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Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Continuous-time Principal-Agent problem, and second order backward SDEs Nizar Touzi Ecole Polytechnique, France with Jaksa Cvitanić, Dylan, Junjian Yang, Possamaï, Yiqin Lin, Zhenjie Ren Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Outline 1 Finite horizon Principal-Agent problem 2 Solving the representation problem by 2BSDEs Case of un-controlled diffusion General controlled diffusion case 3 Infinite horizon Principal-Agent problem Principal-Agent Problem formulation in infinite horizon Random horizon backward SDEs Random horizon backward SDEs Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Moral hazard • Adam Smith (1723-1790) : moral hazard is a major risk in economics : Situation where an agent may benefit from an action whose cost is supported by others Should not count on agents’ morality... Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem (Static) Principal-Agent Problem • Principal delegates management of output process X , only observes X pays salary defined by contract ξ(X ) • Agent devotes effort a =⇒ X a , chooses optimal effort by VA (ξ) := max E UA ξ(X a ) − c(a) =⇒ â(ξ) a • Principal chooses optimal contract by solving max E UP X â(ξ) − ξ(X â(ξ) ) under constraint VA (ξ) ≥ R ξ Non-zero sum Stackelberg game Easier to address in continuous time... Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem (Static) Principal-Agent Problem • Principal delegates management of output process X , only observes X pays salary defined by contract ξ(X ) • Agent devotes effort a =⇒ X a , chooses optimal effort by VA (ξ) := max E UA ξ(X a ) − c(a) =⇒ â(ξ) a • Principal chooses optimal contract by solving max E UP X â(ξ) − ξ(X â(ξ) ) under constraint VA (ξ) ≥ R ξ =⇒ Non-zero sum Stackelberg game Easier to address in continuous time Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem (Static) Principal-Agent Problem • Principal delegates management of output process X , only observes X pays salary defined by contract ξ(X ) • Agent devotes effort a =⇒ X a , chooses optimal effort by VA (ξ) := max E UA ξ(X a ) − c(a) =⇒ â(ξ) a • Principal chooses optimal contract by solving max E UP X â(ξ) − ξ(X â(ξ) ) under constraint VA (ξ) ≥ R ξ =⇒ Non-zero sum Stackelberg game Easier to address in continuous time Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem (Static) Principal-Agent Problem • Principal delegates management of output process X , only observes X pays salary defined by contract ξ(X ) • Agent devotes effort a =⇒ X a , chooses optimal effort by VA (ξ) := max E UA ξ(X a ) − c(a) =⇒ â(ξ) a • Principal chooses optimal contract by solving max E UP X â(ξ) − ξ(X â(ξ) ) under constraint VA (ξ) ≥ R ξ =⇒ Non-zero sum Stackelberg game Easier to address in continuous time Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal-Agent problem formulation Fix Ω := C 0 ([0, T ], Rd ), paths space for output process X , and let Z h V0A (ξ) := sup EP KT ξ(X ) − 0 P∈P T i Rt ν Kt ct (νtP )dt , Kt := e − 0 ks ds where P ∈ P weak solution of Output process : dXt = bt (X , νt )dt + σt (X , νt )dWtP P − a.s. • Given solution P∗ (ξ), Principal solves the optimization problem h i Rt ∗ V0P := sup E P (ξ) HT U(`(X ) − ξ(X )) , Ht := e − 0 hs ds ξ∈ΞR where ΞR := {ξ(X. ), such that V0A (ξ) ≥ R} Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal-Agent problem formulation Fix Ω := C 0 ([0, T ], Rd ), paths space for output process X , and let Z h V0A (ξ) := sup EP KT ξ(X ) − P∈P 0 T i Rt ν Kt ct (νtP )dt , Ktν := e − 0 ks ds where P ∈ P weak solution of Output process : dXt = σt (X , βt ) λt (X , αt )dt + dWtP P − a.s. • Given solution P∗ (ξ), Principal solves the optimization problem h i Rt ∗ V0P := sup E P (ξ) HT U(`(X ) − ξ(X )) , Ht := e − 0 hs ds ξ∈ΞR where ΞR := {ξ(X. ), such that V0A (ξ) ≥ R} Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal-Agent problem formulation Fix Ω := C 0 ([0, T ], Rd ), paths space for output process X , and let Z h V0A (ξ) := sup EP KT ξ − P∈P T 0 i Rt ν Kt ct (νtP )dt , Ktν := e − 0 ks ds where P ∈ P weak solution of Output process : dXt = σt (X ,βt ) λt (X ,αt )dt + dWtP , P − a.s. • Given solution P∗ (ξ), Principal solves the optimization problem h i Rt ∗ V0P := sup E P (ξ) KT U(`(X ) − ξ(X )) , Ht := e − 0 hs ds ξ∈ΞR where ΞR := {ξ, such that V0A (ξ) ≥ R} Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal-Agent problem formulation Fix Ω := C 0 ([0, T ], Rd ), paths space for output process X , and let Z h V0A (ξ) := sup EP KT UA (ξ) − P∈P 0 T i Rt ν Kt ct (νtP )dt , Ktν := e − 0 ks ds where P ∈ P weak solution of Output process : dXt = σt (X ,βt ) λt (X ,αt )dt + dWtP , P − a.s. • Given solution P∗ (ξ), Principal solves the optimization problem h i Rt ∗ V0P := sup E P (ξ) KT U(`(X ) − ξ(X )) , Ht := e − 0 hs ds ξ∈ΞR where ΞR := {ξ, such that V0A (ξ) ≥ R} Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Continuous time Principal-Agent literature Holström & Milgrom 1987 .. . Cvitanić & Zhang 2012 : book Sannikov 2008 Uncontrolled diffusion, 1−dimensional state... closed form solution Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem A subset of revealing contracts • Path-dependent Hamiltonian for the Agent problem : Ht (ω, y , z, γ) := supu bt (ω, u) · z + 12 σt σt> (ω, u) : γ −kt (ω, u)y − ct (ω, u) • For Y0 ∈ R and Z , Γ FX − prog meas, define YtZ ,Γ Z = Y0 + 0 T 1 Zt ·dXt + Γt : dhX it −Ht (X , YtZ ,Γ , Zt , Γt )dt, P − q.s. 2 Proposition VA YTZ ,Γ = Y0 . Moreover P∗ is optimal iff νt∗ = ArgmaxHt (Yt , Zt , Γt ) • Hence νt∗ = ν̂(Yt , Zt , Γt ) Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem A subset of revealing contracts • Path-dependent Hamiltonian for the Agent problem : Ht (ω, y , z, γ) := supu bt (ω, u) · z + 12 σt σt> (ω, u) : γ −kt (ω, u)y − ct (ω, u) • For Y0 ∈ R and Z , Γ FX − prog meas, define YtZ ,Γ Z = Y0 + 0 T 1 Zt ·dXt + Γt : dhX it −Ht (X , YtZ ,Γ , Zt , Γt )dt, P − q.s. 2 Proposition VA YTZ ,Γ = Y0 . Moreover P∗ is optimal iff νt∗ = ArgmaxHt (Yt , Zt , Γt ) • Hence νt∗ = ν̂(Yt , Zt , Γt ) Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Proof : classical verification argument ! RT For all P ∈ P, denote JA (ξ, P) := EP KT ξ − 0 Kt ctν dt . Then JA YTZ ,Γ , P P = Y0 − E hZ 0 T Kt Ht (Yt , Zt , Γt ) − Zt · σtν dWtP i 1 +ctν + ktν Yt − btν · Zt − σσ > : Γt dt 2 ≤ Y0 with equality iff ν ∗ achieves the max of Hamiltonian Only requirement for (Z , Γ) : Z ∈ ∩P∈P H2 (P) Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Proof : classical verification argument ! RT For all P ∈ P, denote JA (ξ, P) := EP KT ξ − 0 Kt ctν dt . Then JA YTZ ,Γ , P P = Y0 − E hZ 0 T Kt Ht (Yt , Zt , Γt )−Zt · σtν