Optimization problem under change of regime of interest rate

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1 Opimizaion problem under change of regime of ineres rae Bogdan Ifimie Buchares Universiy of Economic Sudies, and Simion Soilow Insiue of Romanian Academy homas Lim Laboraoire d Analyse e Probabiliés, Universié d Evry-Val d Essonne and ENSIIE lim@ensiie.fr Monique Jeanblanc Laboraoire d Analyse e Probabiliés, Universié d Evry-Val d Essonne monique.jeanblanc@univ-evry.fr Hai-Nam Nguyen Laboraoire d Analyse e Probabiliés, Universié d Evry-Val d Essonne hainam.nguyen@univ-evry.fr Absrac In his paper, we sudy he classical problem of maximizaion of he sum of he uiliy of he erminal wealh and he uiliy of he consumpion, in a case where a sudden jump in he risk-free ineres rae creaes incompleeness. he value funcion of he dual problem is proved o be soluion of a BSDE and he dualiy beween he primal and he dual value funcions is exploied o sudy he BSDE associaed o he primal problem. Mahemaics Subjec Classificaion 2: 9B6, 9C46, 9G3, 93E2. Keywords: porfolio opimizaion, power uiliy, sochasic ineres rae, dual problem, backward sochasic differenial equaions BSDEs, enlarged filraion. Inroducion Many sudies in he field of Mahemaical Finance are devoed o porfolio and/or consumpion opimizaion problems. In he case of a complee marke, wih several risky asses and a savings accoun adaped o a Brownian filraion, he problem is fully solved in he monography of Karazas and Shreve [. he siuaion in incomplee markes is more delicae, and i is no easy o give closed form soluions see, e.g., Menoncin [5. he incompleeness of he marke may arise from a number of risky asses smaller han he dimension of he driving noise, from consrains on he porfolio, or from an ineres rae which depends on an exra noise, which will be he case in our seing. he lieraure abou he wo firs cases of incompleeness is imporan, he research of he firs auhor was suppored by he Secorial Operaional Programme Human Resources Developmen SOP HRD, financed from he European Social Fund and by he Romanian Governmen under he conrac number SOP HRD/89/.5/S/ he research of he hree oher auhors is suppored by Chaire risque de crédi, French Banking Federaion

2 on he oher hand he lieraure abou he hird case of incompleeness is reduced. We can cie [5 for he case of a mulidimensional incomplee marke and consan ineres rae and a Brownian filraion under Markovian framework, where he auhor solves he problem using HJB equaion. he case where he measurabiliy of he ineres rae creaes incompleeness is presened in Bauerle and Rieder [ in which he dynamics of he ineres rae is driven by a Markov chain. A classical ool o solve uiliy maximizaion problem is he dual approach. his one consiss in solving an auxiliary opimizaion problem, called he dual problem, which is defined on he se of all equivalen maringale measures. he lis of papers sudying ha problem is long and we quoe only few of hem. his approach is used in he case of incomplee markes generaed by a savings accoun wih consan ineres rae and several socks represened by general semi-maringales for HARA uiliy, by Kramkov and Schachermayer [3. hey sae an exisence and uniqueness resul for he final opimal wealh associaed o an invesmen problem, bu no explici formulas are provided. Rogers [6 formulaes an absrac heorem in which he value funcion of he uiliy maximizaion problem and he value funcion for he associaed dual problem saisfy a bidual relaion. As i is menioned, his procedure can be applied for a wide class of porfolio and/or consumpion opimizaion problems. Casañeda-Leyva and Hernández-Hernández [2 deal wih a combined invesmen and consumpion opimizaion problem wih a single risky asse, in a Brownian framework, and where he coefficiens of he model including he ineres rae are deerminisic funcions of some exernal economic facor process. Here, we are concerned wih he problem of maximizaion of expeced power uiliy of boh erminal wealh and consumpion, in a marke wih invesmen opporuniies in a savings accoun wih a sochasic ineres rae, which suffers an unexpeced shock a some random ime τ, and a sock modeled by a semi-maringale driven by a Brownian moion. he unexpeced shock can for example be due o some serious macroeconomic issue. his one implies ha he marke is incomplee. he problem will be solved in he filraion generaed by prices of sock and savings accoun so ha he change of regime ime τ is a sopping ime, under he immersion hypohesis beween he filraion generaed by he sock and he general filraion. Using sandard resuls of dualiy, he original opimizaion problem called he primal problem is linked o he dual problem, in which he conrol parameers ake value in he se of equivalen maringale measures. hen, we prove, by using a similar approach o he one used in Hu e al.[7 for he case of he primal problem wihou consumpion and more recenly in Cheridio and Hu [3 for he case wih consumpion, ha he value funcion of ha problem is soluion of a paricular BSDE, involving one jump. Using a recen resul of Kharroubi and Lim [2, we show ha his BSDE has a unique soluion. hen, we give he opimal porfolio and consumpion in erms of he soluion of his BSDE, and explici formula for he opimal wealh process. We also esablish a dualiy resul for he dynamic versions of he value funcions associaed o primal and dual opimizaion problems which allows us o prove ha he BSDE associaed o he primal problem has a unique soluion. o he bes of our knowledge, he BSDE mehodology has no been used ye for dual problems in he lieraure. he paper is organized as follows. In Secions 2 and 3, we describe he se up and model. In Secion 4, we characerize he se of he equivalen maringale measures, hen we derive and solve he dual opimizaion problem. Finally, Secion 5 is dedicaed o he link beween he value funcions associaed o he primal and dual opimizaion problems and o he compuaion of explici formulas for he opimal wealh process, opimal rading and consumpion policies. 2

