Approximating Random Variables by Stochastic Integrals

Transcription

1 Projekbereich B Discussion Paper No. B 6 Approximaing Random Variables by Sochasic Inegrals by Marin Schweizer November 993 Financial suppor by Deusche Forschungsgemeinschaf, Sonderforschungsbereich 33 a he Universiy of Bonn, is graefully acknowledged. Annals of Probabiliy 994, This version:

2 Approximaing Random Variables by Sochasic Inegrals Marin Schweizer Universiä Göingen Insiu für Mahemaische Sochasik Lozesraße 3 D-3783 Göingen Germany Absrac: Le X be a semimaringale and Θ he space of all predicable X-inegrable processes ϑ such ha ϑ dx is in he space S of semimaringales. We consider he problem of approximaing a given random variable H L by a sochasic T inegral ϑ s dx s, wih respec o he L -norm. If X is special and has he form X = X + M + α d M, we consruc a soluion in feedback form under he assumpions ha α d M is deerminisic and ha H admis a srong F-S decomposiion ino a consan, a sochasic inegral of X and a maringale par orhogonal o M. We provide sufficien condiions for he exisence of such a decomposiion, and we give several applicaions o quadraic opimizaion problems arising in financial mahemaics. Key words: semimaringales, sochasic inegrals, srong F-S decomposiion, mean-variance radeoff, opion pricing, financial mahemaics 99 Mahemaics Subjec Classificaion: 6G48, 6H5, 9A9 Running head: L -approximaion by sochasic inegrals

3 . Inroducion In his paper, we sudy an approximaion problem arising naurally in financial mahemaics. Le X be a semimaringale on a filered probabiliy space Ω, F, F T, P and denoe by Θ he space of all predicable X-inegrable processes ϑ such ha ϑ dx is in he space S of semimaringales. Given an F T -measurable random variable H L and a consan c IR, we hen consider he opimizaion problem. Minimize E H c ϑ s dx s over all ϑ Θ. If we also vary c, we hus wan o approximae a random variable by he sum of a consan and a sochasic inegral of X. This problem has a very naural inerpreaion in financial mahemaics, in paricular in he heory of opion pricing and opion hedging. Think of X as he discouned price a ime of some risky asse e.g., a sock and of ϑ as a dynamic porfolio sraegy, where ϑ describes he number of shares of X o be held a ime. If we assume ha here also exiss some riskless asse e.g., a bank accoun wih discouned price a all imes, hen every ϑ Θ deermines a self-financing rading sraegy whose value process is given by c + ϑ dx, where c IR denoes he iniial capial a ime. For a more deailed exposiion, we refer o Harrison/Pliska 98. In his conex, he random variable H is hen inerpreed as a coningen claim or random loss o be suffered a ime T, and so. corresponds o T minimizing he expeced square of he ne loss, H c ϑ s dx s, a ime T. This problem was previously sudied in various forms of generaliy in Duffie/Richardson 99, Schäl 994, Schweizer 99, Hipp 993 and Schweizer 993a, 993b. Here we exend heir resuls o he case of a general semimaringale in coninuous ime. Once he basic problem. has been solved and if here is a nice dependence of he soluion ξ c on c, one can readily give soluions o various opimizaion problems wih quadraic crieria. These applicaions are discussed in secion 4; hey conain in paricular he opimal T choice of iniial capial and sraegy, he sraegies minimizing he variance of H c ϑ s dx s eiher wih or wihou he consrain of a fixed mean, and he approximaion of a riskless asse. Throughou he paper, X will be an IR d -valued semimaringale in Sloc. For ease of exposiion, however, we formulae he resuls in his inroducion only for d =. We assume ha X has a canonical decomposiion of he form X = X + M + α d M and call K := α s + α s M s d M s, T he exended mean-variance radeoff process of X. Our main resul in secion hen saes ha. has a soluion ξ c for every c IR if K is deerminisic and if H admis a

4 decomposiion of he form. H = H + ξ H s dx s + L H T P -a.s. wih H IR, ξ H Θ and L H a square-inegrable maringale orhogonal o ϑ dm for every ϑ. Moreover, ξ c is explicily given in feedback form as he soluion of.3 ξ c = ξ H + where α + α M V H := H + V H c ξ c s dx s, T, ξ H s dx s + L H, T is he inrinsic value process of H. An ouline of he proof is given in secion and full deails are provided in secion 3. The argumen exends he echnique inroduced in Duffie/Richardson 99 and Schweizer 99 for a diffusion process o he case of a general semimaringale. The assumpion ha K is a deerminisic process is very srong, bu unforunaely indispensable for boh our proof and he validiy of.3. On he oher hand, a decomposiion of he form. can be obained in remarkable generaliy. By slighly adaping a resul of Buckdahn 993 on backward sochasic differenial equaions, we show in secion 5 ha every F T -measurable H L admis such a decomposiion if K is bounded and has jumps bounded by a consan b <. Secion 6 concludes he paper wih several examples. In he posiive direcion, we consider coninuous processes admiing an equivalen maringale measure and a mulidimensional jump-diffusion model. On he oher hand, a counerexample shows ha.3 in general no longer solves. if K is allowed o be random.. Formulaion of he problem Le Ω, F, P be a probabiliy space wih a filraion IF = F T saisfying he usual condiions of righ-coninuiy and compleeness. T > is a fixed finie ime horizon, and we assume ha F = F T. For unexplained noaion, erminology and resuls from maringale heory, we refer o Dellacherie/Meyer 98 and Jacod 979. Le X = X T be an IR d -valued semimaringale in Sloc ; for he canonical decomposiion X = X + M + A of X, his means ha M M,loc and ha he variaion Ai of he predicable finie variaion par A i of X i is locally square-inegrable for each i. We can and shall choose versions of M and A such ha M i and A i are righ-coninuous wih lef limis RCLL for shor for each i. We denoe by M i he sharp bracke process associaed o M i, and we shall assume ha for each i,. A i M i wih predicable densiy α i = α i T.

5 Throughou he sequel, we fix a predicable increasing RCLL process B = B T null a such ha M i B for each i; for insance, we could choose B = d M i. This implies M i, M j B for all i, j, and we define he predicable marix-valued process σ = σ T by i=. σ ij := d M i, M j db for i, j =,..., d, so ha each σ ij is a symmeric nonnegaive definie d d marix. If we define he predicable IR d -valued process γ = γ T by.3 γ i := α i σ ii for i =,..., d, hen. and. imply ha for each i,.4 A i = γ i s db s, T. Definiion. The space L loc M consiss of all predicable IRd -valued processes ϑ = ϑ T such ha he process ϑ sσ s ϑ s db s T is locally inegrable, where denoes ransposiion. The space L loc A consiss of all predicable IRd -valued processes ϑ = ϑ T such ha he process ϑ sγ s db s Finally, we se Θ := L M L A. T is locally square-inegrable. If ϑ L loc M, he sochasic inegral ϑ dm is well-defined, in M,loc, and.5 ϑ dm, ψ dm = ϑ sσ s ψ s db s, T for ϑ, ψ L loc M. If ϑ L loca, he process.6 ϑ s da s := d ϑ i s da i s = ϑ sγ s db s, T i= 3

