On Continuous Time Contract Theory · 2018-06-07 · The Principal-Agent problem Linear and semilinear representation Fully nonlinear representation OnContinuousTimeContractTheory

The Principal-Agent problemLinear and semilinear representation

Fully nonlinear representation

On Continuous Time Contract Theory

Nizar Touzi

Ecole Polytechnique, France

Mete’s Academic Birthday, June 5, 2018

Nizar Touzi On Continuous Time Contract Theory






FormulationExamplesReduction to standard control problem

Outline

1 The Principal-Agent problemFormulationExamplesReduction to standard control problem

2 Linear and semilinear representationPath dependent heat equationPath dependent semilinear equations

3 Fully nonlinear representationCanonical spacePath dependent fully nonlinear PDEs





(Static) Principal-Agent Problem

• Principal delegates management of output process X ,only observes Xpays salary defined by contract ξ(X )

• Agent devotes effort a =⇒ X a, chooses optimal effort by

VA(ξ) := maxa

EUA

(ξ(X a)− c(a)

)=⇒ a(ξ)

• Principal chooses optimal contract by solving

maxξ

EUP

(X a(ξ) − ξ(X a(ξ))

)under constraint VA(ξ) ≥ ρ

Non-zero sum Stackelberg game








VA(ξ) := maxa

EUA

(ξ(X a)− c(a)

)=⇒ a(ξ)


maxξ

EUP

(X a(ξ) − ξ(X a(ξ))


=⇒ Non-zero sum Stackelberg game








VA(ξ) := maxa

EUA

(ξ(X a)− c(a)

)=⇒ a(ξ)


maxξ

EUP

(X a(ξ) − ξ(X a(ξ))










VA(ξ) := maxa

EUA

(ξ(X a)− c(a)

)=⇒ a(ξ)


maxξ

EUP

(X a(ξ) − ξ(X a(ξ))







(Static) Principal-Agent Problem ==> Continuous time



VA(ξ) := maxa

EUA

(ξ(X a)− c(a)

)=⇒ a(ξ)


maxξ

EUP

(X a(ξ) − ξ(X a(ξ))







Principal-Agent problem formulation

Agent problem :

V A0 (ξ) := sup

P∈PEP[ξ(X )−

∫ T

0ct(νt)dt

]P ∈ P : weak solution of Output process for some ν valued in U :

dXt = bt(X , νt)dt + σt(X , νt)dWPt P− a.s.

• Given solution P∗(ξ), Principal solves the optimization problem

V P0 := sup

ξ∈Ξρ

E P∗(ξ)[U(`(X )− ξ(X )

)]where Ξρ :=

ξ(X.) : V A

0 (ξ) ≥ ρ

Possible extensions : random (possibly ∞) horizon, heterogeneousagents with possibly mean field interaction, competing Principals...





Principal-Agent problem formulation

Agent problem :

V A0 (ξ) := sup

P∈PEP[ξ(X )−

∫ T

0ct(νt)dt


dXt = bt(X , νt)dt + σt(X , νt)dWPt P− a.s.


V P0 := sup

ξ∈Ξρ

E P∗(ξ)[U(`(X )− ξ(X )

)]where Ξρ :=

ξ(X.) : V A

0 (ξ) ≥ ρ






Principal-Agent problem formulation : non-degeneracy

Agent problem :

V A0 (ξ) := sup

P∈PEP[ξ(X )−

∫ T

0ct(νt)dt


dXt = σt(X , βt)[λt(X , αt)dt + dW P

t

]P− a.s.


V P0 := sup

ξ∈Ξρ

E P∗(ξ)[U(`(X )− ξ(X )

)]where Ξρ :=

ξ(X.) : V A

0 (ξ) ≥ ρ






Fund managers compensation [Cvitanić, Possamaï & NT ’16]

Fund managers portfolio value

dXt = πt · (αtdt + dWt)

Manager’s problem

V A(ξ) := supα,π

E[ξ(X )−

∫ T

0ct(αt , πt)dt

]=⇒ ν? := (α?, π?)

Principal problem

V P := supξ:V A(ξ)≥R

E[U(Xν?(ξ)T − ξ(X ν?(ξ))

)]





Maker-Taker Fees [El Euch, Mastrolia, Rosenbaum & NT ’18]





Modeling Makers-Takers fees

• Fundamental price Ptt≥0, Market Maker posts bid-ask prices

pbt = Pt − δbt and pat = Pt + δat , with dPt = σdWt

• MM inventory Qt = Nbt − Na

t , where Nbt ,N

at point process with

unit jumps representing the arrivals of bid-ask order with intensities

λbt = λ(δbt ) and λat = λ(δat ), with λ(x) = Ae−kσx

• MM chooses the departure from fundamental price :

