Dynamic Programming Approach to Principal-Agent Problems

Transcription

1 Dynamic Programming Approach o Principal-Agen Problems Jakša Cvianić, Dylan Possamaï and Nizar ouzi Sepember 23, 215 Absrac We consider a general formulaion of he Principal-Agen problem from Conrac heory, on a finie horizon. We show how o reduce he problem o a sochasic conrol problem which may be analyzed by he sandard ools of conrol heory. In paricular, Agen s value funcion appears naurally as a conrolled sae variable for he Principal s problem. Our argumen relies on he Backward Sochasic Differenial Equaions approach o non-markovian sochasic conrol, and more specifically, on he mos recen exensions o he second order case. Key words. Sochasic conrol of non-markov sysems, Hamilon-Jacobi-Bellman equaions, second order Backward SDEs, Principal-Agen problem, Conrac heory. 1 Inroducion Opimal conracing beween wo paries Principal she and Agen he, when Agen s effor canno be conraced upon, is a classical problem in Microeconomics, so-called Principal- Agen problem wih moral hazard. I has applicaions in many areas of economics and finance, for example in corporae governance and porfolio managemen see Bolon and Dewaripon 25 for a book reamen. In his paper we develop a general approach o solving such problems in Brownian moion models, in he case in which Agen is paid only a he erminal ime. he firs paper on coninuous-ime Principal-Agen problems is he seminal paper by Holmsrom and Milgrom hey consider Principal and Agen wih exponenial uiliy funcions and find ha he opimal conrac is linear. heir work was generalized by Schäler Calech, Humaniies and Social Sciences, M/C , 12 E. California Blvd. Pasadena, CA 91125, USA; cvianic@hss.calech.edu. Research suppored in par by NSF gran DMS CEREMADE, Universiy Paris-Dauphine, place du Maréchal de Lare de assigny, Paris, France; possamai@ceremade.dauphine.fr CMAP, Ecole Polyechnique, Roue de Saclay, Palaiseau, France; nizar.ouzi@polyechnique.edu. his auhor graefully acknowledge he financial suppor of he ERC Rofirm, he ANR Isoace, and he Chairs Financial Risks Risk Foundaion, sponsored by Sociéé Générale and Finance and Susainable Developmen IEF sponsored by EDF and CA. 1

2 and Sung 1993, 1997, Sung 1995, 1997, Müller 1998, 2, and Hellwig and Schmid 22. he papers by Williams 29 and Cvianić, Wan and Zhang 29 use he sochasic maximum principle and Forward-Backward Sochasic Differenial Equaions FBSDEs o characerize he opimal compensaion for more general uiliy funcions, under moral hazard, also called he hidden acions case. Cvianić and Zhang 27 and Carlier, Ekeland and ouzi 27 sudy he adverse selecion case of hidden ype, in which Principal does no observe Agen s inrinsic ype. A more recen seminal paper in moral hazard seing is Sannikov 28, who finds a racable model for solving he problem wih a random ime of reiring he agen and wih coninuous paymens o he agen, raher han a lump-sum paymen a he erminal ime. We leave for fuure research a sudy of Sannikov s model using our approach. he main approach aken in he lieraure is o characerize Agen s value funcion and his opimal acions given an arbirary conrac payoff, and hen o analyze he maximizaion 1 problem of he principal over all possible payoffs. his approach does no always work, because i may be hard o solve Agen s sochasic conrol problem given an arbirary payoff, possibly non-markovian, and i may also be hard for Principal o maximize over all such conracs. Furhermore, Agen s opimal conrol depends on he given conrac in a highly nonlinear manner, rendering Principal s opimizaion problem even harder. For hese reasons, in is mos general form he problem was approached in he lieraure also by means of he calculus of variaions, hus adaping he ools of he sochasic version of he Ponryagin maximal problem; see Cvianić and Zhang 212. Sill, none of he sandard approaches can solve he problem when Agen also conrols he diffusion coefficien if i has he dimension a leas wo, and no jus he drif. 2 Our approach is differen, and i works in grea generaliy, including he laer problem. We resric he family of admissible conracs o he conracs for which Agen would be able o solve his problem by dynamic programming. For such conracs, i is easy for Principal o idenify wha he opimal policy for Agen is - i is he one ha maximizes he corresponding Hamilonian. Moreover, he admissible family is such ha Principal can apply sandard mehods of sochasic conrol. Finally, we show ha under mild echnical condiions, Agen s supremum over our admissible conracs is equal o he supremum over all possible conracs. We accomplish ha by represening he value funcion of Agen s problem by means of he so-called second order BSDEs as inroduced by Soner, ouzi and Zhang 211, [28], see also Cheridio, Soner, ouzi and Vicoir 27. I urns ou we also need o use he recen resuls of Possamaï, an and Zhou 215, o bypass he regulariy condiions in [28]. We successfully applied his approach o he above menioned seup in which he agen conrols he diffusion vecor of he oupu process in Cvianić, Possamai and ouzi 215, a problem previously no solved in he lieraure. he res of he paper is srucured as follows: We describe he model and he Principal- Agen problem in Secion 2. We inroduce he resriced family of admissible conracs in Secion 3. Finally, we show ha he resricion is wihou loss of generaliy in Secion 4. 1 For a recen differen approach, see Evans, Miller and Yang 215. For each possible Agen s conrol process, hey characerize conracs ha are incenive compaible for i. However, heir seup is less general, and i does no allow for volailiy conrol, for example. 2 An excepion is Sung 215, which sudies a paricular case of he Principal-Agen problem in he presence of ambiguiy regarding he volailiy componen. 2

