On convergence to the exponential utility problem

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1 Sochasic Processes and heir Applicaions 7 27) On convergence o he exponenial uiliy problem Michael Kohlmann, Chrisina R. Niehammer Deparmen of Mahemaics and Saisics, Universiy of Konsanz, D78464 Konsanz, Germany Received 28 November 25; received in revised form 3 Ocober 26; acceped 9 March 27 Available online 25 March 27 Absrac We provide a mehod for solving dynamic expeced uiliy maximizaion problems wih possibly no everywhere increasing uiliy funcions in an L p -semimaringale seing. In paricular, we solve he problem for uiliy funcions of ype e x exponenial problem) and x ) -h problem). he convergence of he -h problems o he exponenial one is proved. Using his resul an explici porfolio for he exponenial problem is derived. c 27 Elsevier B.V. All righs reserved. MSC: 9B2; 6H; 9B6; 6G48 Keywords: Convex analysis; Sochasic dualiy; Exponenial uiliy funcion; Minimal enropy maringale measure; Convergence of q-opimal maringale measures; Wealh and porfolios. Inroducion Besides he conrol-heoreical ineres here is an economic moivaion for he use of exponenial uiliy funcions. Opimizing he invesmen decisions for a cerain ime horizon of an invesor wih iniial wealh x can be described by maximizing he expeced exponenial uiliy of a erminal value Y of a wealh process Y = x + Ỹ : V exp,ξ x) = max E e x+ỹ ξ) ), ) where ξ represens a financial obligaion he invesor faces in. In a semimaringale model he problem of finding an opimal erminal value of he exponenial problem was compleely Corresponding auhor. el.: ; fax: addresses: michael.kohlmann@uni-konsanz.de M. Kohlmann), chrisina.niehammer@mah.uni-giessen.de C.R. Niehammer). URL: hp:// kohlmann M. Kohlmann) /$ - see fron maer c 27 Elsevier B.V. All righs reserved. doi:.6/j.spa

2 84 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) solved, including coningen claims ξ, in [5] and [5], for differen classes of wealh processes. Varians of he concep appeared before; see Remark 2. in [5] for furher references. Moreover, a backward sochasic differenial equaion BSDE) approach is found in [3] and [3]. he second aricle avoids dual relaions and applies maringale properies of he value funcion, insead. More generally, Schachermayer [32] compleely solved he uiliy maximizaion for a wider class of uiliy funcions. However, hese approaches do no cover no everywhere increasing uiliy funcions. Furhermore, he explici form of he opimal porfolio has no been derived, excep in very special cases for he exponenial uiliy problem; see e.g. [3] and [5]. On he oher hand, for isoelasic uiliy funcions wih parameer α > explici porfolios are known. We herefore presen a complee relaion beween various ypes of maringale measures dual problem), he isoelasic, and he exponenial problem heorem 7). his new approach conains convergence of he erminal values leading o an explici porfolio of he exponenial problem. We furher propose a mehod for solving dynamic uiliy maximizaion problems for possibly no everywhere increasing uiliy funcions. As we consider p-inegrable sraegies see Definiion 2.), erminal values of allowed wealh processes are elemens of L p. We reformulae he dynamic opimizaion problem over wealh processes as a consrain saic problem over L p -random variables. his is implicily done for increasing uiliy funcions in [6]. We presen an exension of his resul also applicable o no everywhere increasing, concave uiliy funcions. For he same class of funcions, we furher sugges a mehod for solving he consrain problem using resuls from convex analysis. In paricular, we obain he opimal soluion for uiliy funcions of he form x ). he opimal erminal value urns ou o be a funcion of he -opimal maringale measure, which is he soluion of he corresponding dual problem from convex analysis. I is known ha he q-opimal maringale measure converges o he minimal enropy measure up o a scaling consan he opimal soluion of he dual exponenial problem. We use hese resuls o show ha he erminal values and he value funcions of he uiliy problem corresponding o he sequence x ) ) m converge o he erminal value of he exponenial uiliy funcion. his convergence hen yields he convergence of he porfolios and provides an explici porfolio for he exponenial uiliy problem in he same seing wih a deerminisic erminal rade-off. Exensions are possible, bu raher echnical and go beyond he scope of his paper. Furher noe ha proofs are given in he case of a rivial claim ξ. Forunaely, resuls remain valid in he non-rivial case leading o an ineresing resul; see Remark. he paper is organized as follows. In Secion 2, we explain he marke model wih L p - sraegies and formulae he main problem in a dynamic and a saic version. Using echniques from convex analysis, Secion 3 describes a mehod for solving he dual problem, a consrain saic problem. We herefore cie some resuls on differen possible dual soluions he minimal enropy maringale measure, he minimal maringale measure, and q-opimal maringale measures for q > in Secion 3.2. Using hese resuls, we derive he main resul of his paper in Secion 4: he convergence of he erminal values and he value funcions of he -h problems o he corresponding values and funcions in he exponenial problem. As an applicaion, Secion 5 gives he corresponding convergence resul for he porfolios and presens he opimal porfolio in he exponenial case. 2. he marke model and problem formulaion We work wih a semimaringale model: Le Ω, F, P) be a probabiliy space,, ) a ime horizon, and F = F ) [, ] a filraion saisfying he usual condiions, i.e. righ-coninuiy