3 2 Se up hroughou his paper Ω, G, P is a probabiliy space on which is defined a one dimensional Brownian moion B [, where < is he erminal ime. We denoe by F := F [, he naural filraion of B augmened by he P-null ses and we assume ha F G. On he same probabiliy space is given a finie posiive G-measurable random variable τ which is inerpreed as a random ime associaed o some unprediced evoluion wih respec o he filraion F in he dynamics of he ineres rae or o a swiching regime. Le H be he càdlàg process equal o before τ and afer τ, i.e., H := τ. We inroduce he filraion G which is he smalles righ-coninuous exension of F ha makes τ a G-sopping ime. More precisely G := G [,, where G is defined for any [, by G := ɛ> G +ɛ, where G := F σh u, u [,, for any [,. hroughou he sequel, we assume he following classical hypoheses. H Any F-maringale is a G-maringale, i.e., F is immersed in G. H2 he process H admis an absoluely coninuous compensaor, i.e., here exiss a nonnegaive G-adaped process λ G, called he G-inensiy, such ha he compensaed process M defined by M := H λ G s ds, is a G-maringale. Noe ha he process λ G vanishes afer τ, and we can wrie λ G = λ F <τ where λ F is an F-adaped process, called he F-inensiy of he process H. We assume ha λ G is uniformly bounded, hence λ F is also uniformly bounded. he exisence of λ G implies ha τ is no an F-sopping ime in fac, τ avoids F-sopping imes and is a oally inaccessible G-sopping ime. We recall in his framework he sandard decomposiion of any G-predicable process ψ which is given by Jeulin [8, Lemma 4.4. Lemma. Any G-predicable process ψ can be decomposed under he following form ψ = ψ τ + ψ τ >τ, where he process ψ is F-predicable, and for fixed non-negaive u, he process ψ u is F- predicable. Furhermore, for any fixed [,, he mapping ψ is F B[, - measurable. Moreover, if he process ψ is uniformly bounded, hen i is possible o choose bounded processes ψ and ψ u. Remark. he process expab 2 a2 [, being an F-coninuous maringale for every real number a, he immersion propery implies ha i is a G-coninuous maringale, hence B is a G-Brownian moion. I follows ha he sochasic inegral ϑ s db s is well defined for a G-adaped process ϑ up o inegrabiliy condiions, e.g. if ϑ is bounded and ha his inegral is a G local-maringale. We define he following spaces which will be used hroughou his paper. 3

4 SF u, resp. S G u, denoes he se of F resp. G-progressively measurable processes X which are essenially bounded on [u,, i.e., such ha ess sup X < ; [u, S,+ F u, resp. S,+ G u, denoes he subse of SF u, resp. S G u, such ha X C for a posiive consan C; HF 2u, resp. H2 G u, denoes he se of square inegrable F resp. G-predicable processes X on [u,, i.e., X 2 H 2 u, := E u X 2 d < ; HG 2 M denoes he se of G-predicable processes X on [, such ha X 2 H 2 G M := E λ G X 2 d <. 3 Model he financial marke consiss in a savings accoun wih a sochasic ineres rae wih dynamics ds = r S d, S =, where r is a non-negaive G-adaped process, and a risky asse whose price process S follows he dynamics ds = S ν d + σ db. Our assumpions abou he marke are he following H3 r is a G-adaped process of he form r = r <τ + r τ τ, where r is a non-negaive uniformly bounded F-adaped process, and for any fixed nonnegaive u, r u is a non-negaive uniformly bounded F-adaped process, and for fixed [,, he mapping r is F B[, -measurable. H4 ν and σ are F-adaped processes, and here exiss a posiive consan C such ha ν C and C σ C, [,, P - a.s. hroughou he sequel, we use he noaion R for he discoun facor defined by R := e rsds for any [,. We now consider an invesor acing in his marke, saring wih an iniial amoun x > and we denoe by π and π he par of wealh invesed in he savings accoun and in he risky asse, and by c he associaed insananeous consumpion process. Obviously we have he relaion π = π. We denoe by X x,π,c he wealh process associaed o he sraegy π, c and he iniial wealh x, and we assume ha he sraegy is self-financing, which leads o he equaion { X x,π,c = x, dx x,π,c = X x,π,c [ r + π ν r d + π σ db c d. We consider he se Ax of he admissible sraegies defined below. 4