3 2 he Principal-Agen problem 2.1 he canonical space of coninuous pahs Le > be a given erminal ime, and Ω := C [, ], R d he se of all coninuous maps from [, ] o R d, for a given ineger d >. he canonical process on Ω is denoed by X, i.e. X x = x = x for all x Ω, [, ], and he corresponding raw canonical filraion by F := {F, [, ]}, where F := σx s, s, [, ]. We denoe by P he Wiener measure on Ω, F, and for any F sopping ime τ, by P τ he regular condiional probabiliy disribuion of P w.r.. F τ see Sroock and Varadhan 1979, which is acually independen of x Ω by independence and saionariy of he Brownian incremens. We say ha a probabiliy measure P on Ω, F is a semi-maringale measure if X is a semimaringale under P. hen, on he canonical space Ω, here is a F progressively measurable process see e.g. Karandikar 1995, denoed by X = X, which coincides wih he quadraic variaion of X, P a.s. for all semi-maringale measure P. We nex inroduce he d d non-negaive symmeric marix σ such ha σ 2 := lim sup ε X X ε, [, ]. ε A map Ψ : [, ] Ω E, aking values in any Polish space E will be called F progressive if Ψ, x = Ψ, x, for all [, ] and x Ω. 2.2 Conrolled sae equaion A conrol process ν = α, β is an F adaped process wih values in A B for some subses A and B of finie dimensional spaces. he conrolled process akes values in R d, and is defined by means of he conrolled coefficiens: λ : R + Ω A R n, bounded, wih λ, α F progressive for any α A, σ : R + Ω B M d,n R, bounded, wih σ, β F progressive for any β B, for a given ineger n, and where M d,n R denoes he se of d n marices wih real enries. For all conrol process ν, and all, x [, ] Ω, he conrolled sae equaion is defined by he sochasic differenial equaion driven by an n dimensional Brownian moion W, X,x,ν s = x + σ r X,x,ν, β r [ λ r X,x,ν, α r dr + dw r ], s [, ], 2.1 and such ha Xs,x,ν = xs, s [, ]. A weak soluion of 2.1 is a probabiliy measure P on Ω, F such ha P[X = x ] = 1, and X σ r X, β r λ r X, α r dr, and X X 3 σ r σ r X, β r dr,

4 are P, F maringales on [, ]. For such a weak soluion P, here is an n dimensional P Brownian moion W P, and F adaped, A B valued processes α P, β P such ha 3 X s = x + In paricular, we have he nex definiion involves an addiional map σ r X, βr P [ λ r X, αrdr P + dwr P ], s [, ], P a.s. 2.2 σ 2 = σ σ X, β P, d dp a.s. c : R + Ω A B R +, measurable, wih c, u F progressive for all u A B, which represens Agen s cos of effor. hroughou he paper we fix a real number p > 1. Definiion 2.1. A conrol process ν is said o be admissible if SDE 2.1 has a weak soluion, and for any such weak soluion P we have [ ] E P sup c s X, a, βs P p ds <. 2.3 a A We denoe by U, x he collecion of all admissible conrols, P, x he collecion of all corresponding weak soluions of 2.1, and P := x Ω P, x. Noice ha we do no resric he conrols o hose for which weak uniqueness holds. Moreover, by Girsanov heorem, wo weak soluions of 2.1 associaed wih α, β and α, β are equivalen. However, differen diffusion coefficiens induce muually singular weak soluions of he corresponding sochasic differenial equaions. For laer use, we inroduce an alernaive represenaion of ses P, x. We firs denoe for all, x [, ] Ω: Σ x, b := σ σ x, b, b B, and B x, Σ := { b B : σ σ x, b = Σ }, Σ S + d. For an F progressively measurable process β wih values in B, consider hen he SDE driven by a d dimensional Brownian moion W X,x,β s = x + Σ 1/2 r X, β r dw r, s [, ], 2.4 wih Xs,x,β = x s for all s [, ]. A weak soluion of 2.4 is a probabiliy measure P on Ω, F such ha P[X = x ] = 1, and X and X X Σ r X, β r dr, are P, F maringales on [, ]. hen, here is an F adaped process β P and some d dimensional P Brownian moion W P such ha X s = x + Σ 1/2 r X, β P r dw P r, s [, ], P a.s Brownian moion W P is defined on a possibly enlarged space, if σ is no inverible P a.s. We refer o [22] for he precise saemens. 4