3 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) and compleeness. his enables us o use righ-coninuous wih lef limis RCLL) versions for all P, F)-semimaringales represening our socks. As only special semimaringales are considered, so ha a Doob Meyer decomposiion holds, we simply call hem semimaringales. All expecaions and spaces wihou a subscrip are defined wih respec o he measure P. K denoes a generic posiive consan. hroughou his paper a coninuous R n+ -valued P, F)- semimaringale S, ) is given, where S = S ) [, ] wih unique decomposiion S = S +M+A ino a local maringale M and a predicable process of bounded variaion A. S represens a vecor of n risky asses and sands for a riskless asse wih consan discouned price, i.e. he riskless asse serves as a numéraire. A self-financing sraegy x, N) is given by he iniial wealh x and he number of shares N = N,..., N n ) of he socks held a ime [, ]. We require ha our sraegies are predicable and saisfy an inegrabiliy condiion: Definiion 2.. he se of L p -rading or p-inegrable sraegies is defined as follows: where A p := L p M) L p A) L p M) = {N P n N L p M) < }, L p A) = {N P n N L p A) < } wih N L p M) := Nd M N ) 2 L p, N L p A) := N da L p and P n he se of all predicable R n -valued processes. See [4] or [29] for undefined noaion and he sandard heorems concerning he heory of inegraion wih respec o semimaringales. Self-financing sraegies in A p hen define a wealh process x + NdS for [, ]. he inegrabiliy assumpion implies ha he se of erminal values of allowable wealh processes is a subse of L p P): G p x) := {Y Y Wx)} L p P) 2) where Wx) := {Y Y = x + NdS, N Ap } H p P) is he class of all wealh processes generaed by he class of L p -rading sraegies, i.e. Y can be hedged by he iniial wealh x R and a rading sraegy N. For a definiion of H p see [29]; briefly i is he space of all canonical decomposiion S = M + A + S such ha S H p := [M] /2 + da s L p + S L p is finie.) he chosen class excludes doubling sraegies, by he Burkholder Davis Gundy inequaliy. o exclude arbirage opporuniies noe: here he no arbirage noion is he noion of no free lunch wih vanishing risk) we assume ha he space of all equivalen maringale measures wih L q - inegrable densiies is nonempy, i.e. M q e, where M q e = {Q dq = Z dp, Z D q e } L q P), p + q = wih D q e = {Z U q EZ ) =, Z >, SZ M loc } and U q is he class of uniformly inegrable maringales M wih E q M q ) <. We add a subscrip Z when densiies are mean, e.g. M q e,z. Spaces wih subscrip a insead of e only require Z, whereas spaces like M S denoe he class of signed local maringale measure, i.e. Z does no have o be nonnegaive. If M q e is a singleon, we call he marke complee; oherwise i is incomplee. When he noaion is clear from he conex, we wrie Z insead of Z and add a superscrip o Z when denoing a densiy process. Before concluding his secion, we come back o he se of allowable rading sraegies. Delbaen and Schachermayer [6] consider simple p-admissible sraegies and define he

4 86 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) corresponding inegral. Since S is assumed o be coninuous and M q e, he closure of he space of hese inegrals K p x) is equal o he closure of G p x); see Lemma 2. in [2]. For K p x), Delbaen and Schachermayer [6] show a hedging resul for L p claims, which hen also holds for G p x), if i is already closed. he closedness is rue under Assumpion 2. he reverse Hölder inequaliy; see [], heorem 4..) So using his assumpion, we have he hedging resul menioned above: Every f L p saisfying E Q f ) = x for every Q M q a is in G p x), i.e. f can be replicaed wih iniial wealh x. We sar by inroducing he reverse Hölder inequaliy R q Q): Definiion 2.2. A process Z saisfies he reverse Hölder inequaliy, R q Q), if here exiss a K q) > such ha Z q ) sup E Q τ Z F τ < K q) 3) τ where is he class of sopping imes τ. M q e for some q > is hen a consequence of he following sronger assumpion used in [3]: Assumpion 2.. A) All F, P)-local maringales are coninuous. B) here exiss an equivalen maringale measure Q such ha is densiy process saisfies he reverse Hölder inequaliy R q P) for some fixed q >. dq Under his assumpion he unique soluion of min q Q M E S dp )q ) exiss in M q e. I is called he q-opimal maringale measure Q q. Moreover, he densiy process of Q q, denoed by Z q), saisfies he reverse Hölder inequaliy R q P) for some fixed q >, if Assumpion 2.B) holds and S is coninuous see heorem 4. []). o include non-increasing uiliy funcions, we exend he class of wealh processes o { } W C x) = Y Y = x + NdS C, N A p, C K p where K p he class of increasing righ-coninuous processes wih dc C L p. Noe: W C is a subse of he se of p-inegrable wealh processes. We consider he following dynamic opimizaion problem: V x) ξ,c sup E[UY ξ)], x R 4) Y W C x) where U is a concave, no necessarily increasing funcion, ξ an F -measurable, L p -inegrable random variable, and E[UX ξ)] <. From a proof analogous o ha of heorem 2.. in [7] J = ess sup Q M q e E Q X F ) is a righ-coninuous Q-supermaringale for every Q M q e. By he opional decomposiion heorem in [8] and some very echnical esimaions, J is in W C sup Q M q e E Q X)). Hence, if E[UX ξ)] < problem 4) is equivalen o he following saic problem: V x) ξ,c sup E[UX ξ)], x R. 5) X L p F ), Q M q e E Q X) x

5 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) As menioned above, a proof for increasing uiliy funcions can be found in [6]. In he sequel, we ackle he saic problem using mehods from convex analysis. We explicily solve he problem for he exponenial uiliy funcion U exp x) = e αx, α > and is no everywhere increasing approximaing sequence U x) = αx ). We show ha heir soluions converge. Noe ha for simpliciy we se α = ; a generalizaion is sraighforward. Remark. i) We use he following noaion hroughou he paper: For m N, we le p = and hence from q + p = we have q =, m. So when he noaion seems more convenien and unambiguous, we wrie Z q) = Z ) for processes used in he q = siuaion for he -problem and Z q = Z := Z ) for is erminal values. ii) he coninuiy assumpion of S is no necessarily needed o solve he -h problem. However, he q-opimal maringale measure is only proved o be a signed local maringale measure, i.e. in M q S ; see e.g. [2]. Furher, he reformulaion of he dynamic o he saic h problem becomes a bi more complicaed. Since we need coninuiy for our convergence resul, we sick o his coninuiy of S hroughou he paper. Remark 2. We consider he exponenial uiliy problem as a limi of he -problems. So he seing of he exponenial problem may be derived from he specific convergence properies derived below. On he oher hand, we could also define he seing of he exponenial conrol problem and hen derive he desired properies from our convergence resuls. A sufficien se of assumpions should imply ha he gains process is bounded from below and is maringale par should be a BMO maringale. he main addiional assumpion in a Brownian seing would be bounding he full rank par of σ away from and. For deails, see e.g. [3], and for he properies of BMO maringales, [28] and [25]. However, we would lose some of he generaliy of he approach here. In Secion 5 we herefore derive a porfolio in he case of a deerminisic erminal value of he rade-off process see he definiion below). his sraegy is conained in all menioned spaces, bu wha is more i is an elemen of he following quie canonical space exending he class of L p -rading sraegies: { A exp = N } A p : Ee α NdS <. p> 3. Solving saic uiliy opimizaion problems 3.. General approach Using heorem 2 in [22] p. 22), we obain: Corollary 3. Suppose here exiss a y R +, a Z op M q a,z, and an X O p := {X L p P) : EUX) < } such ha he Lagrangian LX, y Z) := EUX)) yez X) x) possesses a saddle poin a X, y Z op ), i.e. LX, y Z) LX, y Z op ) LX, y Z op ), for all X O p, y R +, Z Mq a,z or y Z = λ D := R + Mq a,z. hen X solves max EUX)), s.. Q M q a : E Q X) x, X O p. 6)