dWtP i 1 +ctν + ktν Yt − btν · Zt − σσ > : Γt dt 2 ≤ Y0 with equality iff ν ∗ achieves the max of Hamiltonian Only requirement for (Z , Γ) : Z ∈ H2 (P) := ∩P∈P H2 (P) Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Proof : classical verification argument ! RT For all P ∈ P, denote JA (ξ, P) := EP KT ξ − 0 Kt ctν dt . Then JA YTZ ,Γ , P P = Y0 − E hZ 0 T Kt Ht (Yt , Zt , Γt )−Zt · σtν dWtP i 1 +ctν + ktν Yt − btν · Zt − σσ > : Γt dt 2 ≤ Y0 with equality iff ν ∗ achieves the max of Hamiltonian Only requirement for (Z , Γ) : Z ∈ H2 (P) := ∩P∈P H2 (P) Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Proof : classical verification argument ! RT For all P ∈ P, denote JA (ξ, P) := EP KT ξ − 0 Kt ctν dt . Then JA YTZ ,Γ , P P = Y0 − E hZ 0 T Kt Ht (Yt , Zt , Γt )−Zt · σtν dWtP i 1 +ctν + ktν Yt − btν · Zt − σσ > : Γt dt 2 ≤ Y0 with equality iff ν ∗ achieves the max of Hamiltonian Only requirement for (Z , Γ) : Z ∈ H2 (P) := ∩P∈P H2 (P) Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal problem restricted to revealing contracts Dynamics of the pair (X , Y ) under “optimal response” 1 dXt = ∇z Ht (X , YtZ ,Γ , Zt , Γt )dt + 2∇γ Ht (X , YtZ ,Γ , Zt , Γt ) 2 dWt dYtZ ,Γ = Zt · dXt + 12 Γt : dhX it − Ht (X , YtZ ,Γ , Zt , Γt )dt Principal’s value function under revealing contracts : VP ≥ V0 (X0 , Y0 ) := h i sup E U `(X ) − YTZ ,Γ , ∀ Y0 ≥ R (Z ,Γ)∈V n o where V := (Z , Γ) : Z ∈ H2 (P) andP ∗ YTZ ,Γ 6= ∅ Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal problem restricted to revealing contracts Dynamics of the pair (X , Y ) under “optimal response” 1 dXt = ∇z Ht (X , YtZ ,Γ , Zt , Γt )dt + 2∇γ Ht (X , YtZ ,Γ , Zt , Γt ) 2 dWt dYtZ ,Γ = Zt · dXt + 12 Γt : dhX it − Ht (X , YtZ ,Γ , Zt , Γt )dt Principal’s value function under revealing contracts : VP ≥ V0 (X0 , Y0 ) := h i sup E U `(X ) − UA−1 (YTZ ,Γ , ∀ Y0 ≥ R (Z ,Γ)∈V n o where V := (Z , Γ) : Z ∈ H2 (P) andP ∗ YTZ ,Γ 6= ∅ Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Main result Theorem (Cvitanić, Possamaï & NT ’15) Assume V = 6 ∅. Then V0P = sup V0 (X0 , Y0 ) Y0 ≥R Given maximizer Y0? , the corresponding optimal controls (Z ? , Γ? ) induce an optimal contract ? ξ = Y0? Z + 0 T ? ? 1 Zt? · dXt + Γ?t : dhX it − Ht (X , YtZ ,Γ , Zt? , Γ?t )dt 2 Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem On the function V X̄ := (X , Y ) satisfies d X̄t = µ̄(X̄t , Zt , Γt )dt + σ̄(X̄t , Zt , Γt )dWt : 1 dXt = ∇z Ht (X , YtZ ,Γ , Zt , Γt )dt + 2∇γ Ht (X , YtZ ,Γ , Zt , Γt ) 2 dWt dYtZ ,Γ = Zt · dXt + 12 Γt : dhX it − Ht (X , YtZ ,Γ , Zt , Γt )dt h i and recall V0 (X̄0 ) := sup E U(`(XT ) − YTZ ,Γ ) Z ,Γ V0 (X̄0 ) = V (0, X̄0 ) ; V solution of Hamilton-Jacobi-Bellman eq. n o ∂t V0 + supz,γ µ̄(., z, γ)DV0 + 12 σσ > (., z, γ) : D 2 V0 = 0 V0 (T , x, y ) = U(`(x) − y ) Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Sufficient conditions To prove the main result, it suffices that for all ξ ∈?? ∃ (Y0 , Z , Γ) s.t. ξ = YTZ ,Γ , P − q.s. OR, weaker sufficient condition : for all ξ ∈?? ∃ (Y0n , Z n , Γn ) s.t. “YTZ Nizar Touzi n ,Γn −→ ξ” Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Sufficient conditions To prove the main result, it suffices that for all ξ ∈?? ∃ (Y0 , Z , Γ) s.t. ξ = YTZ ,Γ , P − q.s. OR, weaker sufficient condition : for all ξ ∈?? ∃ (Y0n , Z n , Γn ) s.t. “YTZ Nizar Touzi n ,Γn −→ ξ” Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Case