen of he unique special soluion of he BSDE 51, 52. heorem 5 looks a firs glance like heorem 13 of Mania and Schweizer 19. he imporan difference, however, is ha we do no suppose ha he filraion F is coninuous, i.e., ha all local P-maringales are coninuous. If F is coninuous, hen 1 γ loge L = L/γ γ 2 L/γ and heorem 5 corresponds o heorem 13 of Mania and Schweizer 19. Since is allowed o be unbounded in heorem 5, here are some differences in he inegrabiliy properies. owever, recovering he laer resul in precise form and almos full srengh from heorem 5 requires some addiional work which we discuss a he end of his secion. he derivaion in 19 uses he maringale opimaliy principle, he exisence of an opimal sraegy for he indifference value process, and a comparison heorem for BSDEs. Our proof is compleely differen; i is based on our resuls for he FER and is relaion o he indifference value. heorem 4.4 of Becherer 2 is anoher similar resul. Insead of a coninuous filraion, he framework in 2 has a coninuous price process driven by Brownian moions, and a filraion generaed by hese and a random measure allowing he modeling of non-predicable evens. Again, o regain from heorem 5 he same saemen as in heorem 4.4 of Becherer 2, some addiional work is necessary. In Corollary 3.6 of he earlier paper 1, Becherer gives a characerizaion of dqe dq E in a locally bounded semimaringale model. heorem 5 can be viewed as a dynamic exension of ha resul o a general semimaringale model. = Pe, f /, and so heo- Proof of heorem 5. By Lemma 1, 13 implies ha P e, f rem 3 and 9 yield h = k k = h + 1 γ log E N E N + η s ηs dss,, where N,η,k and N,η,k are he FER and FER ; see Proposiion 2 for heir properies. hen ψ := η η is in LS and ψ ds is a Q-
General exponenial uiliy indifference valuaion 27 maringale for every Q P e, f = P e, f. By Bayes formula, E N / E N is he Q E -densiy process of QE, and so i is a posiive QE -maringale and is sochasic logarihm L, defined by E L = E N / E N, is a local Q E -maringale null a. Moreover, dql = E L dqe = dqe shows QL = QE. ence S is a QL-sigmamaringale and Sv is saisfied because Q E has finie relaive enropy wih respec o P. Since h = by definiion, we see ha h is he firs componen of a special soluion of he BSDE 51, 52. o prove uniqueness, le Γ,ψ,L be any special soluion of 51, 52. Denoe by N,η,k he unique FER, and define We claim ha N := N + L + N,L, η := η + ψ and k := k +Γ. 54 N,η,k is he unique FER. 55 For he proof, we firs noe ha E N E L = E N + L + N,L = E N by Yor s formula. Using 51, 52 and 7 for = hus yields = 1 γ log E N E L + = 1 γ loge N + η s ds s + k. η s + ψ s dss + k +Γ herefore N,η,k saisfies 7 for, and i is enough o show ha he assumpions on N and η for FER are fulfilled. By Bayes formula, E N = E N E L is a posiive P-maringale, because E L is a posiive Q E -maringale by Siii and E N is he P-densiy process of Q E. Wriing nex N dq E = N dq E = E N / E N = E L, we see ha PN = QL which implies ha I PN dq E P = EP E L log dq E E L < by Sv and ha S is a PN-sigma-maringale by Siii. Because N,η,k is he FER, η ds = η ds + ψ ds is by Proposiion 2 and Siv a Q-maringale for every Q P e, f = P e, f, hence also for PN and QE, and so N,η,k is an FER saisfying c from Proposiion 2. his implies 55. Uniqueness of he FER and 54 now imply ha Γ, ψ are unique; so is L due o E L = E N / E N, and finally also Γ by 51. his ends he proof. he above argumen shows in paricular a close link beween he FER and he BSDE 51, 52. Provided we have he FER, we can consruc FER from he special soluion of 51, 52, and vice versa. his is familiar from ex-
28 Chrisoph Frei and Marin Schweizer ponenial uiliy indifference valuaion; indeed, knowing FER corresponds o knowing he minimal -enropy measure Q E. Remark 4. If S is locally bounded and is bounded, here is anoher way o prove uniqueness of he firs componen of a special soluion of he BSDE 51, 52, which we briefly skech here. If Γ,ψ,L is a special soluion of 51, 52, he idea is o show ha Γ equals he indifference value process h, which hen yields he desired uniqueness resul. Le, and replace in he definiion of A he condiion ha ϑ ds is a Q-supermaringale for every Q P e, f by assuming ha i is a Q-maringale for every Q P e, f. We do he analogous change for A and noe ha his does no affec he values of V and V, as menioned afer he proof of heorem 3. We now apply Proposiion 3 of Mania and Schweizer 19 o obain Using 51, 52 gives h = 1 log essinf E γ Q E exp ϑ A γ γ ϑ s ds s F. 56 γ = γγ + loge L + γ ψ s ds s = γγ + log E L + γ ψ s ds s, E L which we plug ino 56 o obain h = Γ + 1 log essinf E γ QL exp γ ϑ A ψ s ϑ s ds s F =: Γ + 1 γ logλ, where he probabiliy measure QL is defined by dql dq E := E L. o show ha Λ = 1, we firs noe ha QL P e, f by Sv, P e, f = Pe, f by Lemma 1, and ψ ds as well as ϑ ds are Q-maringales for every Q P e, f = Pe, f by Siv and because ϑ A. Jensen s inequaliy hen yields Λ 1, and we obain Λ 1 by he choice ϑ := ψ A. Noe ha also for his uniqueness proof, we have used he assumpion ha Γ,ψ,L is a special soluion of he BSDE 51, 52, i.e., ha i also saisfies Siv, Sv. We have seen in Secion 3 ha he difference beween FER and he unique FER is an issue of inegrabiliy. he same hing happens here: he nex example shows ha he BSDE 51, 52 may have many soluions if we omi he requiremen Siv which corresponds o d in Proposiion 2. Example 2. As in Example 1, ake independen P-Brownian moions W and W, heir P-augmened filraion F and d = 1, S = W,. hen Q E = P and,, is he unique special soluion of 51, 52. As in Example 1, ake N = W and use Proposiion 1 of Emery e al. 8 o find for any c R a process ψc in LS such ha 1 γ loge N c = ψ s cds s P-a.s.