VA(ξ) := supδb,δa

EUA

(ξ +

∫ T

0pat dN

at − pbt dN

bt + QTPT

)• Given optimal response δ∗(ξ), Platform chooses optimal contract

VP = supξ∈ΞR

EUP

(−ξ + c(Na

T + NbT ))









t , where Nbt ,N






EUA

(ξ +

∫ T

0pat dN

at − pbt dN

bt + QTPT


VP = supξ∈ΞR

EUP

(−ξ + c(Na

T + NbT ))









t , where Nbt ,N






EUA

(ξ +

∫ T

0pat dN

at − pbt dN

bt + QTPT


VP = supξ∈ΞR

EUP

(−ξ + c(Na

T + NbT ))









t , where Nbt ,N






EUA

(ξ +

∫ T

0pat dN

at − pbt dN

bt + QTPT


VP = supξ∈ΞR

EUP

(−ξ + c(Na

T + NbT ))





Some references

Adam Smith (1723-1790), “Scottish economist, moral philosopher,pioneer of political economy

Moral hazard is a major risk in economics

Jean Tirole (Nobel Prize 2014), Industrial organization, regulation

Bengt Holmström and Oliver Hart (Nobel Prize 2016)

Cvitanić & Zhang (’12, Book)

Sannikov ’08 : new approach in continuous time...





A subset of revealing contracts

• Path-dependent Hamiltonian for the Agent problem :

Ht(ω, z , γ) := supu∈U

bt(ω, u) · z +

12σtσ

>t (ω, u) :γ − ct(ω, u)

• For Y0 ∈ R, Z , Γ FX − prog meas, define P−a.s. for all P ∈ P

Y Z ,Γt = Y0 +

∫ t

0Zs · dXs +

12

Γs : d〈X 〉s − Hs(X ,Zs , Γs)ds

Proposition VA

(Y Z ,ΓT

)= Y0. Moreover P∗ is optimal iff

ν∗t = Argmaxu∈U

Ht(Zt , Γt) = ν(Zt , Γt)





A subset of revealing contracts

• Path-dependent Hamiltonian for the Agent problem :

Ht(ω, z , γ) := supu∈U

bt(ω, u) · z +

12σtσ

>t (ω, u) :γ − ct(ω, u)

• For Y0 ∈ R, Z , Γ FX − prog meas, define P−a.s. for all P ∈ P

Y Z ,Γt = Y0 +

∫ t

0Zs · dXs +

12

Γs : d〈X 〉s − Hs(X ,Zs , Γs)ds

Proposition VA

(Y Z ,ΓT

)= Y0. Moreover P∗ is optimal iff

ν∗t = Argmaxu∈U

Ht(Zt , Γt) = ν(Zt , Γt)





Proof : classical verification argument !

For all P ∈ P, denote JA(ξ,P) := EP[ξ − ∫ T0 c

νt dt]. Then

JA(Y Z ,ΓT ,P

)= EP

[Y0+

∫ T

0Zt ·dXt +

12

Γt:d〈X 〉t−Ht(Zt , Γt)dt −∫ T

0cνt dt

]= Y0+EP

∫ T

0

bνt ·Zt +

12σσ>:Γt − cνt − Ht(Zt , Γt)

dt + Zt ·σνt dW P

t

≤ Y0

with equality iff ν = ν∗ maximizes the Hamiltonian







νt dt]. Then

JA(Y Z ,ΓT ,P

)= EP

[Y0+

∫ T

0Zt ·dXt +

12


0cνt dt

]= Y0+EP

∫ T

0

bνt ·Zt +


dt + Zt ·σνt dW P

t

≤ Y0








νt dt]. Then

JA(Y Z ,ΓT ,P

)= EP

[Y0+

∫ T

0Zt ·dXt +

12


0cνt dt

]= Y0+EP

∫ T

0

bνt ·Zt +


dt+Zt ·σνt dW P

t

≤ Y0








νt dt]. Then

JA(Y Z ,ΓT ,P

)= EP

[Y0+

∫ T

0Zt ·dXt +

12


0cνt dt

]= Y0+EP

∫ T

0

bνt ·Zt +


dt+Zt ·σνt dW P

t

≤ Y0 by definition of H








νt dt]. Then

JA(Y Z ,ΓT ,P

)= EP

[Y0+

∫ T

0Zt ·dXt +

12


0cνt dt

]= Y0+EP

∫ T

0

bνt ·Zt +


dt+Zt ·σνt dW P

t

≤ Y0 by definition of H






Principal problem restricted to revealing contracts

Dynamics of the pair (X ,Y ) under “optimal response”

dXt = ∇zHt(X ,YZ ,Γt ,Zt , Γt)︸︷︷︸

bt(X ,ν(Yt ,Zt ,Γt))

dt +2∇γHt(X ,Y

Z ,Γt ,Zt , Γt)