6 is well-defined as a Riemann-Sieljes inegral and has locally square-inegrable variaion ϑ da = ϑ γ db. For any ϑ Θ, he sochasic inegral process G ϑ := ϑ s dx s, T is herefore well-defined and a semimaringale in S wih canonical decomposiion.7 Gϑ = ϑ dm + ϑ da. We remark ha he sochasic inegral ϑ dm canno be defined as he sum d i= ϑ i dm i in general; his is why we refrain from using he noaion ϑ dm. On he oher hand, he noaion ϑ da makes sense due o.6. Having se up he model, he basic problem we now wan o sudy is.8 Given H L and c IR, minimize E [ H c GT ϑ over all ϑ Θ. In order o solve.8, we shall have o impose addiional assumpions on X and H. We firs inroduce he predicable marix-valued process J = J T by seing.9 J ij := <s A i s A j s for i, j =,..., d, where U := U U denoes he jump of U a ime for any RCLL process U. By.4, J can be wrien as. J ij = κ ij s db s, T, where he predicable marix-valued process κ = κ T is given by κ ij := γ i γ j B, T, for i, j =,..., d. Since B is increasing, each κ ij is a symmeric nonnegaive definie d d marix. The following erminology was parly inroduced in Schweizer 993c: Definiion. We say ha X saisfies he srucure condiion SC if here exiss a predicable IR d -valued process λ = λ T such ha. σ λ = γ P -a.s. for all [, T and. K := λ sγ s db s < P -a.s. for all [, T. 4

7 We hen choose an RCLL version of K and call i he mean-variance radeoff MVT process of X. Definiion. We say ha X saisfies he exended srucure condiion ESC if here exiss a predicable IR d -valued process λ = λ T such ha.3 σ + κ λ = γ P -a.s. for all [, T and.4 K := λ sγ s db s < P -a.s. for all [, T. We hen choose an RCLL version of K and call i he exended mean-variance radeoff EMVT process of X. Remarks. If A is coninuous, hen κ by.9 and.; hence condiions SC and ESC are equivalen in ha case. The exac relaion beween SC and ESC is shown in Lemma, and sufficien condiions for SC are provided in Schweizer 993c. For insance, every coninuous adaped process admiing an equivalen local maringale measure saisfies SC. For d =, he name mean-variance radeoff can be heurisically explained in he following way: since σ, λ, α, γ are all scalars, equaion. reduces o by.3. Thus we can choose λ = α = σ λ = σ α da = E[dX F d M Var[dX F ; of course, he las erm is no rigorously defined. 3 Inuiively, boh K and K measure he exen o which X deviaes from being a maringale. More precisely, a process X saisfying ESC is a maringale if and only if K T = P -a.s. In fac, he only if par is immediae if one noices ha K = λ da by.4 and.6, and he if par can be proved by using he definiions of K, λ and κ. In he same way, a process X saisfying SC is a maringale if and only if K T = P -a.s. The nex resul summarizes some elemenary properies of λ and λ; as hey are sraighforward o verify from he definiions, we omi he proof. Lemma. X saisfies SC if and only if X saisfies ESC and K d K s < P -a.s. for all [, T ; s 5

8 in paricular, we hen have K < P -a.s. for all [, T. If X saisfies SC, λ and λ can be consruced from each oher by λ = and K, K are hen relaed by K = λ K, λ = K d K s, K = s λ + K, + K s d K s. Suppose ha X saisfies SC. Then he process K does no depend on he choice of λ and is locally bounded. Any λ saisfying. and. is in L loc M, and he sochasic inegral λ dm is well-defined, in M,loc and does no depend on he choice of λ. Finally, we hen have K = λ dm. 3 Suppose ha X saisfies ESC. Then he process K does no depend on he choice of λ and is locally bounded. Any λ saisfying.3 and.4 is in L loc M, and he sochasic inegral λ dm is well-defined, in M,loc and does no depend on he choice of λ. Finally, we hen have K [ = λ dm + λ da. For some purposes, i is useful o have an alernaive descripion of he space Θ. Recall ha LX denoes he se of all IR d -valued X-inegrable predicable processes. Lemma. If X saisfies., hen { Θ = ϑ LX ϑ dx S } =: Θ. If in addiion X saisfies SC and K T is bounded, hen Θ = L M. Proof. Since he variaion of ϑ da is given by ϑ γ db, i is clear ha Θ conains L M L A. Conversely, X is special and ϑ dx is special for any ϑ Θ ; hence ϑ dm and ϑ da boh exis in he usual sense by Théorème of Chou/Meyer/Sricker 98, and ϑ dx S hus implies ha ϑ L M L A. Finally, ϑ sγ s db s = KT ϑ sσ s λs dbs ϑ sσ s ϑ s λ s σ s λs ϑ sσ s ϑ s db s db s shows ha L M L A if K T is bounded. q.e.d. 6

9 . The main heorem Throughou his secion, we shall assume ha X is given as in secion. In order o formulae our cenral resul on he soluion of.8, we inroduce he following Definiion. We say ha a random variable H L admis a srong F-S decomposiion if H can be wrien as. H = H + ξ H s dx s + L H T P -a.s., where H IR is a consan, ξ H Θ is a sraegy and L H = L H T is in M wih E [ L H = and srongly orhogonal o ϑ dm for every ϑ L M. Remarks. If X is a locally square-inegrable maringale, hen such a decomposiion always exiss. In fac,. is hen he well-known Galchouk-Kunia-Waanabe decomposiion obained by projecing H on he space G T L X which is closed in L since he sochasic inegral is an isomery by he local maringale propery of X. For more deails, see Kunia- Waanabe 967, Galchouk 975 and Meyer 977. Under some addiional assumpions on X, i was shown by Föllmer/Schweizer 99 and Schweizer 99 ha H admis a decomposiion. if and only if here exiss a locally risk-minimizing rading sraegy for H. A more general decomposiion of he ype. was hen sudied by Ansel/Sricker 99 whose erminology we adop and adap here. In paricular, hese auhors prove he uniqueness of such a generalized decomposiion and give sufficien condiions for is exisence in he case d =. For he case where X is coninuous, heir resuls were exended o he mulidimensional case d > in Schweizer 993c. Using a recen resul of Buckdahn 993 on backward sochasic differenial equaions, we shall provide sufficien condiions for a srong F-S decomposiion in secion 5. 3 In a discree-ime framework, a srong F-S decomposiion exiss for any H L if X has a bounded MVT process; see Proposiion.6 of Schweizer 993b. In ha case, Theorem. of Schweizer 993b even shows ha G T Θ is closed in L alhough he sochasic inegral is no an isomery in general. Boh hese resuls are proved by backward inducion in discree ime and hus sugges an approach using backward sochasic differenial equaions. We shall provide an analogue of he firs resul in secion 5 under an addiional condiion on he jumps of K; he quesion of closedness of G T Θ in L remains open so far. Theorem 3. Suppose ha X saisfies ESC and ha he EMVT process K of X is deerminisic. If H L admis a srong F-S decomposiion, hen.8 has a soluion ξ c Θ for any c IR. I is given as he soluion of he equaion. ξ c = ξ H + λ V H c G ξ c where he process V H = V H T is defined by.3 V H := H +, T, ξ H s dx s + L H, T. 7