6 88 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) For a proof, le P := {X L p P) : Z M q a,z, y, X, F y,z ) := Ey Z X) } where, ) denoes he obvious dual pairing, and so D := P = {λ : λ = y Z, y R +, Z M q a,z }, GX) = X x, and hx) = EUX)) in heorem 2 [22]. Before proving exisence in he exponenial and he -cases heorem 6), we give a mehod for searching for a saddle poin of he Lagrangian L in he absrac seing given above. he proof is hen given by applying his mehod and proving he necessary inegrabiliy condiions. We sar by reaing he second inequaliy of Corollary 3: X λ ) = arg max X LX, λ ) for an arbirary λ. From he Lagrangian, he convex dual Ǔy) := sup x D [Ux) xy] canonically arises. D denoes he domain of U. If U is sricly concave and coninuously differeniable no necessarily increasing hen Ǔy) = UI y)) I y)y, where I := U ). he minimizer I y) is unique. And so for a fixed λ = y Z and all X L p, LX, λ ) EǓλ )) + xy, and equaliy holds if and only if X λ ) = I λ ) = I Z y ). he problem of finding a λ ha also saisfies he firs inequaliy, i.e. λ D : LX λ ), λ ) LX λ ), λ ), is equivalen o he following dual problem: min φy, Z ) y,z where φy, Z ) = EUI Z y )) Z y I Z y ))+xy. In he sequel, λ = y Z op denoes he opimal soluion of 7). So λ, X λ )) is a saddle poin, provided ha X λ ) O p. Hence, X λ ) = I y Z op ) O p is he opimal soluion of he primal problem. Suppose he dual soluion exiss. o explicily solve he dual problem, we perform a second minimizaion: Zy ) = arg min Z M q a φy, Z). Puing his ino he dual problem, he dual soluion is eiher, Z)) or y, Zy )), where y is he soluion of X Zy )y ) = EZy )I Zy )y )) = x. 9) Denoe he unique soluion of 9) by Yx). I urns ou ha for large enough m dependen on x) he opimizaion problem for x ) and he exponenial uiliy funcion is independen of he iniial wealh. Yx) exiss and is posiive. So he soluion of 9) in he case of he h problem Y ) and he exponenial problem Y exp ) can easily be derived by invering X and X exp respecively. his leads o he soluions of he dual problems: he Y x) imes -opimal maringale measure and Y expx) imes he minimal enropy maringale measure, respecively q-opimal maringale measures and he minimal enropy maringale measure he erm relaive enropy is used in informaion heory. One looks for a maringale measure ha in an inuiive sense carries mos informaion abou P: Q min = arg min HQ P), Q P f P) where P f P) := {Q Ma : HQ P) < } wih HQ P) = E P dq dq dp log dp )) if Q P and oherwise. If P f P), he unique exisence follows from heorem 2. in []. If in 7) 8)

7 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) addiion P f,e P) := M e P f P), hen Q min M e, i.e. Q min is equivalen o P heorem 2.2). Q min is known as he minimal enropy maringale measure. By Assumpion 2., S is coninuous and herefore saisfies he srucure condiion and admis he decomposiion S = S + M + d M ˆλ, where M a coninuous local maringale and ˆλ a predicable R n -valued process, as defined in [33]. he process ˆK = ˆλ dm = ˆλ d M ˆλ is called he mean variance rade-off process. If he Doléans Dade exponenial Ẑ = E λdm) is a maringale, he minimal maringale measure is defined as d ˆQ = Ẑ dp. For a definiion offering more inerpreaion in he original case, we refer he reader o Föllmer and Schweizer [9]. he minimal enropy maringale measure can be described by a backward sochasic differenial equaion BSDE hereafer). From heorem in [33], we know ha every equivalen maringale measure can be represened as dq dp = Z Q, Z Q = E M Q ), M Q M loc. Furher, using he noaion E M Q ) = E M Q ), Mania e al. [24] prove he following characerizaion of E M Q ) he minimal enropy maringale measure heorem 3.): heorem 4. Le all F, P) local maringales be coninuous and P f,e P). hen he value process V, given by V = ess inf E Qlog E M Q ) F ), Q P f,e P) is a special semimaringale wih V = m + A + V, where m M 2 loc, M2 loc) denoes he space of all local) maringales M wih sup M 2 L < ) and A a locally bounded variaion predicable process. herefore he Galchouk Kunia Waanabe GKW) decomposiion exiss: m = φ s dm s + m, m, M =. Furhermore V is he soluion of he following BSDE: ) Y = Y ess inf Q P f,e P) 2 M Q + M Q, L + L, Y =. ) Moreover, Q min is he minimal enropy maringale measure if and only if dq min dp = E M Q min ), M Q min = ˆλ s dm s m. ) Suppose, in addiion, he minimal maringale measure exiss, i.e. Ẑ is a maringale, and saisfies he log reverse Hölder inequaliy; for a definiion see e.g. [2]. hen, V uniquely solves he above BSDE ) and is bounded. A similar characerizaion is proven for he q-opimal maringale measure in [23]: heorem 5. If Mq e and all P-local maringales are coninuous, hen he following asserions are equivalen: ) he maringale measure Q q is q-opimal; 2) Q q is a maringale measure saisfying where dq q = E M Q q )dp, M Q q = ˆλ s dm s q 2) V s q) d m sq). 3)