5 Definiion 2.2. A diffusion conrol process β is said o be admissible if he SDE 2.4 has a weak soluion, and for all such soluion P, we have [ ] E P sup c s X, a, βs P p ds <. a A We denoe by B, x he collecion of all diffusion conrol processes, P, x he collecion of all corresponding weak soluions of 2.4, and P := x Ω P, x. We emphasize ha ses P, x are equivalen o ses P, x, in he sense ha P, x consiss of probabiliy measures which are equivalen o corresponding probabiliy measures in P, x, and vice versa. Indeed, for α, β U, x we claim ha β B, x. o see his, denoe by P α,β any of he associaed weak soluions o 2.1. hen, here always is a d n roaion marix R such ha, for any s, x, b [, ] Ω B, σ s x, b = Σ 1/2 s x, br s x, b. 2.6 Since d n, and in addiion Σ may be degenerae, noice ha here may be many and even infiniely many choices of R, and in his case we may choose any measurable one. We nex define P β by dp β := E dpα,β λ s X, αs Pα,β dws Pα,β By Girsanov heorem, X is hen a P β, F maringale, which ensures ha β B, x. In paricular, he polar ses of P and P are he same. Conversely, le us fix β B, x and denoe by A he se of A valued and F progressively measurable processes. hen, we claim ha for any α A, we have α, β U, x. Indeed, le us denoe by P β any weak soluion o 2.4 and define dp α,β dp β := E Pβ R s X, β s λ s X, α s dws Pβ. hen, by Girsanov heorem, we have X s = = Σ 1/2 r X, σ r X, Pβ Pβ β r R r X, β β Pβ r λ r X, α r dr + r λ r X, α r dr + Σ 1/2 r X,. Σ 1/2 r X, Pβ β r dw Pβ r, Pβ β r dw Pβ r where W Pβ is a d dimensional P α,β, F Brownian moion. Hence, α, β U, x. Moreover, seing W Pβ Pβ = R X, β W Pα,β + R s X, Pβ β s λ s X, α s ds, defines a Brownian moion under P α,β. Since P β and P α,β are equivalen, we have ΣX, β Pα,β = ΣX, β Pβ, d P β a.e. or d P α,β a.e., 5

6 ha is βs Pα,β Pβ and β s boh belong o B s X, σ sx, 2 d P β a.e. We can summarize everyhing by he following equaliy P, x = α A 2.3 Agen s problem { E R s X, β } s P λ s X, α s dws P P : P P, x, R saisfying In he following discussion, we fix, x, P [, ] Ω P, x, ogeher wih he associaed conrol ν P := α P, β P. In our Principal-Agen problem, he canonical process X is called he oupu process, and he conrol ν P is usually referred o as Agen s effor or acion. Agen is in charge of conrolling he disribuion of he oupu process by opimally choosing he effor process ν P in he sae equaion 2.1, while subjec o cos of effor a rae cx, α P, β P. Furhermore, Agen has a fixed reservaion uiliy denoed by R R, i.e., he will no accep o work for Principal unless he conrac is such ha his value funcion is above R. he conrac agreemen holds during he ime period [, ]. Agen is only cares abou he compensaion ξ received from Principal a ime. Principal does no observe Agen s effor, only he oupu process. Consequenly, he compensaion ξ, which akes values in R, can only be coningen on X, ha is ξ is F measurable. Random variable ξ is called a conrac on [, ], and we wrie ξ C if he following inegrabiliy condiion is saisfied: We now inroduce Agen s objecive funcion: where J A, x, P, ξ := E P[ K νp, Xξ sup E P [ ξ p ] < P P K,sX ν := exp K νp,sxc s X, ν P s ds ], P P, x, ξ C, 2.9 k r X, ν r dr, s [, ], is a discoun facor defined by means of a bounded measurable funcion k : R + Ω A B R, wih k, u F progressive for all u A B. Noice ha J A is well-defined for all, x [, ] Ω, ξ C and P P, x. his is a consequence of he boundedness of k, he non-negaiviy of c, as well as he condiions 2.8 and 2.3. Remark 2.3. If Agen is risk-averse wih uiliy funcion U A, hen we replace ξ wih ξ = U A ξ in J A, and we replace ξ by U 1 A ξ in Principal s problem below. All he resuls remain valid. Agen s goal is o choose opimally he amoun of effor, given he compensaion conrac ξ promised by Principal: V A, x, ξ := sup J A, x, P, ξ. 2.1 P P,x 6