10 Skech of proof. Since he acual argumen is raher lenghy, we give here only he idea of he proof and provide full deails in he nex secion. The firs sep is o show by sandard argumens and esimaes for sochasic differenial equaions ha. has indeed a soluion ξ c and ha ξ c Θ. Since G T Θ is a linear subspace of he Hilber space L, he projecion heorem implies ha a sraegy ξ Θ solves.8 if and only if.4 E [ H c G T ξ G T ϑ = for all ϑ Θ. By.3 and., H = VT H funcion f : [, T IR by P -a.s.; o prove.4, we hus fix ξ, ϑ Θ and define he f := E [ V H c G ξ G ϑ, T. Then he heorem will be proved if we show ha ft = for ξ = ξ c and arbirary ϑ. Now he produc rule and some compuaions give f = E ϑ s + E σ s + κ s ξs H ξ s + γ s V H s c G s ξ db s γ s ξ H s ξ s G s ϑ db s ; insering ξ = ξ c hence yields by.,.3 and.4 f = E Vs H c G s ξ c G s ϑ d K s = fs d K s, since K is deerminisic. Thus f for any ϑ Θ if ξ = ξ c, so ξ c solves.8. Remarks. The above scheme of proof is essenially due o Duffie/Richardson 99. In a model where X is geomeric Brownian moion, hey considered he random variable H = kxt and inroduced he funcion f wih V H replaced by a racking process Z, i.e., a process wih Z T = H P -a.s. For heir special choice of H, Z is easy o guess direcly. In he same framework for X, heir approach was exended o general random variables H by Schweizer 99 who poined ou he possibiliy of sysemaically choosing V H as he racking process. The presen work now considers he case where X is a general semimaringale in Sloc and provides a large class of examples where he condiions of Theorem 3 are saisfied. In a discree-ime framework, problem.8 was also considered by Schäl 994 and Schweizer 993a, 993b. Whereas Schäl 994 worked under he assumpion ha he MVT process is deerminisic, he resuls of Schweizer 993b show ha.8 can be solved in discree ime under he sole assumpion ha he EMVT process is bounded. I is a presen an open quesion wheher his resul can be exended o he coninuous-ime case in full generaliy. 8

11 3. Proof of he main heorem In his secion, we give a deailed proof of Theorem 3. We shall assume hroughou he secion ha X is given as in secion. More specific assumpions abou X and H will be saed when hey are necessary. 3.. Consrucion of he sraegy ξ c The firs sep of he proof consiss in showing ha ξ c is well-defined by. and in Θ. Proposiion 4. Suppose ha X saisfies ESC and ha he EMVT process K of X is deerminisic. If H admis a srong F-S decomposiion, hen for every c IR, here exiss a sraegy ξ c Θ wih 3. ξ c = ξ H + λ V H c G ξ c wih equaliy in L M, where V H is given by.3. Proof. By.3 and.4, λ sσ s λs db s λ sγ s dbs = λ sγ s db s = K T, and since K T is deerminisic, hence bounded, we conclude ha λ is in Θ. Thus he processes Z := Y := λ u dx u, T ξu H + λ u Vu H c dx u, T are well-defined and semimaringales. By Theorem V.7 of Proer 99, he equaion 3. U = Y + U s dz s, T has herefore a unique srong soluion U which is also a semimaringale. Since ξ H Θ and L H M by he srong F-S decomposiion of H, i is clear from.3 ha sup Vu H c L. Since K is deerminisic, hence bounded, his implies ha u T 3.3 sup E [ Y <. T In fac, he definiion of Y yields Y ξu H dx u + 4 Vu H c λ u dm u Vu H c λ u da u

12 and herefore sup E [ Y E sup T T + 4 sup E T ξ H u dx u + 4 sup E T Vu H c d K u Vu H c λ u σ u λu db u by.5,.6 and.4. Bu he firs erm on he righ-hand side is finie since ξ H Θ, and he hird is dominaed by 4E KT T sup u T V H u c d K u Finally, he second erm is majorized by 4E 4 KT E [ sup u T V H u c <. sup Vu H c λ uσ u + κ u λ u db u 4 [ KT E sup Vu H c <, u T u T because κ is nonnegaive definie. This proves From 3.3 and he fac ha K is deerminisic, we obain 3.4 sup E [ U <. T To see his, define he funcion h on [, T by h := E [ U. Then 3. and he definiions of Y and Z imply as in sep h E [ Y + 4E U s λs dm s + 4E U s λ s da s E [ Y + 4 E [ Us d Ks + 4 KT E [ Us d Ks E [ Y KT hs d K s, where he second inequaliy uses Fubini s heorem and he fac ha K is deerminisic. From Gronwall s inequaliy, we conclude ha h exp 4 + KT KT E [ Ys, and so 3.4 follows from 3.3. sup s

13 4 Again since K is deerminisic, 3.4 implies ha 3.5 ϑ := λ V H c U Θ. In fac,.4 yields ϑ sγ s db s = Vs H c U s d K s KT and herefore ϑ L A by 3.4, since K is deerminisic. Furhermore, ϑ sσ s ϑ s db s = T V H s c U s λ s σ s λs db s V H s c U s d K s V H s c U s d Ks by.4 and.3, since κ is nonnegaive definie. Because K is deerminisic, 3.4 implies ha ϑ L M, hence ϑ Θ. 5 Due o 3.5, we can now define a sraegy ξ c Θ by seing Then he definiions of Y and Z imply ha G ξ c = ξ c := ξ H + λ V H c U. ξ s c dx s = Y + U s dz s = U P -a.s. for all [, T by 3. so ha Gξ c saisfies he sochasic differenial equaion G ξ c = Y + G s ξ c dz s = G ξ H + for [, T. Hence he special semimaringale λ s Vs H c G s ξ c dx s Gξ c Gξ H λ V H c G ξ c dx = ξ c ξ H λ V H c G ξ c dx is indisinguishable from, and his implies in paricular ha is inegrand mus be in L M, hus proving 3.. q.e.d. Remark. A closer look a he preceding proof reveals ha we do no really need he full srengh of he assumpion ha K is deerminisic. The same argumen sill works if here exiss a deerminisic funcion k : [, T IR such ha k K is P -a.s. increasing. However,

14 his condiion is no sufficien o prove Theorem 3 by our mehods, and so we have refrained from saing Proposiion 4 in his slighly more general form. 3.. An auxiliary echnical resul The following lemma is a echnical ool which is crucial in he proof of Theorem 3. I allows us o resric aenion o bounded sraegies ϑ in he definiion of he funcion f, and i also les us exploi sopping echniques in he subsequen argumens. We denoe by P B he Doléans measure of he process B on he produc space Ω [, T, and we recall ha an increasing sequence T m m IN of sopping imes is called saionary if P -a.s. he sequence Tm ω is consan from some m m IN ω on. Lemma 5. For fixed H L, c IR and ξ Θ, he following saemens are equivalen: a ξ solves.8. b E [ H c G T ξ G T ϑ = for all ϑ Θ. c E [ H c G T ξ G T ϑ = for all bounded ϑ Θ. d For every bounded ϑ Θ, here exiss a saionary sequence T m m IN of sopping imes such ha T m T P -a.s. and E [ H c G T ξ G T ϑi,tm = for all m IN. Proof. Since ξ is in Θ and G T Θ is a linear subspace of he Hilber space L, he equivalence of a and b follows direcly from he projecion heorem, and i is clear ha b implies c and c implies d. Consider now any sequence ϑ m m IN of IR d -valued predicable processes wih he following properies: 3.6 ϑ m ϑ P B -a.e. for some ϑ Θ, 3.7 sup ϑ m s ϑ s γ s db s L m IN and 3.8 sup ϑ m s ϑ s σ s ϑ m s ϑ s db s L. m IN Then G T ϑ m ends o G T ϑ in L. In fac, 3.6 implies ha boh ϑ m ϑ γ and ϑ m ϑ σϑ m ϑ converge o P B -a.e. Then 3.7 yields by dominaed convergence firs ϑ m s ϑ s γ s db s P -a.s.,