5 Definiion. he se Ax of admissible sraegies π, c consiss in G-predicable processes π, c such ha E π sσ s 2 ds <, c and X x,π,c > for any [,. We are ineresed in solving he classical problem of uiliy maximizaion defined by V x := sup E π,c Ax Uc s ds + UX x,π,c, 2 where he uiliy funcion U is Ux = x p /p wih p,. 4 Dual approach o prove ha here exiss an opimal sraegy o he problem 2, we use he dual approach inroduced by Karazas e al. [ or Cox and Huang [4. For ha, we inroduce he convex conjugae funcion Ũ of he uiliy funcion U, which is defined by Ũy := supux xy, y >. x> he supremum is aained a he poin Iy := U y and a direc compuaion shows ha Iy = y p and Ũy = yq p q where q := p <. We also have he conjugae relaion Ux = inf Ũy + xy, x >. 3 y> Before sudying he dual problem, we characerize he se of equivalen maringale measures which is used o inroduce he dual problem. 4. Characerizaion of he se of equivalen maringale measures he se MP of equivalen maringale measures e.m.m. is MP := {Q Q P, RS is a Q, G local maringale }. he dynamics of he discouned price of he risky asse S := RS is given by d S = σ S db + θ d, 4 where θ := ν r σ is he risk premium. Le Q be a probabiliy measure equivalen o P, defined by is Radon-Nikodym densiy dq G = L Q dp G, where L Q is a posiive G-maringale wih L Q =. According o he Predicable Represenaion heorem see Kusuoka [4, and using he fac ha L Q is posiive, here exiss a pair a, γ of G-predicable processes saisfying γ > for any [, such ha dl Q = L Q a db + γ dm. 5

6 From Girsanov s heorem, he process B defined by B := B a s ds is a Q, G-Brownian moion, and he process M defined by M := M γ s λ G s ds = H + γ s λ G s ds is a Q, G-disconinuous maringale, orhogonal o B. Using 4, we noice ha if a probabiliy measure Q is an e.m.m., hen a = θ for any [,. Lemma 2. he se MP is deermined by all he probabiliy measures Q equivalen o P, whose Radon-Nikodym densiy process has he form L Q = exp θ s db s 2 θ s 2 ds + ln + γ s dh s γ s λ G s ds, where γ is a G-predicable process saisfying γ >. o alleviae he noaions, for any Q MP, we wrie L γ for L Q where γ is he process associaed o Q, i.e., dl γ = L γ θ db + γ dm, L γ =. For any Q MP, we remark ha RX x,π,c +. R sc s ds is a posiive Q, G-local maringale, hence a supermaringale, so we have E Q R X x,π,c + R s c s ds x, π, c Ax, where E Q denoes he expecaion w.r.. he probabiliy measure Q or equivalenly E R L γ Xx,π,c + R s L γ s c s ds x, π, c Ax Dual opimizaion problem We now define he dual problem associaed o 2 according o he sandard heory of convex dualiy. For ha, we consider he se Γ of dual admissible processes. Definiion 2. he se Γ of dual admissible processes is he se of G-predicable processes γ such ha here exiss wo consans A and C saisfying < A γ C for any [, τ and γ = for any τ,. I is ineresing o work wih his admissible se Γ hroughou he sequel since, for any γ Γ, he process L γ is a posiive G-maringale indeed, due o he bounds on γ, he process L γ is a rue maringale, and i saisfies he following inegrabiliy propery which simplifies some proofs in he sequel. Moreover, we consider ha γ is null afer he ime τ since he value of γ afer τ does no inerfere in he calculus, hus i is possible o fix any value for γ afer τ. 6

7 Lemma 3. For any γ Γ, he process L γ saisfies [ E L γ q <. Proof. From Iô s formula, we ge sup [, dl γ q = L γ q[ 2 qq θ 2 qλ G γ +λ G +γ q d qθ db + +γ q dm. his can be wrien under he following form L γ q = K E. where K is he bounded process defined by K := exp qθ s db s +. herefore, here exiss a posiive consan C such ha [ E sup [, + γs q dm s 2 qq θ s 2 qλ G s γ s + λ G s + γs q ds. L γ q [ CE sup [, E We conclude by using he Burkholder-Davis-Gundy inequaliy.. qθ s db s. From he conjugae relaion 3, we ge for any η >, γ Γ and π, c Ax [ E Uc s ds + UX x,π,c E + ηe ŨηR s L γ s ds + ŨηR L γ R s L γ s c s ds + R L γ Xx,π,c Using 5, he previous inequaliy gives for any η >, γ Γ and π, c Ax [ E Uc s ds + UX x,π,c E ŨηR s L γ s ds + ŨηR L γ + ηx. herefore, he following inequaliy holds for any π, c Ax E Uc s ds + UX x,π,c We hus obain,. inf E ŨηR s L γ s ds + ŨηR L γ + ηx. η>,γ Γ [ sup E Uc s ds + UX x,π,c inf E ŨηR s L γ s ds + ŨηR L γ + ηx. 6 π,c Ax η>,γ Γ EY denoes he Doléans-Dade sochasic exponenial process associaed o a generic maringale Y. 7