7 An admissible conrol P P, x will be called opimal if V A, x, ξ = J A, x, P, ξ. We denoe by P, x, ξ he collecion of all such opimal conrols P. In he economics lieraure, he dynamic value funcion V A is called coninuaion uiliy or promised uiliy, and i urns ou o play a crucial role as he sae variable of Principal s opimizaion problem; see Sannikov 28 for is use in coninuous-ime models for and furher references. 2.4 Principal s problem We now define he opimizaion problem of choosing he compensaion conrac ξ ha Principal should offer o Agen. A he mauriy, Principal receives he final value of he oupu X and pays he compensaion ξ promised o Agen. Principal only observes he oupu resuling from Agen s opimal sraegy. We resric he conracs proposed by Principal o hose ha admi a soluion o Agen s problem, i.e., we allow only he conracs ξ for which P, x, ξ. Recall also ha he paricipaion of Agen is condiioned on having his value funcion above reservaion uiliy R. For his reason, Principal is resriced o choose a conrac from he se Ξ, x := { ξ C, P, x, ξ, V A, x, ξ R } As a final ingredien, we need o fix Agen s opimal sraegy in he case in which se P, x, ξ conains many soluions. Following he sandard Principal-Agen lieraure, we assume ha Agen, when indifferen beween such soluions, implemens he one ha is he bes for Principal. where In view of his, Principal s problem reduces o J P, x, ξ := V P, x := sup J P, x, ξ, 2.12 ξ Ξ,x sup E P [ K, P XU lx ξ ], P P,x,ξ where funcion U : R R is a given non-decreasing and concave uiliy funcion, l : R d R is a liquidaion funcion, and K,sX P := exp kr P Xdr, s [, ], is a discoun facor defined by means of a bounded measurable funcion such ha k P is F progressive. k P : R + Ω R, 7

8 Remark 2.4. Agen s and Principal s problems are non-sandard. Firs, ξ is allowed o be of non-markovian naure. Second, Principal s opimizaion is over ξ, and is a priori no in a conrol problem ha may be approached by dynamic programming. he objecive of his paper is o develop an approach ha naurally reduces he problems o hose ha can be solved by dynamic programming. We used his approach in our previous paper [4], albei in a much less general seing. 3 Resriced conracs In his secion we idenify a resriced family of conrac payoffs for which he sandard sochasic conrol mehods can be applied. 3.1 Agen s dynamic programming equaion In view of he definiion of Agen s problem in 2.1, i is naural o inroduce he Hamilonian funcional, for all, x [, Ω and y, z, γ R R d S d R: H x, y, z, γ := sup h x, y, z, γ, u, 3.1 u A B h x, y, z, γ, u := c x, u k x, uy + σ x, βλ x, α z σ σ x, β : γ, 3.2 for u := α, β. Remark 3.1. i he map H plays an imporan role in he heory of sochasic conrol of Markov diffusions, see e.g. Fleming and Soner Indeed, suppose ha he coefficiens λ, σ, c, k depend on x only hrough he curren value x, he conrac ξ depends on x only hrough he final value x, i.e. ξx = gx for some funcion g : R d R. hen, under fairly general condiions, he value funcion of Agen s problem is idenified by V A, x, ξ = v, x, where he funcion v : [, ] R d R can be characerized as he unique viscosiy soluion wih appropriae growh a infiniy of he dynamic programming equaion v, x H, x, v, x, Dv, x, D 2 v, x =,, x [, R d, v, x = gx, x R d. ii he recenly developed heory of pah-dependen parial differenial equaions exends he firs iem of his remark o he non-markovian case. See Ekren, ouzi and Zhang 214. Noe ha Agen s value funcion process V a he erminal ime is equal o he conrac payoff, ξ = V. his will moivae us o consider payoffs ξ of a specific form. he main resul of his secion follows he line of he sandard verificaion resul in sochasic conrol heory. Fix some, x [, ] Ω. Le be F predicable processes wih E P [ Z : [, ] Ω R d and Γ : [, ] Ω S d R [ Zs Z s : σ 2 s + Γ s : σ 2 s ] ds p 2 ] < +, for all P P, x, 8

9 We denoe by V, x he collecion of all such pairs of processes Z, Γ. Given an iniial condiion Y R, define he F progressively measurable process Y Z,Γ, P a.s., for all P P, x by Y Z,Γ s := Y H r X, Y Z,Γ r s, Z r, Γ r dr + Z r dx r Γ r :d X r, s [, ]. 3.3 Noice ha Y Z,Γ is well-defined as a consequence of he Lipschiz propery of H in y, resuling from he boundedness of k. he nex resul follows he line of he classical verificaion argumen in sochasic conrol heory, and requires he following condiion. Assumpion 3.2. For any [, ], he map H has a leas one measurable maximizer u = α, β : [, Ω R R d S d R A B, i.e. H. = h., u.. Moreover, for all, x [, ] Ω and for any Z, Γ V, x, he conrol process is admissible, ha is ν,z,γ U, x. νs,z,γ := u s, Y Z,Γ s, Z s, Γ s, s [, ], We are now in a posiion o provide a subse of conracs which, when proposed by Principal, have a very useful propery of revealing Agen s opimal effor. Under his se of revealing conracs, Agen s value funcion coincides wih he above process Y Z,Γ. Proposiion 3.3. For, x [, ] Ω, Y R and Z, Γ V, x, we have: i Y V A, x, Y Z,Γ.. More- ii Assuming furher ha Assumpion 3.2 holds rue, we have Y = V A, x, Y Z,Γ over, given a conrac payoff ξ = Y Z,Γ, any weak soluion P,Y,Z of he SDE 2.1 wih conrol ν,z,γ is opimal for he Agen problem, i.e. P ν,z,γ P, x, Y Z,Γ. Proof. i Fix an arbirary P P, x, and denoe he corresponding conrol process ν P := α P, β P. hen, i follows from a direc applicaion of Iô s formula ha [ ] E P K, νp Y Z,Γ = Y + E P [ K,sX νp k s X, νs P Ys Z,Γ H s X, Ys Z,Γ, Z s, Γ s +Z s σ s X, βs P λx, αs P + 1 ] 2 σ2 s :Γ s ds, where we have used he fac ha Z, Γ V, x, ogeher wih he fac ha he sochasic inegral KνP,sXZ s σ sdw 2 s P defines a maringale, by he boundedness of k and σ. By he definiion of he Hamilonian H in 3.1, we may re-wrie he las equaion as [ ] E P K, νp Y Z,Γ = Y + E P[ K,sX νp c s X, ν P H s X, Y Z,Γ s, Z s, Γ s +h s X, Ys Z,Γ, Z s, Γ s, νs P ] ds [ ] Y + E P K,sXc νp s X, αs, P βs P ds, 9