15 hence also in L again by 3.7, so ha ϑ m s da s by.6. In he same way, 3.8 yields ϑ s da s in L ϑ m s ϑ s σ s ϑ m s ϑ s db s in L by wice using he dominaed convergence heorem. Bu he las convergence means ha ϑ m ends o ϑ in L M, and his implies ϑ m s dm s ϑ s dm s in L by he isomery propery of he sochasic inegral, hence he asserion by.7. 3 To show ha c implies b, we now fix ϑ Θ and define a sequence of bounded predicable processes ϑ m by seing ψ m := m ϑ m and ϑ m := ψ m I { ψ m γ ϑ γ }I {ψ m ϑ σψ m ϑ ϑ σϑ}i {ψ m σψ m ϑ σϑ}. Then ϑ m m IN saisfies , for by he definiion of ϑ m we have since ϑ L A, and sup ϑ m s ϑ s γ s db s ϑ sγ s db s L, m IN sup ϑ m s ϑ s σ s ϑ m s ϑ s db s ϑ sσ s ϑ s db s L, m IN because ϑ L M. Hence implies ha G T ϑ m ends o G T ϑ in L, and since each ϑ m is in Θ, b follows from c. 4 Finally we show ha d implies c. To ha end, fix a bounded ϑ Θ and a sequence T m m IN of sopping imes as in d. If we define predicable processes ϑ m by ϑ m := ϑi,tm, hen ϑ m m IN again saisfies In fac, saionariy and T m T P -a.s. imply ha ϑ m ϑ P -a.s. for all [, T, hence 3.6. Furhermore, he definiion of ϑ m implies ha sup ϑ m s ϑ s γ s db s ϑ sγ s db s L, m IN 3

16 since ϑ L A, and by he nonnegaive definieness of σ, we have sup ϑ m s ϑ s σ s ϑ m s ϑ s db s sup m IN m IN T m ϑ sσ s ϑ s db s ϑ sσ s ϑ s db s L, since ϑ L M. Thus implies ha G T ϑ m ends o G T ϑ in L, and so c follows from d. q.e.d. Remark. I is imporan for laer applicaions ha he sequence T m m IN of sopping imes can depend on ϑ; his is clearly allowed by he formulaion in d Proof ha ξ c is opimal We begin wih a preliminary echnical resul: Lemma 6. Suppose ha L M is srongly orhogonal o ϑ dm for every ϑ L M. For all sraegies ψ, ϑ Θ, we hen have [ E [Gψ + L, Gϑ = E ϑ sσ s + κ s ψ s db s, T. Proof. By he bilineariy of he square bracke, we have 3.9 [ [Gψ + L, Gϑ = [ + ψ dm, ψ dm + L, [ ϑ dm + ϑ da + ψ da, [ [ ϑ da + L, ψ da, ϑ dm. ϑ dm Since ψ dm and ϑ dm are boh in M, [ ψ dm, ϑ dm ψ dm, ϑ dm is a maringale null a and herefore [ E[ ψ dm, ϑ dm = E[ ψ dm, ϑ dm = E ϑ sσ s ψ s db s by.5. Furhermore, ψ da and ϑ da are boh of finie variaion; his implies ha [ ψ da, ϑ da = <s = d = <s i,j= ϑ sκ s ψ s db s 4 ψ da s ψ i s A i s A j sϑ j s ϑ da s

17 by.9 and.. Since L M is srongly orhogonal o [ ϑ dm for every ϑ L M, L, ϑ dm is a maringale null a for every ϑ L M. Thus i is enough o show ha he fourh and fifh erm on he righ-hand side of 3.9 are boh maringales null a. Now ake any Y M and any predicable finie variaion process C null a wih C T L. Then we claim ha [Y, C is a maringale null a. In fac, [Y, C = Y s C s, T <s is a local maringale null a by Yoeurp s lemma, and sup [Y, C Y s T <s T [Y T C s <s T [Y T C T L <s T C s shows ha his local maringale is acually a rue maringale. Applying his resul once wih Y := ψ dm + L, C := ϑ da and once wih Y := ϑ dm, C := ψ da complees he proof. q.e.d. Proof of Theorem 3. Now we can assemble he previous resuls o prove he main heorem. So fix H L and c IR and assume ha he condiions of Theorem 3 are saisfied. Then he sraegy ξ c Θ is well-defined by 3. due o Proposiion 4. Fix any bounded ϑ Θ and define a sequence of sopping imes by { T m := T inf } V H G + ξ c + G ϑ m. Then T m m IN is saionary, increases o T P -a.s., and V H, G ξ c and G ϑ are all bounded on [[, T m for each m. Define he funcion f : [, T IR by [ f := E V H c G ξ c G ϑi,tm, T. If we can show ha ft = for each m, hen Lemma 5 will imply ha ξ c solves.8, since VT H = H P -a.s. by.3 and., and ϑ was arbirary. Fix m IN. Since by.3, he produc rule implies ha 3. V H c Gξ c = H c + Gξ H ξ c + L H V H c G ξ c G ϑi,tm = Vs H c G s ξ c I,Tm sϑ s dx s G s ϑi,tm G s ϑi,tm dl H s ξ H s ξ s c dx s [ Gξ H ξ c + L H, G ϑi,tm 5.

18 Bu V H and G ξ c are bounded on [[, T m and ϑ is in Θ; hus he process V H c G ξ c I,Tm ϑ dm is a maringale null a. Moreover, G ϑi,tm is bounded due o our choice of T m, and so he processes G ϑi,tm ξ H ξ c dm and G ϑi,tm dl H are also maringales null a. Taking expecaions in 3. and using Lemma 6 herefore yields f = E Vs H c G s ξ c I,Tm sϑ s da s + G s ϑi,tm + = E + E I,Tm sϑ sσ s + κ s ξ H s I,Tm sϑ s G s ϑi,tm ξ c s db s γ s V H s c G s ξ c + σ s + κ s ξ H s ξ H s ξ c s γ s db s ξ s c ξ H s ξ s c da s db s by.4. Bu now 3. and.3 show ha he firs erm vanishes by our choice of ξ c, and again using 3. o rewrie he second one, we obain f = E Vs H c G s ξ c G s ϑi,tm λ s γ s db s = E [ Vs H c G s ξ c G s ϑi,tm d K s by.4 and Fubini s heorem, since K is deerminisic. I is now no difficul o show ha [ 3. E Vs H c G s ξ c G s ϑi,tm = fs for each s, T. In fac, V H u, G u ξ c and G u ϑi,tm converge o V H s, G s ξ c and G s ϑi,tm, respecively, as u increases o s, and as sup Vu H u T, sup u T G u ξ c, sup u T G u ϑi,tm are all in L, 3. follows from he dominaed convergence heorem. Thus f saisfies he inegral equaion f = fs d K s, T ; since his has a unique soluion by Theorem V.7 of Proer 99 recall ha K is RCLL, hence a semimaringale, we conclude ha f, and so he proof of Theorem 3 is complee. q.e.d. 6

19 4. Applicaions In his secion, we use Theorem 3 o solve several opimizaion problems wih quadraic crieria. Unless explicily saed oherwise, we always assume ha X is given as in secion and saisfies he assumpions of Theorem 3. We also fix a random variable H in L admiing a srong F-S decomposiion. 4.. Explici compuaions and auxiliary resuls Lemma 7. For any c IR, 4. E [ V H c G ξ c = H ce K, T. Proof. Since V H c Gξ c = H c+gξ H ξ c +L H by.3 and since ξ H ξ c dm, L H are maringales, we have [ h := E V H c G ξ c = H c + E = H c E ξ H s ξ c s da s [ Vs H c G s ξ c d K s by 3.,.4 and Fubini s heorem, since K is deerminisic. A similar argumen as for 3. shows ha [ E Vs H c G s ξ c = hs ; hence h saisfies he inegral equaion h = H c hs d K s, T, and so 4. follows from Theorem II.36 of Proer 99. q.e.d. Lemma 8. For any c IR, 4. E [ V H c G ξ c = H c E K + g,, T, where g : [, T IR is he unique RCLL soluion of he equaion 4.3 g = E [ L H + E [ L H 7 gs d K s, T.