of un-controlled diffusion General controlled diffusion case Outline 1 Finite horizon Principal-Agent problem 2 Solving the representation problem by 2BSDEs Case of un-controlled diffusion General controlled diffusion case 3 Infinite horizon Principal-Agent problem Principal-Agent Problem formulation in infinite horizon Random horizon backward SDEs Random horizon backward SDEs Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Case of un-controlled diffusion General controlled diffusion case Reduction of the representation problem Suppose σt (ω, b) = σt (ω). Then Hamiltonian reduces to 1 Ht (ω, y , z, γ) = Ft (ω, y , z) + Tr[σt σt> (ω)γ], 2 with Ft (ω, z) := sup − ct (ω, u) − kt (ω, u)y + σt (ω)λt (ω, a) · z u Representation problem reduces to Z T 1 Z ,Γ ξ = YT = Y0 + Zs · dXs + Γs : dhX is − Hs (X , YsZ ,Γ , Zs , Γs )ds 2 0 Z T = Y0 + Zs · dXs − Fs (X , YsZ ,Γ , Zs )ds 0 = YTZ ,0 =: YTZ Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Case of un-controlled diffusion General controlled diffusion case Reduction of the representation problem Representation problem is now Z T Z ξ = YT = Y0 + Zs · dXs − Fs (X , YsZ ,Γ , Zs )ds 0 must hold P − a.s. for all P solution of controlled SDE or, equivalently, P0 − a.s. for some P solution of controlled SDE Backward SDE, Pardoux & Peng ’91 : F ∈ Lip, ξ ∈ Lp , p > 1 : ∃ F−solution (Y , Z ) with kY kL2 +kZ kL2 < ∞ Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Case of un-controlled diffusion General controlled diffusion case Reduction of the representation problem Representation problem is now Z T Z ξ = YT = Y0 + Zs · dXs − Fs (X , YsZ ,Γ , Zs )ds 0 must hold P − a.s. for all P solution of controlled SDE or, equivalently, P0 − a.s. for some P solution of controlled SDE Backward SDE, Pardoux & Peng ’91 : F ∈ Lip, ξ ∈ Lp , p > 1 : ∃ F−solution (Y , Z ) with kY kL2 +kZ kL2 < ∞ Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Case of un-controlled diffusion General controlled diffusion case Reduction of the representation problem Representation problem is now Z T Z ξ = YT = Y0 + Zs · dXs − Fs (X , YsZ ,Γ , Zs )ds 0 must hold P − a.s. for all P solution of controlled SDE or, equivalently, P0 − a.s. for some P solution of controlled SDE Backward SDE, Pardoux & Peng ’91 : F ∈ Lip, ξ ∈ Lp , p > 1 : ∃ F−solution (Y , Z ) with kY kL2 +kZ kL2 < ∞ Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Case of un-controlled diffusion General controlled diffusion case The case of controlled diffusion : difficulties • Similar to the Markov case, very difficult to access to the Hessian component... Need a relaxation of the C 2 −regularity • The Pν ’s (measures induces by the controlled state) are defined on different supports for different values of ν, so can not reduce the analysis to one single measure =⇒ Quasi-sure stochastic analysis : stochastic analysis under a non-dominated family of singular measure Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Case of un-controlled diffusion General controlled diffusion case From fully nonlinear HJB equation to semilinear • Ht (ω, y , z, γ) non-decreasing and convex in γ, Then Ht (ω, y , z, γ) = sup a≥0 Denote σ̂t2 := `t dhX it dt , n1 2 o a : γ − Ht∗ (ω, y , z, a) and : 1 := Ht (Yt , Zt , Γt ) − σ̂t2 : Γt + Ht∗ (Yt , Zt , σ̂t2 ) ≥ 0 2 The required representation ξ = YTZ ,Γ , P−q.s. is