8 82 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) V q) = V q) + mq) + Aq) is equal o ess inf q Q Me EE M Q )) q F ); i uniquely solves he following BSDE: [ ] Y = Y ess inf qq ) Y s d M Q s + q M Q, L + L, <, M Q Mq e 2 Y =. mq) denoes he orhogonal par of he GKW decomposiion of mq): m q) = φ s q)dm s + m q). 4) If E ˆλ s dm s) is a maringale, i.e. he minimal maringale measure exiss and in addiion i saisfies he reverse Hölder condiion, hen he value process V q) above is he unique soluion of he above BSDE and here exis posiive consans k and K such ha almos surely for all [, ]: k V q) K. A simple consequence of wo corollaries of heorems 4 and 5, Corollary 3.4 in [24] and Corollary 3 in [23] also see [3]), is ha if ˆK is deerminisic, he minimal enropy maringale measure, he minimal maringale measure, and he q-opimal maringale measures q > coincide almos surely. Under he weaker Assumpion 2., Sanacroce [3] esablishes ha ) mq) E q m, q. 5) Furhermore, E sup Z q) Z min, q, 6) L and, in paricular, Z q) Z min q), q, where Z ) and Z min) ) are densiy processes of he q-opimal maringale measures and he minimal enropy maringale measure, respecively. he las asserion, using a dualiy approach, is also proven in [2]. Assumpions are more or less he same; he obained convergence is weaker. Nex, we see ha he dual soluion of he opimizaion problem wih uiliy funcion x ) is he -opimal maringale measure imes Y x) and he dual of he exponenial problem is he minimal enropy maringale measure imes Y exp x). So he above consideraions already show ha he dual measures converge Exponenial uiliy funcion and is approximaing sequence Using he above approach, we solve he saic problem given in 6) wih UX) = e αx and an arbirary p >. Alhough our saic problem is quie general, we can show ha he opimal value X exp) coincides wih he usual opimal erminal value of he dynamic exponenial problem characerized e.g. in [5,5]. While here are quie resriced classes of sraegies in hese papers, our approach leaves considerable space o define a wide class of porfolios, e.g. A exp. In Secion 5, he opimal X exp) will urn ou o be he limi of he opimal soluions of he h problem. Using his, we give he problem a dynamic componen by developing, under some

9 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) weak assumpions, a sraegy ha reaches X exp). In he sequel and if i is clear from he conex, we denoe by X he opimal soluion of he considered opimizaion problem, e.g. X = X exp). Wihou loss of generaliy we se α =. By 8) we obain X Z y ) = I Z y ) = logz y ) 7) where Z y ) is he minimizer of φy, Z ) = EUI Z y )) Z y I Z y )) + xy = y + xy + y log y + y HQ P) wih Z = dq dp. We have y, so as above, we sar by deriving Zy ): Zy ) = arg min φy, Z). Z Clearly, Zy ) is equal o he densiy relaed o he minimal enropy maringale measure Q min = arg min Q HQ P) and independen of y, and herefore also independen of he iniial wealh x. o deermine y, we apply he resul in Eq. 9), i.e. X Zy )y ) = x: X Zy )y ) = X Zmin y ) = EZ min I Z min y )) = EZ min logz min y ))) = EZ min log Z min ) log y = HQ min P) log y. We calculae he inverse of X and finally obain he soluion: Yx) = exp{ HQ min P) x} X x) = X Yx)) = I Z min Yx)) = log Z min + HQ min P) + x. 8) By plugging he opimal soluion ino sup X O p E e X ), we obain a dualiy under an arbirary probabiliy measure P wih P f,e P) : sup E exp X)) = e X O p x min Q P f,e P) HQ P). 9) Noe ha we sill have o prove ha X O p for all p >. Under he assumpions of heorem 4, we have EM Q min) 2p ) < ; hence E log p Z min = E M Q min ) p 2 [M Q min ] 2 p E M Q min ) p + 2 ) p ) [M Q min ] p K p) EM Q min ) 2p ) < 2) by he inequaliies of Burkholder, Davis and Gundy and of Doob, and K p) a posiive consan exp) X dependen on p. Furher, Ee ) = e HQmin P) x <. We urn o he soluion of he -h problem w.r.. he uiliy funcion u x) = x ) ): he sricly monoonic derivaive of u is x ). So we have I y) := u ) y) = y ),

10 822 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) and he dual problem, described in 7), convers o min )y EZ ) yez), y R +,Z M a,z which has he same soluion as min Z Ma,Z EZ ), he -opimal maringale measure. Recall ha we denoe is densiy process by Z ) and he relaed densiy by Z ) =: Z. I is independen of y. So X Z,y) = EZ I y Z )) = y Consequenly, Y Z),x) = x) EZ ). )) ) E Z 2) and x) = Z x E X ) Z ). 22) Finally, we have o check wheher X ) is in L. his is clear from X ) x) K x m) Z < as Z L q, where q = and K xm) is a consan depending on m and x. Summarizing, we hus have heorem 6. If M q e, he Lagrangian in Corollary 3 wih Ux) = x ) possesses a saddle poin a X ) x), Y x)z ). he corresponding -h saic problem 6) has a soluion see 22)). If Ux) = exp x), under he assumpions of heorem 4, a saddle poin exiss and is given by X exp x), Y exp x)z min ), where X exp x) see 8)) is he soluion of he saic exponenial problem. 4. Convergence of he erminal values and he value funcions his secion is devoed o he convergence of he erminal values and he value funcions of he -h problem o he exponenial one. Afer some esimaions, he fac ha I y) = y ) converges o I exp y) = log y, and he convergence of he -opimal measures o he minimal enropy maringale measure yield Z I y m Z ) P Z min I y Z min ) for an arbirary real sequence y m ) m wih limi y. Afer esablishing his, we show ha Y Z )x) converges o Y Zmin,expx) or equivalenly heir corresponding inverse funcional X, for large enough m o ensure ha Y x) is sricly posiive). ogeher, all his yields: X ) x) = I Z Y x)) P/a.s. I exp Z min Y exp x)) = X exp x).