20 Proof. By Theorem V.7 of Proer 99, 4.3 has indeed a unique soluion. Now define h : [, T IR by [ h := E V H c G ξ c L H. Since L H and ξ H ξ c dm are srongly orhogonal, we obain E [ L H G ξ H ξ c = E L H ξ H s ξ c s da s = E L H s ξ H s ξ c s da s by Theorem VI.6 of Dellacherie/Meyer 98 and an approximaion argumen o accoun for he fac ha L H is no bounded, bu only in M. Thus.3 implies ha [ h = E L H G ξ H ξ c + E [ L H = E L H s ξs H ξ s c da s + E [ L H + E [ L H = E [ L H + E [ L H E [ Vs H c G s ξ c L H s d K s, where he las equaliy uses 3.,.4 and he fac ha K is deerminisic. argumen as for 3. shows ha [ E Vs H c G s ξ c L H s = hs ; A similar hence h saisfies he inegral equaion h = E [ L H + E [ L H hs d K s, T and herefore by uniqueness coincides wih g. Now he same argumens as in he proof of Theorem 3 yield for arbirary ϑ Θ [ E V H c G ξ c G ϑ =, T, and so we deduce from.3 and 4. ha [ [ E V H c G ξ c = E V H c G ξ c H c + G ξ H ξ c + L H = H c E K + h, hence 4.. q.e.d. Equaion 4.3 for he funcion g no only has a unique soluion; here also exiss an explici expression for g which can for insance be found in Théorème 6.8 of Jacod

21 [ H This allows us o give an explici formula for he minimal risk E c GT ξ c as a funcion of he iniial capial c. The resul generalizes a previous compuaion of Duffie/Richardson 99 and provides he coninuous-ime analogue of he resuls of Schäl 994 and Schweizer 993b. For ease of exposiion, we only rea here he case where K <. This is no severe resricion since we have K in any case. In fac,.4,.3,.5,. and.6 imply ha K = λ σ + κ λ = λ σ λ B + d i,j= λ i κ ij λ j B = λ dm + K is a real soluion of he equaion x = c + x wih c. Since he soluions of his equaion are ± 4 c and since here exiss a real soluion, we conclude ha c 4 Corollary 9. Suppose ha Then we have for any c IR 4.4 K = λ A < P -a.s. for [, T. [ H min E c GT ϑ [ = E H c G T ξ c ϑ Θ = E K T H c + E [ L H + E K d E [ L H s. s If K is coninuous, 4.4 simplifies o and x. 4.5 [ E H c G T ξ c = e K T H c + E [ L H + E e K T K s d L H s. Proof. By Theorem 3 and Lemma 8, i is clearly enough o compue he value gt. Since K <, Théorème 6.8 of Jacod 979 implies ha g is given by E K E [ L H + E K d E [ L H s s [ E K d E [ L H, K s for every [, T. Because E [ L H and K are boh RCLL and of finie variaion, [ E [ L H, K = E [ L H s Ks = K s d E [ L H s <s 9 s

22 by Theorem VIII.9 of Dellacherie/Meyer 98. Furhermore, E K s = E K s + E K E K s s = E K s K s by he definiion of he sochasic exponenial, and hus we obain 4.4. If K is coninuous, hen E K = exp K and 4.4 simplifies o 4.6 gt = e K T E [ L H + e K T K s d E [ L H s. Now ake any sequence τ n n IN of pariions of he inerval [, T whose mesh size τ n := max i+ i ends o. Due o he coninuiy of K, Theorem I.49 of Proer 99 i, i+ τ n implies ha and e K T K s d E [ L H s = lim e KT Ki n i τ n E [ L H i+ E [ L H i e K T K s d L H s = lim e KT Ki n i τ n L H i+ L H i P -a.s. Since K is increasing and L H M, he sums on he righ-hand side of he las equaion are bounded by L H T L. Hence we obain e K T K s d E [ L H s = E e K T K s d L H s by he dominaed convergence heorem, and combining his wih 4.6 yields 4.5. q.e.d. 4.. The opimal choice of iniial capial and sraegy As a firs applicaion, consider now he problem 4.7 Minimize E [ H V G T ϑ over all pairs V, ϑ IR Θ. This can be inerpreed as choosing an iniial capial V and a self-financing rading sraegy ϑ so as o minimize he expeced ne quadraic loss a ime T. In paricular, V is hen he Θ-approximaion price of H as defined in Schweizer 993d.

23 Corollary. Under he assumpions of Theorem 3, he soluion of 4.7 is given by he pair H, ξ H. Proof. Since he funcion g defined by 4.3 does no depend on c, i is clear from Lemma 8 ha he mapping c E [ H c GT ξ c is minimized by c = H. For any pair c, ϑ, he definiions of ξ c and c herefore imply ha [ H E c GT ϑ [ [ E H c G T ξ c E H c G T ξ c The variance-minimizing sraegy Consider nex he problem 4.8 Minimize Var[H G T ϑ over all ϑ Θ. q.e.d. In a very special case for boh X and H, his was solved by Richardson 989 and Duffie/ Richardson 99; he nex resul gives he soluion in our general framework. Noe ha in conras o Duffie/Richardson 99, our argumen remains he same wheher X is a maringale or no. Corollary. Under he assumpions of Theorem 3, he soluion of 4.8 is given by he sraegy ξ H. Proof. Wih he same noaions as in he proof of Corollary, we have for every ϑ Θ [ H Var[H G T ϑ = E E[H GT ϑ G T ϑ [ E H E[H G T ϑ G T ξ E[H G T ϑ [ E H c G T ξ c [ Var H c G T ξ c [ = Var H G T ξ c, where he firs inequaliy uses he definiion of ξ c wih c := E[H G T ϑ and he second he definiion of c. q.e.d The mean-variance fronier The hird problem we address is 4.9 Given m IR, minimize Var[H G T ϑ over all ϑ Θ saisfying he consrain E[H G T ϑ = m.

24 We firs show ha for every c IR, ξ c is H-mean-variance efficien in he sense ha [ Var H G T ξ c Var[H G T ϑ for every ϑ Θ such ha E[H G T ϑ = E [ H G T ξ c. To see his, le m = E [ H G T ξ c, ake any ϑ Θ wih E[H G T ϑ = m and use he definiion of ξ c o obain Var[H G T ϑ = Var[H c G T ϑ [ H = E c GT ϑ m c [ [ E H c G T ξ c E H c G T ξ c [ = Var H c G T ξ c [ = Var H G T ξ c. Like 4.8, also 4.9 was solved by Richardson 989 and Duffie/Richardson 99 in a very special case, and we now generalize heir resul o our siuaion. Noe ha he assumpion K T below is equivalen o assuming ha X is no a maringale; see secion. Corollary. Assume he condiions of Theorem 3 and suppose ha K T. For every m IR, he soluion of 4.9 is hen given by ξ c m wih 4. c m = m H E K T E K T. Proof. Fix m IR. By he H-mean-variance efficiency of ξ c, i is enough o show ha here exiss c IR wih E [ H G T ξ c = m, since he corresponding sraegy ξ c will hen solve 4.9. Bu Lemma 7 implies ha for every c IR E [ H G T ξ c = H E K T + c E K T, and his equals m if c is given by c m in 4.; noe ha c m is well-defined since E K T by he assumpion ha K T. q.e.d Approximaion of a riskless asse As a las applicaion, consider now he problem.8 in he special case where H and c =. The sraegy ξ c = ξ by definiion hen solves he problem 4. Minimize E [ GT ϑ over all ϑ Θ.