8 We inroduce he dual problem for any η > Ṽ η = inf E ŨηR s L γ s ds + ŨηR L γ γ Γ = ηq q inf E R s L γ s q ds + R L γ q. γ Γ We hus consider he following opimizaion problem inf E R s L γ s q ds + R L γ q. γ Γ o solve his problem we use a similar approach o he one used in Cheridio and Hu [3 which is linked o he dynamic programming principle. More precisely, we look for a family of processes {J d γ [, : γ Γ}, called he condiional gains, saisfying he following condiions i J d γ = R L γ q + R sl γ s q ds, for any γ Γ. ii J d γ = J d γ 2, for any γ, γ 2 Γ. iii J d γ is a G-submaringale for any γ Γ. iv here exiss some γ Γ such ha J d γ is a G-maringale. Under hese condiions, we have Indeed, using i and iii, we have J d J d γ = inf E R s L γ s q ds + R L γ q. γ Γ [ γ E[ J d γ = E R s L γ s q ds + R L γ q, 7 for any γ Γ. hen, using i and iv, we have J d γ = E [ J d γ = E R s L γ s q ds + R L γ q. 8 herefore, from ii, 7 and 8, we ge for any γ Γ [ E R s L γ s q ds + R L γ q = J d γ = J d γ E R s L γ s q ds + R L γ q. We can see ha J d γ = inf E R s L γ s q ds + R L γ q. γ Γ We now consruc a family of processes {J d γ [, : γ Γ} saisfying he previous condiions using BSDEs. For ha we look for J d γ under he following form, which is based on he dynamic programming principle, J d γ = R s L γ s q ds + R L γ q Φ, [,, 9 8

9 where Φ, ϕ, ϕ is soluion in SG, H2 G, H2 G M o Φ = fs, Φ s, ϕ s, ϕ s ds ϕ s db s ϕ s dh s, where f is o be deermined such ha iii and iv above hold. In order o deermine f, we wrie J d γ as he sum of a maringale and a non-decreasing process ha is null for some γ Γ. Applying inegraion by pars formula leads us o d[r L γ q = R L γ q[ 2 qq θ 2 + λ + γ q qλ G γ + r d qθ db + + γ q dm. aking ino accoun and applying inegraion by pars formula for he produc of processes RL γ q and Φ, we ge dj d γ = R L γ q A γ d+r L γ q ϕ qθ Φ db + [ Φ + ϕ +γ q Φ dm, where he predicable finie variaion par of J d γ is given by R sl γ s q A γ s ds, where A γ := λ G a γ + + f, Φ, ϕ, ϕ qr Φ + 2 qq θ 2 Φ λ G Φ qθ ϕ, wih a x := Φ + ϕ + x q qφ x, [,. 2 In order o obain a non-negaive process A γ for any γ Γ o saisfy he condiion iii and ha is null for some γ Γ o saisfy he condiion iv, i is obvious ha he family {A γ [, : γ Γ} has o saisfy min γ Γ A γ =. Assuming ha here exiss a posiive consan C such ha Φ C and Φ + ϕ C for any [, τ, we remark ha he minimum is aained for γ defined by so ha γ Φ := q Φ + ϕ a := min x> a x = qφ p Φ + ϕ p + qφ. his leads o he following choice for he generaor f f, y, z, u = 4.3 Soluion of he BSDE qr 2 qq θ 2 + qλ G y + qθ z qλ G y + u p y p. 3 We remark ha he obained generaor 3 is non sandard since i involves in paricular he erm y + u p y p. We shall prove he following resul 9

10 heorem. he BSDE Φ = qrs 2 qq θ s 2 + qλ G s Φs + qθ s ϕ s qλ G s Φ s + ϕ s p Φ p s ds ϕ s db s ϕ s dh s, admis a soluion Φ, ϕ, ϕ belonging o S,+ G, H 2 G, H2 G M, such ha Φ + ϕ. We use he decomposiion procedure inroduced in [2 o prove heorem. For ha, we ransform he BSDE 4 ino a recursive sysem of Brownian BSDEs. In a firs sep, for each u [,, we prove ha he following BSDE has a soluion on he ime inerval [u, dφ u = [ qr u 2 qq θ u 2 Φ u + qθ u ϕ u Φ u =, 5 and ha he iniial value Φ uu of his BSDE is F u -measurable. hen, in a second sep, we prove ha he following BSDE has a soluion on he ime inerval [, [ qr dφ = 2 qq θ 2 + qλ F Φ + qθ ϕ qλ F Φ p Φ p d + ϕ db, 6 Φ =, where Φ is par of he soluion of he BSDE 5. 4 d + ϕ udb, Proposiion. For any u [,, he BSDE 5 admis a unique soluion Φ u, ϕ u S F u, H2 F u,. Furhermore, Φ u C for any [u, where C is a consan which does no depend on u. Proof. Le us fix u [,. Since he BSDE 5 is linear wih bounded coefficiens, he soluion Φ u, ϕ u SF u, H2 F u, is given by Φ u = E Γ u + Γ suds F, [u,, 7 where for a fixed [u,, Γ su s sands for he adjoin process defined by s Γ su = exp qr v u + 2 qq θ vu 2 dv E qθvudb v. o prove ha Φ is uniformly bounded, we inroduce he probabiliy measure P u, defined on F, for, by is Radon-Nikodym densiy Z u := E qθ vudb v, which is a rue maringale, and we denoe by E u he expecaion under his probabiliy. hen, by virue of he formula 7 and Bayes rule, we ge Φ u = E u[ exp qr s u + 2 qq θ su 2 F ds + E u s exp qr v u + 2 qq θ vu 2 dv ds F. s