11 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) Noe ha he kind of convergence depends on he given assumpions and is specified laer. Convergence in probabiliy can be srenghened by esablishing uniform inegrabiliy of I Z Y x))) m. Esablishing all hese seps yields our main heorem: heorem 7. In our model wih an F -adaped coninuous semimaringale S = S + M + A, le one of he following assumpions be saisfied: ) Assumpion 2.. 2) he erminal value of he mean variance rade-off process ˆK = ˆλ dm ) is deerminisic. hen, he soluion of -h problem converges in L o he soluion of he exponenial problem, i.e., x) = Z x E X ) Z ) L X exp x) = log Z min + HQ min P) + x. 23) Moreover, he values of he dual problems converge: lim φ y m, Z ) = φ exp y, Z min ), m and so also do he value funcions of he primal problem: lim Eu X ) m x))) = lim V x) = V exp x) = EU exp X exp m x))). If he second assumpion holds rue, e.g. in a Brownian seing wih deerminisic coefficiens, he dual problems of he -h and he exponenial problems have he same soluion up o he consan Y i x), he densiy of he minimal enropy maringale measure imes Y i x) for i =, exp. he erminal values in 23) converge P almos surely and in L p for all p. Noe ha boh assumpions imply M q e. Furher, under assumpion 2) he erminal value of he rade-off process is bounded and so assumpion ) holds. o prove heorem 7, we need o esablish he following hree seps under assumpion ) L -convergence) or 2) a.s.- convergence): ) a) Z Z y m ) )) m is uniformly inegrable. b) Z y m ) ) m is uniformly inegrable. L 2) Z /a.s. Z min, m, Z := Z ) ). 3) For every posiive, real sequence y m ) m wih limi y, y m Z I Z y m ) = y m Z Z y m ) ) L /a.s. y Z min logz min y) = y Z min I exp Z min y). Uniform inegrabiliy using assumpion )) or almos sure convergence in iem 3 assumpion 2)), he fac ha Z min and Z are sricly posiive for all m, and iem 2 yield I Z y m ) L /a.s. I exp Z min y) for any posiive real sequence y m ) wih limi y. Furher, i is well known ha for

12 824 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) a sequence ξ n ) wih Eξ n < converging in probabiliy o ξ, we have ha Eξ n Eξ if and only if ξ n ) n is uniformly inegrable. So o prove ha X Z )y) converges o X exp y), we need ha Z I y Z )) m is uniformly inegrable and bounded from above. Noe ha x x ) is bounded by 2 from above; see 24) below. Since X ) m converges, Y ) m does oo, and so X ) x) = I Z Y x)) a.s./l I exp Z min Y exp x)) = X exp x). By iem 3, we have convergence of he dual funcions: φ Y x), Z ) = E Y x)z Z Y x)) )) Y x) E Y exp x)z min logz min Y x))) Y exp x) = φ exp Y exp x), Z min ). By dualiy on he -h levels and in he exponenial case, we have convergence of he primal value funcionals: lim V x) = lim φ Y x), Z ) = φ exp Y exp x), Z min ) = V exp x). m m Finally, in he deerminisic rade-off case X ) x) converges o X exp) x) almos surely and in L. As for all m and for all q > Z = Ẑ L q since ˆK is deerminisic), X ) x) L p for all m and for all p. Hence, we find ha X ) x) converges in all L p, p his obviously also holds for p ). We sar o prove iem )a): Proof. We consider he funcion x x ). For every ɛ >, here exiss an m, choose m = 2ɛ + 2, such ha for all m m, x x ) 2 x x u du x,)) + 2 x x u du x ) 2 x log x) x,)) + x2 )x )) x ) ɛ x ɛ x. 24) Noe ha ɛ = and x log x) x,)).4. In he case of assumpion 2), his implies uniform inegrabiliy, since every consan sequence of a non-negaive inegrable random variable in his case Z +ɛ min ) m = Zmmm +ɛ ) m) is uniformly inegrable. Under assumpion ), by 24) i is sufficien o show ha Z +ɛ is uniformly inegrable. his is esablished by he de la Vallée Poussin heorem VP). As in [3] proof of heorem ) also using a resul in [6], we have for a posiive consan K and some µ > ha sup E Z q) + µ ) < K. 25) <q q Nex, we apply he VP o he funcion G) = +ɛ 2, where ɛ 2 > is sill arbirary, which obviously fulfills he assumpions in he VP. We would like o prove ha Z +µ q) ) q q is uniformly inegrable for a µ >. So we have o show ha sup q q EG Z q +µ )) = sup q q E Z q +µ ) +ɛ 2 ) <.