10 and he resul follows by arbirariness of P P, x. ii Le ν := ν,z,γ for simpliciy. Under Assumpion 3.2, he exac same calculaions as in i provide for any weak soluion P ν [ ] [ ] E Pν K νpν,,x Y Z,Γ = Y + E Pν K,s νpν Xc s X, αs Pν, βs P ds., ogeher wih i, his shows ha Y = V A, x, Y Z,Γ 3.2 Resriced Principal s problem, and P ν P, x, Y Z,Γ. Recall he process u x, y, z, γ = α, β x, y, z, γ inroduced in Assumpion 3.2. In his secion, we denoe λ x, y, z, γ := λ x, α x, y, z, γ, σ x, y, z, γ := σ x, β x, y, z, γ. 3.4 Noice ha Assumpion 3.2 says ha for all, x [, ] Ω and for all Z, Γ V, x, he sochasic differenial equaion, driven by a n dimensional Brownian moion W X,x,u s = x + σrx,x,u, Yr Z,Γ, Z r, Γ r [ λ rx,x,u, Yr Z,Γ ], Z r, Γ r dr + dw r, s [, ], X,x,u s = xs, s [, ], 3.5 has a leas one weak soluion P,Z,Γ. he following resul on Principal s value funcion V P when he conrac payoff is ξ = Y Z,Γ is a direc consequence of Proposiion 3.3. Proposiion 3.4. For all, x [, ] Ω, we have V P, x sup Y R V, x, Y, where, for Y R: V, x, Y := E P,Z,Γ[ K, P U lx Y Z,Γ ]. 3.6 sup Z,Γ V,x sup ˆP Z,Γ P,x,Y Z,Γ We coninue his secion by a discussion of he bound V, x, y which represens Principal s value funcion when he conracs are resriced o he F measurable random variables Y Z,Γ wih given iniial condiion Y. In he secions below, we idenify condiions under which ha resricion is wihou loss of generaliy. Clearly, V is he value funcion of a sandard sochasic conrol problem wih conrol processes Z, Γ V, x, and conrolled sae process X, Y Z,Γ, he conrolled dynamics of X being given in weak formulaion by 3.5, and hose of Y Z,Γ given by 3.3: dys Z,Γ = Z s σsλ s Γ s :σsσ s H s X, Ys Z,Γ, Z s, Γ s ds + Z s σsx, Ys Z,Γ, Z s, Γ s dws P. 3.7 In view of he conrolled dynamics , he relevan opimizaion erm for he dynamic programming equaion corresponding o he conrol problem V is defined for, x, y [, ] R d R by: G, x, y, p, M { := sup z,γ R S d R σ λ x, y, z, γ p x + z σ λ + 12 γ :σ σ H x, y, z, γp y σ σ x, y, z, γ: M xx + zz M yy + σ σ x, y, z, γz M xy }, 1