11 From H3 and H4, and since q <, here exiss a posiive consan C which is independen of u such ha Φ u C for any [u,. Proposiion 2. he BSDE 6 admis a unique soluion Φ, ϕ S,+ F, HF 2,. Proof. he generaor of he BSDE 6 is no defined on he whole space [, Ω R R and he generaor is no classical. So he proof of his proposiion will be performed in several seps. We firs inroduce a modified BSDE where he erm y p is replaced by y m p where m is a posiive consan which is defined laer o ensure ha he generaor is well defined on he whole space [, Ω R R. We hen prove via a comparison heorem ha he soluion of he modified BSDE saisfies he iniial BSDE. In he las sep, we prove he uniqueness of he soluion. Sep. Inroducion of he modified BSDE. We consider { dy = ḡ, Y, ŷ d ŷ db, 8 Y =, where he generaor ḡ is given by ḡ, y, z := + 2 qq θ 2 qr qλ F y qθ z + qλ F Φ p y m p, wih m := exp q Λ, and Λ is a consan such ha λ F Λ for any [,. Since p,, here exiss a posiive consan C such ha y m p C + y. We also have Φ. is uniformly bounded, and using assumpions H2, H3 and H4 we obain ha ḡ has linear growh uniformly w.r.. y. I follows from Fan and Jiang [6 ha he BSDE 8 has a unique soluion Y, ŷ SF, H2 F,. For he convenience of he reader, we recall he Fan and Jiang condiions, which, in our seing, are obviously saisfied. he soluion of he BSDE is unique if: dy = f, Y, ŷ d ŷ db, Y = he process f,, [, L 2,, 2 dp d a.s., y, z fω,, y, z is coninuous, 3 f is monoonic in y, i.e., here exiss a consan µ, such ha, dp d a.s., y, y 2, z, fω,, y, z fω,, y 2, z y y 2 µy y 2 2, 4 f has a general growh wih respec o y, i.e., dp d.a.s., y, fω,, y, fω,,, + ϕ y where ϕ : R R + is an increasing coninuous funcion, 5 f is uniformly coninuous in z and uniform w.r.. ω,, y, i.e., here exiss a coninuous, non-decreasing funcion φ from R + o iself wih a mos linear growh and φ = such ha dp d a.s., y, z, z 2, fω,, y, z fω,, y, z 2 φ z z 2.

12 Sep 2. Comparison. We now show ha he soluion of he BSDE 8 is lower bounded by m, and his is accomplished via a comparison resul for soluions of Brownian BSDEs. We remark ha he following inequaliy holds ḡ, y, z 2 qq θ 2 qr qλ F y qθ z =: g, y, z. herefore, we inroduce he following linear BSDE { dz = g, Z, ẑ d ẑ db, Z =. 9 In he same way as we proceed wih he BSDE 5, we have an explici form of he soluion of he BSDE 9 given by Z = E Υ F, where Υ s s sands for he soluion of he linear SDE dυ s = Υs[ 2 qq θ s 2 qrs qλ F s ds qθsdb s, Υ =. We can rewrie he soluion of he BSDE 9 under he following form Z = E [ exp 2 qq θ s 2 qrs qλ F F s ds, where E is he expecaion under he probabiliy P defined by is Radon-Nikodym densiy dp F = E qθ vdb v dp F for any [,. By virue of he assumpion H4, i follows ha Z E [ exp qλ F F sds m. From he comparison heorem for Brownian BSDEs, we obain Y Z m, which implies ha Y m = Y for any [,. herefore, Y, ŷ is a soluion of he BSDE 6 in S,+ F, HF 2,. Sep 3. Uniqueness of he soluion. Suppose ha he BSDE 6 has wo soluions Y, Z and Y 2, Z 2 in S,+ F, HF 2,. hus, here exiss a posiive consan c such ha Y c and Y 2 c for any [,. In his case, Y, Z and Y 2, Z 2 are soluions of he following BSDE { dy = h, Y, ŷ d ŷ db, where he generaor h is given by Y =, h, y, z := + qr + 2 qq θ 2 qλ F y qθ z+ qλ F Φ p y c p. From [6, we know ha his BSDE admis a unique soluion, herefore we ge Y = Y 2. 2

13 We are now able o prove heorem. Proof. From Proposiions and 2 and heorem 3. in [2, we obain ha he BSDE 4 admis a soluion Φ, ϕ, ϕ belonging o SG, H2 G, H2 G M given by Φ = Φ <τ + Φ τ τ, ϕ = ϕ τ + ϕ τ >τ, ϕ = Φ Φ τ. 2 Noe ha ϕ and ϕ are G-predicable processes. Moreover, from Proposiions and 2, here exiss a posiive consan C such ha Φ C. We also remark ha which implies ha Φ + ϕ. Φ + ϕ = Φ τ + Φ τ >τ = Φ τ, Remark 2. We remark ha if r and θ are deerminisic, Φ is deerminisic. Moreover, if r, θ and λ F are deerminisic, Φ is deerminisic. More precisely, ϕ u = ϕ =, and he BSDEs 5 and 6 urn ino ODEs [ qr dφ u = u 2 qq θ u 2 Φ u d, and { dφ = Φ =, Φ u =, [ qr 2 qq θ 2 + qλ F Φ qλ F Φ p Φ p d, wih an explici soluion for he firs equaion. Remark 3. If r u = r for any u, here is no change of regime. Our resul is coheren wih ha obvious observaion, since, in ha case, we have ha θ u = θ for any u which implies Φ = Φ for any [,. 4.4 A verificaion heorem We now urn o he sufficien condiion of opimaliy. In his par, we prove ha he family of processes {J d γ [, : γ Γ} defined by J d γ :=. R sl γ s q ds+rl γ q Φ wih Φ defined by 2 saisfies he condiions i, ii, iii and iv. By consrucion, J d γ saisfies he condiions i and ii. As explained previously a candidae o be an opimal γ is a process γ such ha J d γ is a G-maringale, hence his one is Lemma 4. he process γ defined by 2 is admissible. γ Φ := q. 2 Φ + ϕ Proof. By consrucion, γ is G-predicable. Moreover, from heorem, we remark ha here exiss wo consans A and C such ha < A γ C for any [, which implies ha γ Γ. 3