13 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) So choose ɛ 2 > and µ > such ha µ = µ+ɛ 2 +µɛ 2 = +µ)+ɛ 2 ) and he asserion follows from 25) and he VP. Proof. Iem )b)): Since y m Z > and by he second las inequaliy of 24), we have Z y m ) ) 2 logy m Z ) ym Z,) + 2y ɛ m ɛ Z ɛ 2 logy m Z ) ym Z,) + 2y ɛ m ɛ Z Z ) + Z,))) 2 logy m Z ) ym Z,) + 2y ɛ m ɛ Z + ). 26) We know ha Z is uniformly inegrable and so also 2y ɛ m ɛ + Z ) y m converges o a real number). I remains o show ha 2) logy m Z ) ym Z,) is uniformly inegrable. We show ha ) H log Z log Z min) 27) wihou using ha Z L Z min. his yields convergence in probabiliy of 2) logy m Z ) ym Z,) o 2) logy Z min ) y Zmin,) and L -inegrabiliy. Furher, we know ha 2 logy m Z ) ym Z,) is non-negaive for all m. I remains o show ha E 2 logy m Z ) ym Z,)) E 2 logy Z min ) y Zmin,)) 28) o conclude ha he sequence 2 logy m Z ) ym Z,) is uniformly inegrable and herefore also Z y m ) ). o prove 28) i suffices o show ha Elog Z ) converges o Elog Z min ), since his is saisfied if and only if Elogy m Z )) converges o Elogy Z min )) for every real posiive sequence y m ) m converging o y. Furher, log x n ) = log x n ) xn,) converges if and only if logx n ) xn [, )) = log x n ) + and logx n ) converge. We already have convergence in probabiliy and for large enough m: logy m Z ) ym Z [, ) y + K )Z, where y + K )Z is uniformly inegrable. Hence he expecaion of he posiive par converges. Elog Z ) Elog Z min ) follows from 27). o show 27), we have by ) and 3) ha for q = : log Z min = M Q min 2 M Q min = log Z = ˆλ dm s q ˆλ dm s m 2 M Q min, V s q) d m sq) 2 M Q q and M Q min = ˆK 2 ˆλ dm s, m s + m = ˆK + m s since m and M are orhogonal. Similarly M Q q = ˆK + q V s q) d m sq). Finally, le Z ) = EZ F )) and Z min = EZ min F )) ; log Z ) log Z min H = M Q q 2 M Q q M Q min + 2 M Q min H = q V s q) d m sq) + m + 2 M Q min M Q q ) H q V s q) d m sq) + m + H 2 2 M Q min M Q q ). H

14 826 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) he firs erm is equal o E 2 q V s q) d m sq) m ) and converges o zero for q by Corollary 3 in [3]. he same corollary can be applied for he convergence of he second erm. Alernaively, is convergence follows from he uniform boundedness of he maringale norms proved in Corollary 2 in [3]. Noe ha by heorem 4.5 and Proposiion 4.7 in [2] he log reverse Hölder inequaliy LRH) for a definiion see e.g. he paper menioned) for Z min is equivalen o Assumpion 2. B. So under he las assumpion, by heir Lemma 4.6, Condiion S) is saisfied and finally by heir Lemma 2.2 M Q min B M OP). he inequaliies of Kunia and Waanabe and of Hölder yield 2 M Q min M Q q ) = d M Q min M Q q ) H L = d M Q min + M Q q, M Q min M Q q ) L L M Q min + M Q q M Q min M Q q E 2 M Q min + M Q q E 2 M Q min M Q q 2E 2 M Q min M Q q + ) 8E 2 M Q min ) E 2 q V s q) d m sq) m ) K E 2 q V s q) d m sq) m )) + E q V s q) d m sq) m 29) since M Q min B M OP). Noe ha we imiae here a sor of Fefferman s inequaliy see [25]) P which would give he same resul. I follows ha Z Zmin and since Z ) m is uniformly L inegrable, we have Z Z min. Using ha sup Z ) is uniformly inegrable, Doob s inequaliy yields convergence in H of Z q) o Z min. Proof. Iem 3)): For x, y >, we have x = arg max z [x,y] zi z)) ) = arg max z [x,y] zi z) + I z) ) since zi z) + I z)) <, for z >. By an applicaion of he mean value heorem, we have for x < y and m m = 2ɛ + 2, x I x) yi y) = x x ) y y ) x I x) + I x) x y = x x + x ) x y 2 max{, x } + 2ɛ x ɛ +.8 ) x y. 3) x

15 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) See 24) for he second las inequaliy. By 3), we obain Z y m Z y m ) ) Z min y logz min y)) y m Z Z y m ) ) y Z min Z min y) ) + y Z min Z min y) ) y Z min logz min y)) 2 max{, y m Z ), y Z min ) } ) + ɛ max{y Z min, y m Z }) ɛ + Z y m Z min y min{z min y, Z y m } + y Z min Z min y) P ) y Z min logz min y)) 3) for any posiive, real-valued sequence y m ) m wih limi y, e.g. Y x)) for fixed x. Since Z y m Z y m ) ) is uniformly inegrable, convergence in L follows. his complees he proof. Remark 8. Noe ha from 25) he convergence of Z also holds in an L +ɛ -space for an ɛ > and Z min L +ɛ. his follows direcly from uniform inegrabiliy and he convergence in probabiliy. 5. Convergence o he opimal porfolio for an exponenial uiliy funcion We urn o he quesion of convergence of he corresponding porfolios, namely wheher he opimal porfolios N ) of he -problems converge o he opimal porfolio N exp) of he exponenial problem. Here we will resric our consideraions o he case where assumpion 2) of heorem 7 holds ˆK = ˆλ dm is deerminisic); for some ideas on a more general seing see Remark. he basic idea used o derive convergence of he opimal conrols/porfolios consiss in considering X ), X exp) as he erminal values of a BSDE describing he price of he erminal values. he wo componens of he soluions of hese BSDEs are derived and he second pars of he soluions corresponding o he opimal porfolios are shown o converge. Finally, we consider he case of a Brownian marke wih deerminisic coefficiens µ), σ ), and r =. We sar by searching for a porfolio q ) ha reaches X ) by solving he corresponding pricing equaion and show ha hese porfolios converge o a price process wih erminal value X exp). he pricing equaion for he claim X ) is of he following form: dp ) = q ) p ) ) = X ) x) ) d M ˆλ + q ) ) dm + dl ), 32) where L ) is he orhogonal erm appearing in he Föllmer Schweizer decomposiion. q ) represens he porfolio. Since X ) is aainable, L ) vanishes and we have ha 32) is uniquely solvable. he above BSDE is linear so we can look for a porfolio by considering p ) := Ẑ EẐ X ) x) F ) 33)