11 Mxx M where M =: xy Mxy S d+1 R, M xx S d R, M yy R, M xy M d,1 R and p =: M yy px R d R. p y heorem 3.5. Le ϕ x,. = ϕ x,. for ϕ = k, k P, λ, σ, H, and le Assumpion 3.2 hold rue. Assume furher ha he map G : [, R d R d+1 S d+1 R R is upper semiconinuous. hen, V, x, y is a viscosiy soluion of he dynamic programming equaion: { v k P v, x, y + G, x, v, x, y, Dv, x, y, D 2 v, x, y =,, x, y [, R d R, v, x, y = Ulx y, x, y R d R. he las saemen is formulaed in he Markovian case, i.e. when he model coefficiens are no pah-dependen. A similar saemen can be formulaed in he pah dependen case, by using he noion of viscosiy soluions of pah-dependen PDEs inroduced in Ekren, Keller, ouzi & Zhang 214, and furher developed in [9, 1, 23, 24]. However, one hen faces he problem of unboundedness of he conrols z, γ, which ypically leads o a non-lipschiz G in erms of he variables Dv, D 2 v, unless addiional condiions on he coefficiens are inroduced. 4 Comparison wih he unresriced case In his secion we find condiions under which equaliy holds in Proposiion 3.4, i.e. he value funcion of he resriced Principal s problem of Secion 3.2 coincides wih Principal s value funcion wih unresriced conracs. We sar wih he case in which he diffusion coefficien is no conrolled. 4.1 Fixed volailiy of he oupu We consider here he case in which Agen is only allowed o conrol he drif of he oupu process: B = {β o } for some fixed β o U, x. 4.1 Le P βo be any weak soluions of he corresponding SDE 2.4. he main ool for our main resul is he use of Backward SDEs. his requires inroducing filraion F Pβo +, defined as he P β compleion of he righ-limi of F, 4 under which he predicable maringale represenaion propery holds rue. In he presen seing, all probabiliy measures P P, x are equivalen o P βo. Consequenly, he expression of he process Y in 3.3 only needs o be solved under P β, and reduces o Ys Z := Ys Z, = Y Fr X, Y Z,Γ r s, Z r dr + Z r dx r, s [, ], P βo a.s., 4.2 where he dependence on he process Γ simplifies immediaely, and F x, y, z := { sup c x, α, b k x, α, by + σ x, β x λ x, α z }. 4.3 α A 4 For a semimaringale probabiliy measure P, we denoe by F + := s>f s is righ-coninuous limi, and by F P + he corresponding compleion under P. he compleed righ-coninuous filraion is denoed by F P +. 11

12 heorem 4.1. Le Assumpion 3.2 hold. In he seing of 4.1, assume in addiion ha P βo, F Pβo + saisfies he predicable maringale represenaion propery and he Blumenhal zeroone law. hen, V P, x = sup V, x, y, for all, x [, ] Ω. y R Proof. For all ξ Ξ, x, we observe ha condiion 2.8 guaranees ha ξ L p P β. o prove ha he required equaliy holds, i is sufficien o show ha all such ξ can be represened in erms of a conrolled diffusion Y Z,. However, we have already seen ha F is uniformly Lipschiz-coninuous in y, z, since k, σ and λ are bounded, and by definiion of admissible conracs, we have also ha [ ] E Pβo F X,, p <, hen, he sandard heory see for insance [22] guaranees ha he BSDE Y = ξ + F r X, Y r, Z r dr Z r σ r X, β o r dw Pβo r, is well-posed, because we also have ha P βo saisfies he predicable maringale represenaion propery. Moreover, we hen have auomaically Z, V, x. his implies ha ξ can indeed be represened by he process Y which is of he form he general case he purpose of his secion is o exend heorem 4.1 o he case in which Agen conrols boh he drif and he diffusion of he oupu process X. Similarly o he previous secion, he criical ool is he heory of Backward SDEs, bu suiable for pah-dependen sochasic conrol problems. he addiional conrol of he volailiy requires o invoke he recen exension of Backward SDEs o he second order case. his needs addiional noaion, as follows. Le M denoe he collecion of all probabiliy measures on Ω, F. he universal filraion F U = F U is defined by F U := P M F P, [, ], and we denoe by F U +, he corresponding righ-coninuous limi. Moreover, for a subse P M, we inroduce he se of P polar ses N P := { N Ω : N A for some A F wih sup P P PA = }, and we inroduce he P compleion of F F P := F P [, ], wih F P := F U σ N P, [, ], ogeher wih he corresponding righ-coninuous limi F P +. Finally, for echnical reasons, we work under he ZFC se-heoreic axioms, as well as he axiom of choice and he coninuum hypohesis 5. 5 Acually, we do no need he coninuum hypohesis, per se. Indeed, we only wan o be able o use he main resul of Nuz 212, which only requires axioms ensuring he exisence of medial limis. We make his choice here for ease of presenaion. 12