14 From he above resuls, J d γ is a semi-maringale wih a local maringale par and a non-decreasing predicable variaion par and J d γ is a local maringale. Proposiion 3. he process J d γ is a G-submaringale for any admissible process γ Γ and is a G-maringale for γ given by 2. Proof. From and 4, we can rewrie he dynamics of J d γ under he following form dj d γ = R L γ q dm γ + A γ d, where dm γ = ϕ qθ Φ db + + γ q Φ + ϕ Φ dm, and A γ = λ G [ a γ a γ, wih a. defined by 2. From 9, Lemma 3 and since Φ SG,, we remark ha for any γ Γ [ E J d γ <. 22 sup [, For any γ Γ, we have ha. R sl γ s q dm γ s is a G-local maringale. Hence, here exiss an increasing sequence of G-sopping imes n n N valued in [, saisfying lim n n =, P a.s. such ha. n R s L γ s q dm γ s is a G-maringale for any n N. herefore, we obain for any [, [ E J d n γ = J d γ + E [ n R s L γ s q A γ s ds. Since RL γ q A γ, from 22 and using he monoone convergence heorem, we obain From 22 and he previous inequaliy, we have [ E E R L γ q A γ d <. sup [, R s L γ s q dm γ s <. I follows ha he local maringale. R sl γ s q dm γ s is a rue maringale and he process J d γ is a G-submaringale for any γ Γ. We obain wih he same argumens ha he process J d γ is a maringale. 4.5 Uniqueness of he soluion of he BSDE o solve he dual problem i is no necessary o prove he uniqueness of he soluion of he BSDE bu his one is useful o characerize he value funcion of he primal problem in he las par of his paper. o prove he uniqueness we do no use a comparison heorem for BSDE bu he following dynamic programming principle. 4

15 Lemma 5. Le Y be a process wih Y = such ha. R sl γ s q ds + RL γ Y is a G- submaringale for any γ Γ and here exiss γ Γ such ha. R sl γ s q ds + RL γ Y is a G-maringale. hen, we have { } Y = ess inf γ Γ R L γ q E R s L γ s q ds + R L γ G q. Proof. he following inequaliy holds for any γ Γ Y R L γ q E R s L γ s q ds + R L γ G q. Moreover, we know ha Y = R L γ E R s L γ q s q ds + R L γ G q. herefore, we ge { } Y = ess inf γ Γ R L γ q E R s L γ s q ds + R L γ G q. We now prove ha any soluion of he BSDE saisfies he properies of Lemma 5. Lemma 6. Le Φ, ϕ, ϕ S,+ G, HG 2, H2 G M be a soluion of he BSDE. hen, he process. R sl γ s q ds+rl γ Φ is a G-submaringale for any γ Γ and here exiss γ Γ such ha. R sl γ s q ds + RL γ Φ is a G-maringale. Proof. o simplify he noaion we denoe W γ :=. R sl γ s q ds + RL γ Φ. From Iô s formula, we ge for any γ Γ W γ = R L γ q{ λ a γ a γ d+ ϕ qθ Φ db + +γ q Φ + φ Φ dm }, p where a. is defined by 2 and γ Φ := Φ +. φ We know ha E[sup W γ < from Lemma 3 and a γ a γ for any γ Γ by definiion of γ. herefore, using he same argumens as for he proof of Proposiion 3 we can prove ha, for any γ Γ, he process. R sl γ s q ds + RL γ Φ is a G-submaringale and. R sl γ s q ds + RL γ Φ is a G-maringale. We can conclude from Lemmas 5 and 6 ha here exiss a unique soluion of he BSDE in S,+ G, HG 2, H2 G M. 5 Primal problem and opimal sraegy In his secion, we deduce he soluion of he primal problem 2 using he dualiy resul of he previous secion, and we characerize he value funcion associaed o he primal problem by he soluion of a BSDE which is in relaionship wih he BSDE 4 associaed o he dual problem. he following proposiion shows he exisence of an opimal soluion for he primal problem and characerizes his soluion in erms of he soluion of he dual problem. 5