16 828 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) as a possible candidae. Iô s formula and a coefficien comparison hen yield ha his process p ) is in fac equal o he opimal price process Y ) ha reaches X ). Before saring hese calculaions and inroducing an example, again remember ha in he presen siuaion he minimal maringale measure or is densiy) coincides wih he q-opimal maringale measure for all q. Wih Z = Ẑ for all m, we find from 22) X ) x) = Ẑ x ) E Ẑ )). By Novikov s condiion Ẑ q = E q ˆλ dm) is a maringale for every q R and herefore EẐ q ) =. I follows ha Ẑ L q P) for every arbirary q, because { Ẑ q = exp { = exp q ˆλ s dm s 2q q ˆλ s dm s 2 and since ˆλ dm is deerminisic, EẐ q ) = E { exp { q 2 q ) = exp 2q q ˆλ s dm s 2 q 2 ˆλ s d M s ˆλ s } q 2 ˆλ s d M s ˆλ s + q 2q q 2 ˆλ s d M s ˆλ s + q 2q q 2 ˆλ s d M s ˆλ s } 34) q 2 ˆλ s d M s ˆλ s }) ˆλ s d M s ˆλ s } <. 35) By plugging 34) and 35) ino 33) and applying Iô s formula we find ha p ), q ) uniquely solves 32), where ) and p ) := Ẑ EẐ X ) x) F ) = exp =: exp z ) q ) x) = ) ˆλ z ) ˆλ s dm ) s ˆλ 2 ) 2 s d M sλ s ) ˆλ s ) d M sλ s x ) )) β ) x )) opimal -h wealh process) β ) =: N ) )) porfolio process). Wih his we have found he opimal porfolios for he -problems. Nex we urn o he convergence of he soluions of he -level BSDEs o he BSDE of he exponenial problem: Since ˆλ dm is coninuous, we have ha x ˆλz ) β ) ˆλ, P.a.s. 36)

17 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) uniformly in. Since ˆλdM is deerminisic, we furher show for all p >, Esup p ) p p ], m, 37) where p := x + ˆλ ds. Finally, we already know ha p ) L X exp) x), which yields p = X exp) x). Hence he opimal porfolio ha reaches X exp) is equal o N exp) = ˆλ A exp where A exp is defined in Remark 2. Afer esablishing 36), 37) follows from he dominaed convergence heorem, and ˆλ A exp from our assumpion. We hus ge he following heorem: heorem 9. If ˆλ dm is deerminisic, hen ˆλ, ) A exp K is he opimal porfolio of he problem V exp x) = max E e x+ NdS C ) ), 38) N,C) A exp K where K is an arbirary class of righ-coninuous increasing processes. Furher, p E sup p) x + ˆλ ds) ), m, p 39) where p ) is he opimal wealh process of V x) sup E N,C) A p K p x + NdS C ) Finally, we esablish he equaliy X exp = x + ˆλ ds. Before proving he las heorem, we apply hese resuls o a Brownian case: Example. We consider an n-dimensional sock: S = S + S s µs)ds +, x R. 4) S s σ s)dw s, 4) where W is a d-dimensional Brownian moion, n d, σ a deerminisic n d-marix, and S = diags ),..., S n) ). We have ˆλ = µ σ σ ) S and so Ẑ is of he form { Ẑ = exp θ s dw s } θ s 2 ds, 2 where θ = σ σ σ ) µ. All assumpions in Frielli s heorems are saisfied by he minimal maringale measure noe ha all coefficiens are bounded); see [7]. Wih ˆλ = S σ σ ) µ, 32) is uniquely solvable wih p, q ). On he oher hand ) z ) β ) = p) x). 42)

18 83 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) Hence, wih q ) = θ p) p ), q ) ) is he opimal soluion of dp ) where q ) π ) = ), = θ q) d + q ) ) dw, p ) ) = X ) x) ) = q ) ) S σ. So σ σ ) σ θ Y ) ) = Sq Sˆλ = σ σ ) σ θ =: π exp), where π ) he amoun invesed in he socks S. Finally summarizing he above resuls we have he following commuing diagram where = should be read as corresponds o in he above explained sense : Y x)z Y exp x)z min convergence of dual soluions) = = X ) x) X exp) x) convergence of erminal wealhs) = = π ) π exp convergence of porfolios) = = V x) V exp x) convergence of value funcions) Proof of heorem 9. We sar by esablishing 36). For 36), we jus have o consider z ) and β ). Since ˆλ dm is coninuous, we have ha for arbirary ω sup ˆλ dm) K ω). Similarly for ˆλ dm. And so he powers of β ) and z ) converge o zero. By defining g := x z) β ), we ge by he Burkholder Davis Gundy inequaliy p ) E sup q s ) ˆλ s ) ds s K p E sup gs )ˆλ s d M s ˆλ p ) s + sup gs )ˆλ s dm p ) s ) K p E sup g ˆλ s d M s ˆλ s p p + E sup gs ) 2 ˆλ s d M s ˆλ 2 s. p. p K p Esup g p ) ˆλ s dm s + ˆλ s dm 2 s ) K p E sup g p.

19 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) By he dominaed convergence heorem and 36), i remains o show ha max g p is dominaed by an inegrable random variable: exp ) ˆλ s dm s + exp + K z 2). ˆλ s dm s By Doob s inequaliy and 34), we have for all p >, E max g ) p K + K E max ) exp. ). ) ˆλ s 2 dm s exp ˆλ s 2 dm s z 2) p ) = K + K E Ẑ p ) <. Finally, ˆλ A exp, i.e. ˆλ L p M) = ˆλ d M ˆλ) 2 L p <, and ˆλ L p A) = N da = ˆλ d M ˆλ < L p A) L p A) holds since ˆλ d M ˆλ is deerminisic and ˆλ ds = X exp O p. Remark. i) Obviously, our esimaions in he proof of heorem 9 heavily rely on he srong assumpions made here and are more complex in he general seing of he firs secions. We pospone his general case o a forhcoming paper, since we have o develop a more echnical approach, e.g. using a here: localized version of a) generalizaion of he monoone sabiliy Proposiion 2.4 of [7] also see []) o derive he convergence of he porfolios from he convergence of he erminal values of a family of BSDEs. A major difficuly in he general seing is ha of overcoming a boundedness condiion like X exp L ; see e.g. [2]. Neverheless, under some sronger condiions he above mehod also works in he more general siuaion, where he )-maringale measures are all differen, in paricular differen from he minimal maringale measure, which ypically arises in he case where he rade-off ˆK is no deerminisic see [27]). We briefly skech he idea here wihou giving he echnical proofs. Since X ) x) = I Y x)z ) ) is aainable, he soluion of 32) is easily guessed o be p ) = Z ) ) EZ ) X ) x) F ). 43) A lenghy and edious calculaion gives he following resuls, for q = p ) : = Z ) ) EE M Q q )) F )E Z ) Now apply heorem 5 o represen he densiy process Z ) as he exponenial of M Q q = where again q = Z ) = E ˆλ s dm s q ) ) o find ˆλ dm q ) ) ). 44) V s q) d m sq) 45) ) V s q) d m sq). 46)