25 This can be inerpreed as approximaing in L he riskless payoff by he erminal wealh achievable by a self-financing rading sraegy ϑ. Such a quesion is of some ineres in pracice since i may happen ha we have several risky asses X,..., X d, bu no riskless asse a our disposal. The assumpion c = is hen quie naural, since he absence of a riskless asse makes i impossible o ransfer an iniial capial from ime o ime T. Proposiion 3. Under he assumpions of Theorem 3, he soluion of 4. is given by he sraegy 4. ξ = λ E λ dx, T. The corresponding gains process Gξ is 4.3 G ξ = E λ dx, T. For every [, T, ξ also solves he problem 4.4 Minimize E [ G ϑ over all ϑ Θ, and we have 4.5 [ E G ξ = E K, [ Var G ξ = E K E K. Proof. I is obvious ha he srong F-S decomposiion of H is given by H =, ξ H and L H. Since V H,. herefore implies ha Gξ saisfies he equaion hence G ξ = G ξ = E G s ξ λs dx s, T, λ dx, T, and his proves 4.3 and 4.. The same argumen as in he proof of Theorem 3 shows ha ξ solves 4.4. Finally, L H implies ha g by 4.3, so Lemma 7 and Lemma 8 yield [ [ E G ξ = E K = E G ξ and herefore 4.5. q.e.d. 3

26 4.6. The maringale case In his subsecion, we ake a brief look a he simplificaions of he preceding resuls in he case where X is a local maringale, i.e., A. Firs of all, Θ hen coincides wih L M and GΘ is jus he sable subspace of M generaed by M M = X X. Since G T Θ is herefore a closed subspace of L, i is clear ha.8 has a unique soluion for every H L, and every H L admis a srong F-S decomposiion which is given by he wellknown Galchouk-Kunia-Waanabe decomposiion of H wih respec o he local maringale X. The process λ is idenically, and herefore ξ c = ξ H = ξ H for every c IR by 3.. Finally Gϑ is a maringale for every ϑ Θ, so E [H G T ϑ = E[H = H for every ϑ Θ and hus i is clear ha 4.9 can only have a soluion for m = H. 5. Exisence of a srong F-S decomposiion In his secion, we give a sufficien condiion on X o ensure ha every H L admis a srong F-S decomposiion. Basically, his is a consequence of a recen resul by Buckdahn 993 on backward sochasic differenial equaions. To keep he paper self-conained and since our case is no exacly covered by Buckdahn s resuls, we neverheless provide complee proofs here. Unless saed differenly, we shall assume ha X is given as in secion and saisfies SC. Firs of all, we need some noaion: Definiion. R denoes he space of all real-valued adaped RCLL processes U = U T such ha U R := sup U <. T L By I M, we denoe he space of all maringales L M such ha E[L = and L is srongly orhogonal o ϑ dm for every ϑ L M. In oher words, I M is he orhogonal complemen in M of he sable subspace generaed by M. Finally, B denoes he Banach space R L M I M wih any of he equivalen norms U, ϑ, L a := a U R + ϑ sσ s ϑ s db s + L T for a >. Noe ha his definiion coincides wih he one by Buckdahn 993 if he componens of M are pairwise orhogonal. Definiion. Fix a random variable H L, a process ϱ L M and an IR d -valued predicable RCLL process C = C T of finie variaion null a such ha ϑ dc is in R for every ϑ L M. The mapping ψh,ϱ C : B B is hen defined by ψh,ϱu, C ϑ, L := Ũ, ϑ, L, 4 L

27 where Ũ is an RCLL version of 5. Ũ := E H ϱ s + ϑ s dc s F, T, and ϑ and L are given by he Galchouk-Kunia-Waanabe decomposiion H ϱ s + ϑ s dc s = E H ϱ s + ϑ s dc s + ϑ s dm s + L T ; see Jacod 979, Théorème 4.35 and Proposiion 4.6. From he definiion of ψh,ϱ C, i is clear ha Ũ, ϑ, L saisfies he equaion 5. Ũ = H ϱ s + ϑ s dc s ϑ s dm s L T L, T. To find a srong F-S decomposiion of a given H L, we shall herefore look for a fixed poin V H, ξ H, L of he mapping ψh, A, since we hen obain from 5. ha 5.3 H = H + wih H := E [ V H and L H := L + V H E [ V H. ξ H s dx s + L H T P -a.s. Proposiion 4. Suppose ha C has he form C = σν db for some predicable IR d -valued process ν. If C saisfies 5.4 KC T := ν s σ s ν s db s δ < P -a.s. for some consan δ, hen ψ C H,ϱ has a unique fixed poin in B for every pair H, ϱ L L M. Proof. Noe firs ha 5.4 ensures ha ψh,ϱ C is well-defined since by he Cauchy-Schwarz inequaliy, ϑ dc = T ϑ sσ s ν s db s K C T ϑ sσ s ϑ s db s L. Following Buckdahn 993, we now show ha ψh,ϱ C is a conracion on B, a for suiable a. Firs of all, 5. implies ha Ũ Ũ T = E ϑ s ϑ s dc s F T E ϑ s ϑ s σ s ν s db s F 5

28 and herefore Ũ Ũ R ϑ s ϑ s σ s ν s db s L KC T ϑ ϑ L M by he Doob and Cauchy-Schwarz inequaliies. Moreover, 5. shows ha T ϑ s ϑ s dm s + L T L L T + L = = ϑ s ϑ s dc s Ũ + Ũ ϑ s ϑ s dc s E ϑ s ϑ s dc s F and so we obain = E ϑ s ϑ s σ s ϑ s ϑ s db s + L L T L ϑ s ϑ s dm s + L T L L T + L ϑ s ϑ s σ s ν s db s KC T ϑ ϑ L M. L Puing hese esimaes ogeher, we obain ψh,ϱu, C ϑ, L ψh,ϱu C, ϑ, L a = Ũ Ũ, ϑ ϑ, L L a a + KC T ϑ ϑ L M a + δ U, ϑ, L U, ϑ, L a, and so 5.4 implies ha ψ C H,ϱ is indeed a conracion on B, a for < a < δ δ. This complees he proof. q.e.d. Theorem 5. Suppose ha X saisfies SC and ha he MVT process K of X is bounded and saisfies 5.5 sup { K } τ τ sopping ime <. 6

29 Then every H L admis a srong F-S decomposiion. Proof. As in Buckdahn 993, we show by a backward inducion argumen ha ψ A H, has a fixed poin in B for every H L. Since K is bounded, 5.5 implies he exisence of sopping imes = τ < τ <... < τ n = T such ha 5.6 Kτj K τj δ < P -a.s. for j =,..., n and some consan δ. Define he processes C j and D j by seing C j := I τj,t s da s, T, for j =,..., n +, D j := C j C j+ = I τj,τ j s da s, T, for j =,..., n. Due o 5.6, K Dj T = K τj K τj δ < P -a.s. for j =,..., n and so each ψ,ϱ Dj has a unique fixed poin U, ϑ, L B for every ϱ L M by Proposiion 4. Moreover, he definiion of ψ,ϱ Dj shows ha ϑ is given by he inegrand in he Galchouk- Kunia-Waanabe decomposiion of ϱ s + ϑ s dds j = I τj,τ j sϱ s + ϑ s da s, and since his random variable is F τj -measurable, we conclude ha ϑ = on τ j, T. Now fix H L. Due o 5.6, K Cn T = K τn K τn δ < P -a.s., and so Proposiion 4 implies ha ψ Cn H, has a unique fixed poin V n, ξ n, L n in B. Assuming ha ψ Cj H, has a fixed poin V j, ξ j, L j in B, we denoe by U j, ϑ j, R j he unique fixed poin of ψ Dj,ξ j. Since ϑ j = on τ j, T, we obain ξ j dc j + ξ ξ j + ϑ j dd j = j I τj,t + ξ j + ϑ j I τj,τj da = I τj,t ξ j + ϑ j da = ξ j + ϑ j dc j, 7

30 and 5. herefore yields V j + U j = H = H ξ j s dc j s ξs j + ϑ j s dds j ξ j s dm s L j T Lj ξs j + ϑ j s dcs j ϑ j s dm s R j T R j ξs j + ϑs j dm s L j T + Rj T L j R j. By 5., his shows ha V j + U j, ξ j + ϑ j, L j + R j is a fixed poin of ψh, Cj. By inducion, ψh, A = ψc H, herefore has a fixed poin V H, ξ H, L in B, and since Θ = L M by Lemma, we obain he srong F-S decomposiion of H as in 5.3. q.e.d. As an immediae consequence, we deduce Corollary 6. Suppose ha X saisfies ESC and he EMVT process K is deerminisic and saisfies 5.7 sup { K } τ τ sopping ime <. Then.8 admis a soluion ξ c Θ for every H L and every c IR. Proof. By Lemma and 5.7, X saisfies SC and K T is bounded even deerminisic and saisfies 5.5. By Lemma, Θ = L M and so we can apply Theorem 5 and Theorem 3. q.e.d. We conclude his secion by relaing he srong F-S decomposiion o he minimal signed local maringale measure P for X. To ha end, we recall ha X saisfies SC and define he minimal maringale densiy Ẑ M loc by Ẑ := E λ dm. Then Ẑ saisfies dẑ = Ẑ λ dm, and his implies ha ẐL is in M loc for every L I M. Moreover, one can show by using he produc rule, Yoeurp s lemma and SC ha ẐX is in M loc and ẐGϑ is in M,loc for every ϑ Θ. Now assume ha K T = λ dm is bounded. Then Théorème II. of Lepingle/Mémin T 978 implies ha Ẑ is in M, and his allows us o define a signed measure P P on F wih P [Ω = by seing d P dp := ẐT L P. The preceding argumens show ha ẐGϑ is in M P for every ϑ Θ, hence Ê[G T ϑ = for every ϑ Θ, 8

31 and so P is a signed Θ-maringale measure in he sense of Schweizer 993d. Moreover, he facs ha ẐX M locp and ẐL M P for every L I M jusify calling P he minimal signed local maringale measure for X; see Föllmer/Schweizer 99, Ansel/Sricker 99 and Schweizer 993c. If Ẑ is sricly posiive, we can even replace signed by equivalen hroughou. Lemma 7. Suppose ha X saisfies SC, he MVT process K of X is bounded and H L P admis a srong F-S decomposiion. Then he process ẐV H is in M P, where V H is given by.3. In paricular, we have If Ẑ is sricly posiive, hen we also have H = Ê[H. 5.8 V H = Ê[H F, T. Proof. By definiion, ξ H Θ and L H I M ; hence he preceding argumens yield ẐV H = Ẑ H + Gξ H + L H M P. Since V H T = H P -a.s., Ẑ = and E [ L H =, we deduce Ê[H = E [ Ẑ V H = H. Finally, he las asserion follows from he Bayes rule. q.e.d. 6. Examples In his secion, we illusrae he preceding resuls by means of several examples. 6.. Coninuous processes admiing an equivalen maringale measure Consider firs any coninuous adaped IR d -valued process X. If we assume ha X admis an equivalen local maringale measure, i.e., here exiss a probabiliy measure P P such ha X is a local P, IF -maringale, hen X is in Sloc P and saisfies. and SC; see Ansel/Sricker 99 or Theorem of Schweizer 993c. Moreover, K is coninuous and so 5.5 is rivially saisfied; hus Theorem 5 implies ha every H L P admis a srong F-S decomposiion if K T is bounded. If K is even deerminisic, hen he opimizaion problem.8 admis a soluion ξ c for every pair c, H IR L P. This example generalizes previous resuls of Schweizer 993a, 993c who obained a srong F-S decomposiion under he slighly more resricive assumpion ha K T is bounded and H is in L +ε P for some ε >. On he oher hand, he mehod used here allows o give an explici descripion no only of V H, bu also of he processes ξ H and L H. To see his, we noe ha coninuiy of X and boundedness of K T imply ha he minimal maringale densiy Ẑ is sricly posiive and in M r P for every r <, so P is a probabiliy measure equivalen 9

32 o P, and X is a coninuous local P, IF -maringale. H L +ε P can hen be obained by seing The srong F-S decomposiion of V H := Ê[H F, T as in 5.8 and L H := V H E [ V H ξ H s dx s, T, where ξ H denoes he inegrand wih respec o X in he Galchouk-Kunia-Waanabe decomposiion of H under P. Using he Burkholder-Davis-Gundy inequaliies, one can moreover deduce addiional inegrabiliy properies of ξ H and L H from informaion abou he inegrabiliy of H. For more deails, see Schweizer 993a, 993c. 6.. A mulidimensional jump-diffusion model As a second class of examples, we consider a fairly general jump-diffusion model where X is given as he soluion of he sochasic differenial equaion 6. dx i = X i µ i d + n j= v ij dw j + m k= ϕ ik dn k, T for i =,..., d, wih all X i >. Wihou special menion, all processes will be defined for [, T. In 6., W = W,..., W n is an n-dimensional Brownian moion and N = N,..., N m is an m-variae poin process wih deerminisic inensiy ν = ν,..., ν m ; his is equivalen o saying ha N,..., N m are independen Poisson processes wih inensiies ν,..., ν m, respecively. W and N are hen auomaically independen. We shall ake d n+m so ha in financial erms, here are more sources of uncerainy in he marke han asses available for rade. IF = F T denoes he P -augmenaion of he filraion generaed by W and N, and F = F T. The coefficiens µ = µ,..., µ d, v = v ij i=,...,d;j=,...,n and ϕ = ϕ ik i=,...,d;k=,...,m are assumed o be predicable processes and for simpliciy P -a.s. bounded, uniformly in and ω. We also assume ha ν is bounded uniformly in, 6. ν k >, T, for k =,..., m and 6.3 ϕ ik > P -a.s. for [, T, i =,..., d and k =,..., m. We define he d m marix-valued process ψ by ψ ik he addiional condiion ha := ϕ ik νk for [, T and impose 6.4 he marix Σ := v v + ψ ψ uniformly in and ω, is P -a.s. srongly nondegenerae, 3