16 Proposiion 4. he opimal sraegy is given by c = η R L γ p, π = ϕ + θ σ Φ p where η is defined by η := and γ is given by 2. E R L γ x q d + R L γ q, [,, 23 p, 24 Before proving Proposiion 4, we prove ha he sraegy π, c is admissible. Lemma 7. he sraegy π, c given by 23 is admissible and he wealh associaed o π, c is X x,π,c = η R L γ p Φ. 25 Proof. Using assumpions H3 and H4, and he properies of Φ, ϕ, ϕ given by heorem, we obain ha E π sσ s 2 ds < and π is G-predicable. Moreover, from, he wealh process X x,π,c associaed o he sraegy π, c is defined by he SDE X x,π,c = x, [ dx x,π,c = X x,π,c η R L γ p d. r q θ 2 + θ ϕ Φ d + ϕ Φ q θ db 26 From Proposiion 3, he process. R sl γ s q ds + RL γ q Φ is a G-maringale, which implies Φ = E R L γ q d + R L γ q. From he previous equaliy and 24, we remark ha η p Φ = x. Using Iô s formula and 4, we check ha η RL γ p Φ is a soluion of he SDE 26. Moreover, his SDE admis a unique soluion. herefore, we have Using he fac ha c paricular, η R L γ p X x,π,c and X x,π,c We now prove Proposiion 4. Proof. From 6, we obain = X x,π,c sup E Uc s ds + UX x,π,c inf ηq π,c Ax η>,γ Γ = η R L γ p Φ. 27 > for any [,, we conclude he proof. In is hedgeable. q E By he definiion of γ and η, he previous inequaliy is equivalen o sup E Uc s ds + UX x,π,c π,c Ax xp p 6 E R s L γ s q ds + R L γ q + ηx. R s L γ s q ds + R L γ q p. 28

17 By definiion of π, c and Lemma 7, we remark ha E Uc sds + UX x,π,c = xp p E Since π, c is admissible, from 28 and 29, we obain R s L γ s q ds + R L γ q p. 29 E Uc sds + UX x,π,c = sup E Uc s ds + UX x,π,c. π,c Ax herefore, π, c is an opimal soluion of he primal problem 2. We now characerize he value funcion associaed o he primal problem using he dynamic programming principle. For fixed [, we denoe by π, c a sraegy defined on he ime inerval [, and X,x,π,c s s [, he wealh process associaed o his sraegy given he iniial value a ime is x >. We firs define he se of conrol for a fixed. Definiion 3. he se A x of admissible sraegies π, c from ime consiss in he se of G- predicable processes πs, c s s [, such ha E π sσ s 2 ds <, c s and X,x,π,c s > for any s [,. We define he value funcion a ime for he primal problem as follows V, x := xp p Ψ x, 3 where Ψ x := x p ess sup E π,c A x c s p ds + X,x,π,c p G. For any π, c A x, we define he sraegy ˆπ, ĉ by ˆπ := π and ĉ := c /x. We remark ha ˆπ, ĉ A and X,x,π,c = xx,,ˆπ,ĉ from. Combining he previous relaions wih he definiion of Ψ x, we obain Ψ x = Ψ. For he sake of breviy, we shall denoe Ψ insead of Ψ. he value funcion a ime can be rewrien as follows V, x = x p Ψ /p. From 3 and Proposiion 4, we have V, x = E c s p Xx,π ds + p p,c p = V x. 3 Using dynamic conrol echniques, we derive he following characerizaion of he value funcion. Proposiion 5. For any π, c Ax,. here exiss π, c Ax such ha. c s p p c s p p he proof of his proposiion is given in El Karoui [5. ds + V., Xx,π,c is a G-supermaringale and ds + V., Xx,π,c is a G-maringale. Using hese properies, we can characerize he value funcion wih a BSDE. 7

18 Proposiion 6. he process Ψ saisfies he equaliy Ψ = Φ p. Moreover, he process Ψ is soluion of he BSDE Ψ = pψ q s pr s Ψ s + p 2 p θ s 2 Ψ s + ψ s 2 + Ψ s ψ s db s ψ s dh s. Proof. From 23, 25 and 3, we have c s p ds + V, Xx,π p,c = η q p p p θ ψ s s ds R s L γ s q ds + R L γ q Φ. From Proposiions 3 and 5, he process. c s p p ds + V.,,c Xx,π. and. R sl γ s q ds + RL γ q Φ are G-maringales. herefore, aking he condiional expecaion for he above equaliy, we obain c s p ds + X x,π,c p Ψ = η q R s L γ q ds + R L γ q Φ. Since c = η R L γ p, he following relaion holds for any [, herefore, from 27 and 33, we obain Applying Iô s formula o Φ p, we obain dφ p s 32 X x,π,c p Ψ = η q R L γ q Φ. 33 Φ p = Ψ. = pφ p qr 2 qq θ 2 + λ G q Φ + qθ ϕ qλ G Φ + ϕ p Φ p 2 pφ ϕ 2 d + p ϕ Φ p db + Φ + ϕ p Φ p dh. Seing ψ = p ϕφ p and ψ = Φ + ϕ p Φ p 32., we ge ha Ψ, ψ, ψ saisfies he above resul is no sufficien o characerize he value funcion of he primal problem since i is no obvious ha he BSDE 32 admis a unique soluion. Bu hanks o he uniqueness of he soluion of he BSDE 4 we ge he following characerizaion of he value funcion of he primal problem. heorem 2. Ψ is he unique soluion of he BSDE 32 in S,+ G, HG 2, H2 G M. Proof. Le y, z, u S,+ G define Y := y q From Iô s formula, we ge ha dy =, HG 2, H2 G M be a soluion of he BSDE 32. We z and U := y + u q y q for any [,., Z := qy q [ qr qq θ 2 + qλ G Y qλ G U + Y p Y p 2 +qθ Z d + Z db + U dh. 8

