Download Principal-Agent Problem and Random Horizon Second Order

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