equivalent to Z ξ = Y0 + 0 T Zt · dXt + Ht∗ (Yt , Zt , σ̂t2 )dt − Nizar Touzi Z T `t dt, P − q.s. 0 Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Case of un-controlled diffusion General controlled diffusion case Wellposedness of second order BSDEs The following statement does not refer to the appropriate integrability conditions... Theorem (Soner, NT & Zhang ’10, Possamaï, Tan & Zhou ’15) For all ξ, there exists a unique triple of F−adapted processes (Y , Z , K ), such that Z T Z T Z T 2 ∗ • Yt = ξ − Hs (Ys , Zs , σ̂s )ds − Zs dXs + dLs , P−q.s. t t t • L nondecreasing, L0 = 0, and inf P∈P E P LT = 0 Differs from the required representation only because L is in general not absolutely continuous wrt Lebesgue Main result follows from approximation technique Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Case of un-controlled diffusion General controlled diffusion case Wellposedness of second order BSDEs The following statement does not refer to the appropriate integrability conditions... Theorem (Soner, NT & Zhang ’10, Possamaï, Tan & Zhou ’15) For all ξ, there exists a unique triple of F−adapted processes (Y , Z , K ), such that Z T Z T Z T 2 ∗ • Yt = ξ − Hs (Ys , Zs , σ̂s )ds − Zs dXs + dLs , P−q.s. t t t • L nondecreasing, L0 = 0, and inf P∈P E P LT = 0 Differs from the required representation only because L is in general not absolutely continuous wrt Lebesgue Main result follows from approximation technique Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal-Agent Problem formulation in infinite horizon Random horizon backward SDEs Random horizon backward SDEs Outline 1 Finite horizon Principal-Agent problem 2 Solving the representation problem by 2BSDEs Case of un-controlled diffusion General controlled diffusion case 3 Infinite horizon Principal-Agent problem Principal-Agent Problem formulation in infinite horizon Random horizon backward SDEs Random horizon backward SDEs Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal-Agent Problem formulation in infinite horizon Random horizon backward SDEs Random horizon backward SDEs Infinite horizon Principal-Agent problem Given contract θ := τ, ξ, {Ft , t ≥ 0} , Agent’s problem is : Z τ h Rt i A P V0 (θ) := sup E Kτ ξ + Kt dFt − ct (νtP )dt , Kt := e − 0 ks ds 0 P∈P where τ stop. time, ξ ∈ Fτ , F ∈ FV, and P ∈ P weak solution of dXt = σt (βt ) λt (αt )dt + dWtP , P − a.s. • Given solution P∗ (θ), Principal solves the optimization problem h i Rt ∗ V0P := sup E P (θ) KτP U(` − Fτ − ξ) , Ht := e − 0 hs ds θ∈ΞR where ΞR := {θ, such that V0A (θ) ≥ R} Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal-Agent Problem formulation in infinite horizon Random horizon backward SDEs Random horizon backward SDEs Motivation • One-dimensional output process =⇒ HJB equation for Principal problem in (x, y ), and under additional structure condition =⇒ ODE =⇒ explicit solutions... • Huge majority of economic contributions is in this context... • Methodology applies to this context... but needs to extend the “representation” to random horizon setting “ ξ = YτZ ,Γ , P − q.s. 00 Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal-Agent Problem formulation in infinite horizon Random horizon backward SDEs Random horizon backward SDEs Random horizon backward SDE Given ξ, τ , want to solve dYt = −ft (Yt , Zt )dt + Zt · dWt , and Yτ = ξ, P − a.s. (C1L ) ft (y , z) − ft (y , z 0 ) ≤ L |z − z 0 | (C2µ ) f C 0 in y , and (y − y 0 ) · ft (y , z) − ft (y 0 , z) ≤ −µ |y − y 0 |2 (C3) ft (y , z) ≤ ft0 + Cst. |y | + |z| Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal-Agent Problem formulation in infinite horizon Random horizon backward SDEs Random horizon backward SDEs Wellposedness of random horizon backward SDE I kY k2H2 (P) λ kξk2L2 (P) := EP e λτ |ξ|2 R τ λt λ2 P 2 P sup λt |Y |2 := E e |Y | dt , kY k := E e t t 2 t≤τ 0 S (P) λ Theorem (Pardoux ’98) Let (C1L , C2µ , C3) hold, and assume further sup kξkL2 (P) + kf 0 kH2 (P) < ∞ for some λ > −2µ + L2 Q∈QL λ λ Then Random horizon BSDE has a unique solution in the class sup kY kS2 (P) + kY kH2 (P) + kZ kH2 (P) < ∞ Q∈QL λ λ λ R. QL := Q := E 0 λt · dWt · P for some λ, kλkL∞ ≤ L Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal-Agent Problem formulation in infinite horizon Random horizon backward SDEs Random horizon backward SDEs Wellposedness of random horizon backward SDE II kY k2H2 (P) λ kξk2L2 (P) := EP e λτ |ξ|2 R τ λt λ2 P 2 P sup λt |Y |2 := E e |Y | dt , kY k := E e t t 2 t≤τ 0 S (P) λ Theorem (Z. Ren, NT & Y. Lin ’17) Let (C1L , C2µ , C3) hold, and assume further sup kξkL2 (Q) + kf 0 kH2 (Q) < ∞ for some λ > −2µ + L2 Q∈QL λ λ Then Random horizon BSDE has a unique solution in the class sup kY kS2 (Q) + kY kH2 (Q) + kZ kH2 (Q) < ∞ Q∈QL λ λ λ R. QL := Q := E 0 λt · dWt · P for some λ, kλkL∞ ≤ L Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal-Agent Problem formulation in infinite horizon Random horizon backward SDEs Random horizon backward SDEs Random horizon second order backward SDE o n it ∈ ... , want to solve Given ξ, τ , P = P : X P − loc. mg, dhX dt dYt = −ft (Yt , Zt )dt + Zt · dXt − dKt , Yτ = ξ, P − q.s. and K non-decreasing, K0 = 0, inf P∈P EP [Kτ ] = 0 (C1L ) ft (y , z) − ft (y , z 0 ) ≤ L |σ̂t (z − z 0 )| (C2µ ) f C 0 in y , and (y − y 0 ) · ft (y , z) − ft (y 0 , z) ≤ −µ |y − y 0 |2 (C3) ft (y , z) ≤ ft0 + Cst. |y | + |z| Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal-Agent Problem formulation in infinite horizon Random horizon backward SDEs Random horizon backward SDEs Wellposedness of random horizon 2nd order backward SDE kY k2H2 (P) λ kξk2L2 (P) := EP e λτ |ξ|2 R τ λt λ2 P 2 P sup λt |Y |2 := E e |Y | dt , kY k := E e t t 2 t≤τ 0 S (P) λ Theorem (Z. Ren, NT & Y. Lin ’17) Let (C1L , C2µ , C3) hold, and assume further sup kξkL2 (Q) + kf 0 kH2 (Q) < ∞ for some λ > −2µ P∈PL λ λ Then Random horizon 2BSDE has a unique solution in the class sup kY kS2 (P) + kY kH2 (P) + kZ kH2 (P) < ∞ P∈PL λ λ λ R. PL := P = E 0 λt · dWt · P for some P ∈ P, λ, kλkL∞ ≤ L Nizar Touzi Principal-Agent problem & 2BSDEs Finite horizon Principal-Agent problem Solving the representation problem by 2BSDEs Infinite horizon Principal-Agent problem Principal-Agent Problem formulation in infinite horizon Random horizon backward SDEs Random horizon backward SDEs Conclusion • Non-Markov stochastic control problems appear naturally in stochastic differential games, and are naturally addressed by BSDEs and 2BSDEs • Various extensions following the same methodology : Many interacting agents & 1 Principal Continuum of agent with mean field interaction & 1 Principal 1 Agent & Many Principals Elie, Mastrolia, Possamaï, Ren... Nizar Touzi Principal-Agent problem & 2BSDEs