12︸︷︷︸

σt(X ,ν(Yt ,Zt ,Γt))

dWt

dY Z ,Γt = Zt · dXt + 1

2Γt : d〈X 〉t − Ht(X ,YZ ,Γt ,Zt , Γt)dt

=⇒ Principal’s value function under revealing contracts :

VP ≥ V0(X0,Y0) := sup(Z ,Γ)∈V

E[U(`(X )− Y Z ,Γ

T

)], for all Y0 ≥ ρ

where V :=

(Z , Γ) : Z ∈ H2(P) and P∗(Y Z ,ΓT

)6= ∅





Reduction to standard control problem

Theorem (Cvitanić, Possamaï & NT ’15)

Assume V 6= ∅. Then

V P0 = sup

Y0≥ρV0(X0,Y0)

Given maximizer Y ?0 , the corresponding optimal controls (Z ?, Γ?)

induce an optimal contract

ξ? = Y ?0 +

∫ T

0Z ?t · dXt +

12

Γ?t : d〈X 〉t − Ht(X ,YZ?,Γ?

t ,Z ?t , Γ?t )dt





Recall the subclass of contracts

Y Z ,Γt = Y0 +

∫ t

0Zs · dXs +

12

Γs :d〈X 〉s − Hs(X ,Y Z ,Γs ,Zs , Γs)ds

P− a.s. for all P ∈ P

To prove the main result, it suffices to prove the representation

for all ξ ∈?? ∃ (Y0,Z , Γ) s.t. ξ = Y Z ,ΓT , P− a.s. for all P ∈ P

OR, weaker sufficient condition :

for all ξ ∈?? ∃ (Y n0 ,Z

n, Γn) s.t. “Y Zn,Γn

T −→ ξ”






Y Z ,Γt = Y0 +

∫ t

0Zs · dXs +

12








T −→ ξ”






Y Z ,Γt = Y0 +

∫ t

0Zs · dXs +

12








T −→ ξ”





Gamma, Mete, and I

H.M. Soner, and NT. Super-replication under Gamma constraint,SIAM Journal on Control and Optimization 39, 73-96 (2000).

P. Cheridito, H.M. Soner and NT. The multi-dimensionalsuper-replication problem under Gamma constraints, Annales del’Institut Henri Poincaré, Série C : Analyse Non-Linéaire 22,633-666 (2005)

H.M. Soner, and NT. Hedging under Gamma constraints byoptimal stopping and face-lifting. Mathematical Finance 17, 1,59-80 (2007).

U. Cetin, M. Soner and NT, Option hedging under liquidity costs,Finance and Stochastics, 14, 317-341




Path dependent heat equationPath dependent semilinear equations

Outline








Linear representation of random variables

Predictable Representation Property of BM

Theorem

For all ξ ∈ L2(FWT ), there is a unique (Y ,Z ) FW−prog. meas. s.t.

ξ = Yt +∫ Tt ZsdWs , P− a.s. E

[ ∫ T0 (|Yt |2 + |Zt |2)dt

]<∞,

• For ξ = g(WT ) : Heat equation and Itô’s formula• True for ξ = g(Wt1 , . . . ,Wtn)... conclude by density argument

Heat equation with path dependent boundary condition

Connection with W 1,2−solution of Heat equation







Theorem



[ ∫ T0 (|Yt |2 + |Zt |2)dt

]<∞,










Theorem



[ ∫ T0 (|Yt |2 + |Zt |2)dt

]<∞,








Backward SDE and semilinear PDE

Required representation in differential form

dYt = ZtdWt , t ≤ T , andYT = ξ, P− a.s.

In the Markovian case ξ = g(WT ) =⇒

Yt = v(t,Wt), so that dYt = ∂tv(t,Wt)dt+Dv(t,Wt)·dWt+12

∆v(t,Wt)dt

by Itô’s formula. Direct identification yields

Zt = Dv(t,Wt) and ∂tv(t,Wt) +12

∆v(t,Wt) = 0










∆v(t,Wt)dt



∆v(t,Wt) = 0










∆v(t,Wt)dt



∆v(t,Wt) = 0





Semilinear representation of random variables

Here again, ξ = ξ(Ws , s ≤ T ). Let

f : R× Ω× R× Rd −→ R, Lip in (y , z), unif in (t, ω), E[∫ T

0 |f0s |2ds

]<∞

Theorem (Pardoux & Peng ’92)

For all ξ ∈ L2(FWT ), there is a unique FW−prog. meas. (Y ,Z ),

E[ ∫ T

0 (|Yt |2 + |Zt |2)dt]<∞, s.t

Yt = ξ+∫ T

tfs(Ys ,Zs)ds −

∫ T

tZsdWs , P− a.s.