13 BSDE characerizaion of Agen s problem We now provide a represenaion of Agen s value funcion by means of he so-called second order BSDEs, or 2BSDEs as inroduced by Soner, ouzi and Zhang 211, [28] see also Cheridio, Soner, ouzi and Vicoir 27. Furhermore, we use crucially recen resuls of Possamaï, an and Zhou 215 o bypass he regulariy condiions in [28]. We firs re-wrie he map H in 3.1 as: H x, y, z, γ = sup β B F x, y, z, Σ := sup α,β A B x,σ { F x, y, z, Σ x, β Σ x, β:γ }, { c x, α, β k x, α, βy + σ x, βλ x, α z }. We consider a reformulaion of Assumpion 3.2 in his seing: Assumpion 4.2. he map F has a leas one measurable maximizer u = α, β : [, Ω R R d S + d A B, i.e. F, y, z, Σ = c, α, β k, α, β y + σ, β λ, α z. Moreover, for all, x [, ] Ω, and for all admissible conrols β U, x, he conrol process is admissible, ha is ν,y,z,β U, x. We also need he following condiion. ν,y,z,β s := α s, β s Y s, Z s, Σ s β s, s [, ], Assumpion 4.3. For any, x, β [, ] Ω B, he marix σ σ x, β is inverible, wih a bounded inverse. he following lemma shows ha he ses P, x saisfy naural properies. Lemma 4.4. he family {P, x,, x [, ] Ω} is sauraed, and saisfies he dynamic programming requiremens of Assumpion 2.1 in [22]. Proof. Consider some P P, x and some P under which X is a maringale, and which is equivalen o P. hen, he quadraic variaion of X under P is he same as is quadraic variaion under P, ha is σ sσs X, β s ds. By definiion, P is herefore a weak soluion o 2.4 and belongs o P, x. he dynamic programming requiremens of Assumpion 2.1 in [22] follow from he more general resuls given in [12, 13]. Given an admissible conrac ξ, we consider he following sauraed 2BSDE in he sense of Secion 5 of [22]: Y = ξ + F s X, Y s, Z s, σ sds 2 Z s dx s + dk s, 4.4 where Y is F P + progressively measurable process, Z is an F P predicable process, wih appropriae inegrabiliy condiions, and K is an F P opional non-decreasing process wih K =, and saisfying he minimaliy condiion K = essinf P [ ] E P K F P+,, P a.s. for all P P. 4.5 P P,P,F + 13

14 Noice ha, in conras wih he 2BSDE definiion in [28, 22], we are using here an aggregaed non-decreasing process K. his is a consequence of he general aggregaion resul of sochasic inegrals in [21]. Since k, σ, λ are bounded, and σσ is inverible wih a bounded inverse, i follows from he definiion of admissible conrols ha F saisfies he inegrabiliy and Lipschiz coninuiy assumpions required in [22], ha is for some κ [1, p and for any s, x, y, y, z, z, a [, ] Ω R 2 R 2d S + d F s x, y, z, a F s x, y, z, a C y y + a 1/2 z z, sup E P P P essup P E P [ ] p κ F s X,,, σ s 2 κ F + < +. hen, in view of Lemma 4.4, he well-posedness of he sauraed 2BSDE 4.4 is a direc consequence of heorems 4.1 and 5.1 in [22]. We use 2BSDEs 4.4 because of he following represenaion resul. Proposiion 4.5. Le Assumpions 4.2 and 4.3 hold. hen, we have V A, x, ξ = sup E P [Y ]. P P,x Moreover, ξ Ξ, x if and only if here is an F adaped process β wih values in B, such ha ν := a X, Y, Z, Σ X, β, β U, x, and for any associaed weak soluion P β of 2.4. K =, P β a.s. Proof. By heorem 4.2 of [22], we know ha we can wrie he soluion of he 2BSDE 4.4 as a supremum of soluions of BSDEs, ha is Y = essup P Y P P P,P,F + where for any P P and any s [, ],, P a.s. for all P P, Ys P = ξ + F s X, Y r, Z r, σ rdr 2 Zr P dx r dm P r, P a.s. s s s wih a càdlàg F P +, P maringale M P orhogonal o W P. For any P P, le B σ 2, P denoe he collecion of all conrol processes β wih β B X, σ 2, d P a.e. For all P, α P A, and β BX, σ 2, P, we nex inroduce he backward SDE Y P,α,β s = ξ + s s cr X, α r, β r k r X, α r, β r Y P,α,β r Z P,α,β r dx r s dm P,α,β r, P a.s σ r X, β r λ r X, α r Zr P,α,β dr

15 Le P α,β be he probabiliy measure, equivalen o P, defined by dp α,β dp := E R s X, β s λ s X, α s dws P. hen, he soluion of he las linear backward SDE is given by: [ Y P,α,β = E Pα,β K α,β, Xξ K α,β,s Xc s X, α1s, β s ds F + ], P a.s. Following El Karoui, Peng & Quenez 1997, i follows from he comparison resul for BSDEs ha he processes Y P,α,β induce a sochasic conrol represenaion for Y P see also Lemma A.3 in [22]. his is jusified by Assumpion 4.2, and we now obain ha: his implies ha Y P = essup P α,β A B σ 2,P [ Y P = essup P E Pα,β α,β A B σ 2,P Y P,α,β, P a.s., for any P P. K α,β, Xξ and herefore for any P P, we have P a.s. [ Y = essup P E P α,β P,α,β P,P,F + A B σ 2,P [ = essup P E P K, νp Xξ P P,P,F + K α,β, Xξ K α,β,s Xc s X, αs, β s ds F + K,s νp Xc s X, α P s, βs P ds F + K α,β,s Xc s X, αs, β s ds F + where we have used he connecion beween P and P recalled a he end of Secion 2.1. he desired resul follows by classical argumens similar o he ones used in he proofs of Lemma 3.5 and heorem 5.2 of [22]. By he above equaliies, ogeher wih Assumpion 4.2, i is clear ha a probabiliy measure P P, x is in P, x, ξ if and only if ν = a, β X, Y, Z, Σ, ], ], ] where Σ is such for any associaed weak soluion P β o 2.4, we have K Pβ =, P β a.s. 4.3 he main resul heorem 4.6. Le Assumpions 3.2, 4.2, and 4.3 hold rue. hen V P, x = sup V, x, y for all, x [, ] Ω. y R 15