33 i.e., here exiss a consan ε > such ha for all [, T, x Σ x ε x P -a.s. for all x IR d. This implies ha Σ is P -a.s. inverible for each wih Σ ε ϱ = ϱ,..., ϱ d defined by and ha he process ϱ := Σ µ + ϕ ν = v v + ψ ψ µ + ϕ ν, T is P -a.s. bounded, uniformly in and ω. Finally, we assume ha 6.5 ϕ ϱ k δ P -a.s. for [, T, k =,..., m and some consan δ >. For fuure reference, we inroduce he noaion x y for he coordinaewise produc of wo vecors x, y IR m : x y k := x k y k for k =,..., m. Remark. Since jump-diffusion models for sock prices have recenly been used by several auhors, we provide here a brief comparison of our assumpions o hose made in oher papers and poin ou he relevan differences. We should like o emphasize, hough, ha all hese papers are concerned wih opimizaion problems differen from.8; he overlap only concerns he basic model used for X. The paper by Jeanblanc-Picqué/Ponier 99 considers he case where d = and n = m = so ha here are only one Brownian moion and one independen Poisson process. The marix Σ is hen given by v + ϕ ν v v + ϕ ϕ ν is deerminan is v ϕ v ϕ ν, v v + ϕ ϕ ν, v + ϕ ν and so 6.4 is by 6. equivalen o he condiion.5 of Jeanblanc-Picqué/Ponier 99 ha v ϕ v ϕ α > P -a.s. for [, T and some consan α. A similar compuaion yields ϕ ϱ = µ v µ v v ϕ v ϕ so ha our condiion 6.5 is by 6. a uniform version of heir condiion.6 which is necessary for absence of arbirage. The crucial difference o our siuaion is ha hey assume d = = m + n. This implies ha no only he driving process W, N as explained in he remark below bu also X iself has he maringale represenaion propery. Hence every random variable H L is he sum of a consan and a sochasic inegral wih respec o X, wihou an addiional erm L H T as in.. In he language of financial mahemaics, his means ha X yields a complee marke; see Harrison/Pliska 983. The imporance of he 3 ν

34 assumpion d = m + n is herefore explained by he well-known fac ha mos opimizaion problems are subsanially easier o solve in a complee han in an incomplee siuaion. Shirakawa 99a considers essenially he same basic model as we do and sudies he problem of finding sufficien condiions for he exisence of an equivalen maringale measure for X. He shows in his Theorem 4. ha absence of arbirage in a firs sense implies he exisence of predicable processes π = π,..., π n and χ = χ,..., χ m such ha χ k > for each k and µ + ϕν = vπ + ϕν χ. π and χ are inerpreed as risk premium processes associaed o W and N, respecively. Theorem 4.4 of Shirakawa 99a hen shows ha absence of arbirage in a sronger second sense even implies he exisence of an equivalen maringale measure for X. Our assumpions 6.4 and 6.5 imply he same conclusions; in fac, we can ake π := v ϱ and χ := ν ϕ ϱ ν, he inerpreaion of π and χ as risk premia is provided by 6.7 and 6.8 below, and an equivalen maringale measure will be exhibied below. Thus we see again ha our assumpions are closely relaed o a no-arbirage condiion on X. However, we have no pursued any furher he issue of explicily consrucing an arbirage opporuniy from a violaion of 6.5; for an approach in ha direcion, see Jeanblanc-Picqué/Ponier The problem addressed in Shirakawa 99b is essenially he same as in Jeanblanc- Picqué/Ponier 99, bu for he case where boh W and N are mulidimensional. He also assumes ha d = n+m and his implies ha his assumpions are pracically he same as ours; 6.4 and 6.5 correspond o his Assumpion.4. The clue o seeing his is he observaion ha for d = m + n, a sligh modificaion of his Lemma.3 shows ha Σ = D D, T, where he marix-valued process D is defined by wih ν := v v v Idd d ϕ F ϕ E D := F ϕ E, T,,,..., ν ν m ν E := Id d d v v v v, T and F := ϕ E ϕ, T. Esablishing he correspondences beween his condiions and ours is hen a maer of sraighforward bu edious compuaions. 4 The same model as in Shirakawa 99b is also sudied in Xue 99. His main conribuion is o provide a rigorous proof of he maringale represenaion resul used wihou proof in Jeanblanc-Picqué/Ponier 99 and Shirakawa 99b; see also Galchouk 976. In conras o our siuaion, Xue 99 also considers he complee case d = m + n. Apar from ha, his condiions are almos idenical o ours; he also assumes 6.4, and 6.5 is alhough wihou he bound being uniform implicily used in his consrucion of he equivalen maringale measure by he appeal o his Theorem I.6.. 3

35 Using 6.3, 6.4 and he boundedness of µ, v, ϕ, ν, one can show by a similar argumen as in Xue 99 ha X belongs o he space S p of semimaringales for every p <. The canonical decomposiion X = X + M + A is given by M i = n j= X i s v ij s dw j s + m k= X i s ϕ ik s dn k s ν k s ds, T and A i = X i s µ i s + ϕ s νs i ds, T for i =,..., d. I is easy o see ha X saisfies. and SC, and if we choose B := for all [, T, he processes λ and K are given by and K = λ i = X i ϱ i, T, for i =,..., d µs + ϕ s νs vs v s + ψ s ψ s µ s + ϕ s νs ds, T. For deails of hese compuaions, we refer o Schweizer 993a. Due o he boundedness of µ, ϕ, ν and he nondegeneracy of Σ, K is coninuous and bounded, and Theorem 5 herefore implies ha every H L admis a srong F-S decomposiion. If we assume in addiion ha 6.6 he process µ + ϕ ν v v + ψ ψ µ + ϕ ν T is deerminisic, hen.8 can be solved for every pair c, H IR L. This generalizes Corollary II.8.5 of Schweizer 993a. Remarks. As equivalen maringale measure for X, we can choose he minimal signed local maringale measure P. Using 6.3, 6.4, 6.5 and he boundedness of µ, v, ϕ, ν, one can in fac show ha Ẑ is sricly posiive and in Mr P for every r < ; hence P P, and X is in M p P for every p <. Moreover, Girsanov s heorem implies ha 6.7 Ŵ := W + v sϱ s ds, T is an n-dimensional Brownian moion wih respec o P and IF, and ha N is an m-variae poin process wih P, IF -inensiy 6.8 ν := ν ϕ ϱ ν, T. For deails, see Schweizer 993a. 33

36 For random variables H L +ε P wih some ε >, he exisence of a srong F-S decomposiion was also esablished in Schweizer 993a by a differen mehod. The argumen here used he fac ha wih respec o is own filraion IF, he process W, N has he maringale represenaion propery: every F L P can be wrien as F = E[F + n j= f j s dw j s + m k= g k s dn k s ν k s ds P -a.s. for predicable processes f = f,..., f n and g = g,..., g m saisfying n m E ds + E ν k s ds <. j= f j s k= Applying his resul o F := HẐT allows o give a fairly explici consrucion of he processes V H, ξ H and L H in erms of f, g and H. The somewha lenghy deails can be found in Schweizer 993a. 3 In conras o he case where X is coninuous, he srong F-S decomposiion can here no be obained as he Galchouk-Kunia-Waanabe decomposiion under P, since he corresponding P -maringale L will ypically no be a P -maringale. Consider now he special case m = so ha 6. is he sandard mulidimensional diffusion model inroduced by Bensoussan 984 and generalized by Karazas/Lehoczky/Shreve/ Xu 99. Condiions 6., 6.3 and 6.5 hen disappear, and 6.4 can be relaxed o he assumpion ha 6.9 he marix v v is P -a.s. inverible for every [, T, if we impose in addiion he condiion g k s 6. v sϱ s ds C < P -a.s. for some consan C; his guaranees ha K T is bounded. Condiion 6.9 follows immediaely from he sandard assumpion in Karazas/Lehoczky/Shreve/Xu 99 ha he marix v has full rank d n P -a.s. for every [, T. Condiion 6. is also quie usual; i is for insance saisfied if v ϱ is P -a.s. bounded, uniformly in and ω. Finally, 6.6 reduces o he assumpion ha µ v v µ is deerminisic. T In paricular, if we choose d = one asse available for rade, m = no Poisson componen, n = wo driving Wiener processes and v = v r, v = v r, µ = m wih r, hen 6.4 is equivalen o assuming ha v is bounded away from, uniformly in and ω, and 6.6 ranslaes ino he assumpion ha m is deerminisic. v T 34