19 herefore, Y, Z, U is soluion of he BSDE 4 and, from Subsecion 4.5, we have by uniqueness of he soluion Y = Φ and from Proposiion 6 we ge ha y = Ψ. Remark 4. From heorems and 2 and Proposiion 6, we can conclude ha he BSDE 32, which is associaed o he primal problem 2, admis a unique soluion Ψ, ψ, ψ belonging o S,+ G, HG 2, H2 G M. Solving his BSDE direcly is no eviden because of he erms Ψ q and ψ 2 Ψ. Remark 5. We poin ou ha, in he case where he coefficiens of he model are deerminisic funcions of some exernal economic facors and in a Brownian seing, he opimal conrol processes π, c have he same expressions ha hose obained by Casañeda-Leyva and Hernández-Hernández [2 see Proposiion 3.. in [2. We also remark ha, in a defaul-densiy seing, he opimal conrol processes π, c have he same expressions ha hose obained by Jiao and Pham [9. In hese wo papers, he opimal porfolio is given in erms of he value funcion of he primal problem or in erms of he soluion of he primal BSDE as π ψ = + θ. pσ Ψ ψ We have proved ha p Ψ = ϕ Φ, hence, he wo soluions have he same form. In paricular, afer τ, our seing is ha one of a complee marke, and our formula is raher sandard. In paricular, in he case where r and θ are deerminisic, he invesor is myopic, and he opimal porfolio is π = θ p σ. Before τ, he invesor akes ino accoun he fac ha he ineres rae will change, since he afer defaul value funcion appears in he before defaul value funcion his is he erm Φ in he associaed BSDE. 6 Conclusion In his paper, we have sudied he problem of maximizaion of expeced power uiliy of boh erminal wealh and consumpion in a marke wih a sochasic ineres rae in a model where immersion holds. We have derived he opimal sraegy solving he associaed dual problem. hen, we have given he link beween he value funcions associaed o he primal and dual problems, which has allowed o characerize he value funcion of he primal problem by a BSDE. If one assumes ha B is a G-semi-maringale wih canonical decomposiion of B in G of he form B = B G + µ s ds, wih a bounded process µ and B G a G-Brownian moion, he price dynamics of he risky asse can be rewrien as follows ds = S ν + σ µ d + σ db G, where he coefficiens ν and σ can be chosen G-predicable and bounded. In his case, he e.m.m. can be wrien on he form indeed, a predicable represenaion propery holds for he pair B G, M dl = L a db G + γ dm 9

20 and using he same mehods and argumens, we can obain similar resuls. he real difficuly is ha one has o assume ha he process µ is bounded, and we do no know any condiion on τ which implies ha fac. Wihou any heoreical difficuly, we can generalize his paper o he case where here are several ordered changes of regime of ineres rae. References [ Bauerle, N., Rieder, U. 24 Porfolio Opimizaion wih Markov-modulaed sock prices and ineres raes Auomaic Conrol, IEEE rans 49 3, [2 Casañeda-Leyva, N., Hernández-Hernández, D. 25, Opimal consumpion-invesmen problems in incomplee markes wih sochasic coefficiens, SIAM J. Conrol and Opim. 44 4, [3 Cheridio, P., Hu, Y. 2, Opimal consumpion and invesmen in incomplee markes wih general consrains, Sochasics and Dynamics, [4 Cox, J., Huang, C.F. 989 Opimal consumpion and porfolio policies when asse prices follow a diffusion process, Journal of Economic heory 49, [5 El Karoui, N. 98 Les aspecs probabilises du conrôle sochasique, Lecure Noes in Mahemaics 86, , Springer Verlag. [6 Fan, S.J., Jiang, L. 2 Uniqueness resul for he BSDE whose generaor is monoonic in y and uniformly coninuous in z, C. R. Acad. Sci. Paris 348, [7 Hu, Y., Imkeller, P., Müller, M. 24 Uiliy maximizaion in incomplee markes, Annals of Applied Probabiliy 5, [8 Jeulin,. 98 Semi-maringales e grossissemen d une filraion, Lecure Noes in Mahemaics, 833, Springer. [9 Jiao, Y., Pham, H. 2 Opimal invesmen wih counerpary risk: a defaul densiy model approach, Finance and Sochasics 5, [ Karazas, I., Lehoczky, J.P., Shreve, S. 987 Opimal porfolio and consumpion decisions for a small invesor on a finie horizon, SIAM J. Conrol and Opim. 25, [ Karazas, I., Shreve, S. 998 Mehods of Mahemaical Finance, Springer Verlag, Berlin. [2 Kharroubi, I., Lim,. 2 Progressive enlargemen of filraions and backward SDEs wih jumps, forhcoming in Journal of heoreical Probabiliy. [3 Kramkov, D., Schachermayer, W. 999 he asympoic elasiciy of uiliy funcions and opimal invesmen in incomplee markes, Annals of Applied Probabiliy 9 3, [4 Kusuoka, S. 999 A remark on defaul risk models, Adv. Mah. Econom., [5 Menoncin, F. 22 A class of incomplee markes wih opimal porfolio in closed form, Working paper. [6 Rogers, L. C. G. 23 Dualiy in consrained opimal invesmen and consumpion problems: a synhesis, Paris Princeon Lecures on Mahemaical Finance 22, Lecure noes in Mahemaics 84, 95-32, Springer. 2