20 832 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) Now separae ou of Z ) ) he exponenial maringales z ) by making use of a Novikov condiion o find a represenaion of p ) similar o he above represenaion. Formally, hen urns ou o have he same form as above, which hen proves ha q ) q ) x) = ) ˆλ z ) β ) =: N ) ) is he porfolio process. Finally he convergence has o be shown: x ˆλ z ) β ) ˆλ, P.a.s. 47) When deriving hese calculaions one mus carefully rea he orhogonal erms in he GKW decomposiion and keep in mind ha he erminal values are all aainable. his gives he desired resul in he framework of his aricle. A sufficien very srong) se of assumpions for making his hold in he Brownian seing above is he following: µ is bounded for all and ω and σ is bounded away from zero and bounded above he reverse Hölder inequaliy is saisfied). he resul urns ou o be formally idenical o he resul in he very resriced seing of he above example. hese resuls will be exended by making use of echniques differen from hose in his aricle. ii) By making use of he sandard change of numéraire echniques i is easily seen ha for r he above resul holds wih µ replaced by µ r. By approximaing exp αy ξ)), Y := x + Y by he sequence + αξ ) αy ) we find for he opimal porfolio of he -problem from a BSDE similar o 32) wih erminal value X ) ξ ha π ) = α ) σ σ σ ) θ + p) ξ Y ) ) σ σ σ ) q ) ξ, 48) where p ) ξ, q ) ξ ) = p ξ, q ξ ) is he soluion of he usual BSDE for hedging ξ, for simpliciy a L -random variable. he opimal conrol for he exp hedging problem hus urns ou o be π exp = σ σ σ ) θ α σ σ σ ) q ξ. 49) his resul should be compared o he mean variance hedging resul e.g. in [8,2]: he pure mean variance hedging borrows money o hedge he claim and invess he difference beween he price of he claim and he acual wealh according o a sor of Meron porfolio. Here we have a similar behavior, bu an exra erm appears which ries o drive he wealh higher han he claim. When looking a he srucure of he funcional e αx+ỹ ξ) his obviously makes perfec sense. he disadvanage in using he exponenial hedging however is similar o he well known disadvanages of he mean variance hedging: In he laer case overshooing he claim is punished; in he case under consideraion here, overshooing is rewarded see e.g. he discussion in [4]).

21 M. Kohlmann, C.R. Niehammer / Sochasic Processes and heir Applicaions 7 27) Conclusion he paper provides a new and complee framework for solving he dynamic uiliy maximizaion problem for an exponenial uiliy funcion via an approximaion approach. We consider conrol problems for he sequence of funcions x ) -h problem), which is raher ineresing in iself as a modificaion of he isoelasic conrol problem; see e.g. [3]. We sar by giving a soluion mehod for solving a general dynamic uiliy maximizaion problem wih no necessarily increasing, concave uiliy funcions. In a firs sep, we ransform he dynamic problem ino a saic problem via a hedging argumen. For increasing uiliy funcions his was already proven in [6]. An exension o no everywhere increasing uiliy funcions in an L p seing is now given in his paper for he firs ime. We furher presen a simple mehod for solving he saic problem applying a dualiy approach in Secion 3. Using his mehod, we can easily derive he opimal erminal value of he -h and he exponenial problem. heorem 7 presens our main resul on he relaion beween various kinds of opimal maringale measures, he -h, and he exponenial problem. Under some very weak assumpion in a general semimaringale model, we can prove he convergence of he erminal values and he value funcions of he -h o he exponenial problem. An explici porfolio for he exponenial problem has so far only been compued in very special cases; see e.g. [5] or [3]. Secion 5 herefore esablishes a porfolio for he -h problem in a seing wih a deerminisic erminal value of he rade-off process via a BSDE approach. he above convergence resul hen yields srong convergence of he porfolios and gives an explici porfolio for he exponenial problem. he resuls of his aricle have recenly been generalized o include Lévy process dynamics in [26] and general jump process dynamics in [9]. Acknowledgemens he work of he second auhor presens he main resuls of Chrisina Niehammer s hesis a he Universiy of Konsanz. he final version also conains resuls achieved during her presen Ph.D. sudy a he Universiy of Giessen. hese sudies are suppored by he HypoVereinsbank which is graefully appreciaed. However, his paper does no reflec he opinion of HypoVereinsbank; i is he personal view of he auhors. he auhors would like o hank Chrisian Bender for many helpful discussions relaed o BSDE heory and graefully acknowledge he remarks of he wo referees, especially he very careful reading of he manuscrip by one of he referees who gave very helpful advice on an inaccuracy in inequaliy 29). his improved he firs version of he manuscrip a lo. References [] Ph. Briand, B. Delyon, Y. Hu, E. Pardoux, L. Soica, L p -Soluions of sochasic differenial equaions, Sochasic Process. Appl. 8 23) [2] Ph. Briand, Y. Hu, BSDE wih quadraic growh and unbounded erminal value, Probab. heory Relaed Fields 36 26) [3] V. Bürkel, Linear isoelasic sochasic conrol problems and backward sochasic differenial equaions of Riccai ype, Disseraion, Universiy of Konsanz, 25. [4] R. Con, P. ankov, Rerieving Lévy processes from opion prices: Regularizaion of an ill-posed inverse problem, SIAM J. Conrol Opim. 45 ) 25. [5] F. Delbaen, P. Grandis,. Rheinländer, D. Samperi, M. Schweizer, C. Sricker, Exponenial hedging and enropic penalies, Mah. Finance 2 2) 22)

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