16 Proof. he inequaliy V P, x sup y R V, x, y was already saed in Proposiion 3.4. o prove he converse inequaliy we consider an arbirary ξ Ξ, x and we inend o prove ha Principal s objecive funcion J P, x, ξ can be approximaed by J P, x, ξ ε, where ξ ε = Y Zε,Γ ε for some Z ε, Γ ε V, x. Sep 1: Le Y, Z, K be he soluion of he 2BSDE 4.4 Y = ξ + F s, X, Y s, Z s, σ 2 sds Z s dx s + dk s, where we recall again ha he aggregaed process K exiss as a consequence of he aggregaion resul of Nuz [21], see Remark 4.1 in [22]. By Proposiion 4.5, we know ha for every P P, x, ξ, we have K =, P a.s. For all ε >, define he absoluely coninuous approximaion of K: K ε := 1 ε ε K s ds, [, ]. Clearly, K ε is F P predicable, non-decreasing P q.s. and We nex define for any [, ] he process Y ε := Y K ε =, P a.s. for all P P, x, ξ. 4.6 F s X, Y ε s, Z s, σ 2 sds + Z s dx s dk ε s, 4.7 and verify ha Y ε, Z, K ε solves he 2BSDE 4.4 wih erminal condiion ξ ε := Y ε and generaor F. his requires o check ha K ε saisfies he required minimaliy condiion, which is obvious by 4.6. Sep 2: For, x, y, z [, ] Ω R R d, noice ha he map γ H x, y, z, γ F x, y, z, σ 2 x 1 2 σ2 x : γ is valued in R +, convex, coninuous on he inerior of is domain, aains he value by Assumpion 3.2, and is coercive by he boundedness of λ, σ, k. hen, his map is surjecive on R +. Le K ε denoe he densiy of he absoluely coninuous process K ε wih respec o he Lebesgue measure. Applying a classical measurable selecion argumen, we may deduce he exisence of an F predicable process Γ ε such ha K ε s = H s X, Ȳ ε s, Z s, Γ ε s F s X, Ȳ ε s, Z s, σ 2 s 1 2 σ2 s : Γ ε s. Subsiuing in 4.7, i follows ha he following represenaion of Y ε Y ε = Y H s X, Y ε s, Z s, Γ ε sds + Z s dx s holds: Γ ε s :d X s. Sep 3: he conrac ξ ε := Y ε akes he required form 3.3, for which we know how o solve Agen s problem, i.e. V A, x, ξ ε = Y, by Proposiion 3.3. Moreover, i follows from 4.6 ha ξ = ξ ε, P a.s. 16

17 Consequenly, for any P P, x, ξ, we have E P [ K P, UlX ξ ε ] = E P [ K P, UlX ξ ], which implies ha J P, x, ξ = J P, x, ξ ε. 4.4 Example: coefficiens independen of X In his secion, we consider he paricular case in which σ, λ, c, k, and k P are independen of x. 4.8 In his case, he Hamilonian H inroduced in 3.1 is also independen of x, and we re-wrie he dynamics of he conrolled process Y Z,Γ as: Y Z,Γ s := Y H r Y Z,Γ r s, Z r, Γ r dr + Z r dx r Γ r :d X r, s [, ]. By classical comparison resul of sochasic differenial equaion, his implies ha he flow Y Z,Γ s is increasing in erms of he corresponding iniial condiion Y. hus, opimally, Principal will provide Agen wih he minimum reservaion uiliy R he requires. In oher words, we have he following simplificaion of Principal s problem, as a direc consequence of heorem 4.6. Proposiion 4.7. Le Assumpions 3.2, 4.2, and 4.3 hold rue. hen in he conex of 4.8, we have: V P, x = V, x, R for all, x [, ] Ω. We conclude he paper wih wo addiional simplificaions of ineres. Example 4.8 Exponenial uiliy. i Le Uy := e ηy, and consider he case k. hen under he condiions of Proposiion 4.7, i follows ha V P, x = e ηr V, x, for all, x [, ] Ω. Consequenly, he HJB equaion of heorem 3.5, corresponding o V, may be reduced o [, ] R d by applying he change of variables v, x, y = e ηy f, x. ii Assume in addiion ha he liquidaion funcion lx = h x is linear for some h R d. hen, i follows ha V P, x = e ηh x R V,, for all, x [, ] Ω. Consequenly, he HJB equaion of heorem 3.5, corresponding o V, may be reduced o an ODE on [, ] by applying he change of variables v, x, y = e ηh x R f. Example 4.9 Risk-neural Principal. Le Ux := x, and consider he case k. hen under he condiions of Proposiion 4.7, i follows ha V P, x = R + V, x, for all, x [, ] Ω. Consequenly, he HJB equaion of heorem 3.5, corresponding o V, may be reduced o [, ] R d by applying he change of variables v, x, y = y + f, x. 17

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