Unique fixed point for the Picard iteration

(Y ,Z ) 7−→ Y ′t = ξ +∫ T

tfs(Ys ,Zs)ds −

∫ T

tZ ′sdWs , 0 ≤ t ≤ T

W 1,2−type solution of semilinear heat equation with

path-dependent nonlinearity and boundary data





Semilinear representation of random variables

Here again, ξ = ξ(Ws , s ≤ T ). Let

f : R× Ω× R× Rd −→ R, Lip in (y , z), unif in (t, ω), E[∫ T

0 |f0s |2ds

]<∞

Theorem (Pardoux & Peng ’92)

For all ξ ∈ L2(FWT ), there is a unique FW−prog. meas. (Y ,Z ),

E[ ∫ T

0 (|Yt |2 + |Zt |2)dt]<∞, s.t

Yt = ξ+∫ T

tfs(Ys ,Zs)ds −

∫ T

tZsdWs , P− a.s.

Unique fixed point for the Picard iteration

(Y ,Z ) 7−→ Y ′t = ξ +∫ T

tfs(Ys ,Zs)ds −

∫ T

tZ ′sdWs , 0 ≤ t ≤ T

W 1,2−type solution of semilinear heat equation with

path-dependent nonlinearity and boundary data




Canonical spacePath dependent fully nonlinear PDEs

Outline








Towards fully nonlinear PDEs : quasi-sure stochastic analysis

Ω :=ω ∈ C 0(R+,Rd) : ω(0) = 0

X : canonical process, i.e. Xt(ω) := ω(t)

Ft := σ(Xs , s ≤ t), F := Ft , t ≥ 0

PW : collection of all semimartingale measures P such that

dXt = btdt + σtdWt , P− a.s.

for some F−processes b and σ, and P−Brownian motion W

Class of prob. meas. on Ω : P ⊂ PW , sufficiently rich...

P−quasi-surely MEANS P−a.s. for all P ∈ P






Ω :=ω ∈ C 0(R+,Rd) : ω(0) = 0


Ft := σ(Xs , s ≤ t), F := Ft , t ≥ 0











Ω :=ω ∈ C 0(R+,Rd) : ω(0) = 0


Ft := σ(Xs , s ≤ t), F := Ft , t ≥ 0










Semimartingale measures on canonical space

〈X 〉 : quadratic variation process (defined on R+ × Ω)

〈X 〉t := X 2t −

∫ t0 2XsdXs = P−lim|π|→0

∑n≥1

∣∣Xt∧tπn − Xt∧tπn−1

∣∣2for all P ∈ PW , and set

σ2t := limh0

〈X 〉t+h−〈X 〉th

Example : (d = 1) Let P1 :=Wiener measure, i.e. X is a P1−BM,and define P2 := P1 (2X )−1. Then

σt = 1, P1 − a.s. and σt = 2, P2 − a.s.





Nonlinear expectation operators

P0 : subset of local martingale measures, i.e.

dXt = σtdWt , P− a.s. for all P ∈ P0

and

PL := ∪P∈P0PL(P) where PL(P) :=Q = D(λ) · P : ‖λ‖L∞ ≤ L

D(λ) : the Doléans-Dade exponential dD(λ)t

D(λ)t= λt · dWt , D(λ)0 = 1

=⇒ Nonlinear expectations

Et := supP∈P0

EPt and ELt := sup

P∈PL

EPt





Nonlinearity

Assumptions F : R+ × ω × R× Rd × Sd+ −→ R satisfies

(C1L) Lipschitz in (y , σz) :∣∣F (., y , z , σ)− F (., y ′, z ′, σ)∣∣ ≤ L

(|y − y ′|+

∣∣σ(z − z ′)∣∣

(C2µ) Monotone in y :

(y − y ′) ·[F (., y , .)− F (., y ′, .)

]≤ −µ |y − y ′|2

Denote f 0t := Ft

(0, 0, σt

)





Random horizon 2ndorder backward SDE

For a stop. time τ , and Fτ−measurable ξ :

Yt∧τ = ξ +

∫ τ

t∧τFs(Ys ,Zs , σs)ds −

∫ τ

t∧τZs · dXs+

∫ τ

t∧τdKs , P − q.s.

K non-decreasing, K0 = 0, and minimal in the sense

infP∈P

EP[ ∫ t2∧τ

t1∧τdKt

]= 0, for all t1 ≤ t2





Connection with fully nonlinear PDEs

Rewrite the 2BSDE in differential form

dYt = −Ft(Yt ,Zt , σt)dt + Zt · dXt − dKt , t ≤ τ, andYT = ξ, P − q.s.

Markovian case ξ = g(XT ) and Ft(X , y , z , σt) = f (t,Xt , y , z , σt) =⇒Yt = v(t,Xt)

dYt = ∂tv(t,Xt)dt + Dv(t,Xt)·dXt +12Tr[σ2tD

2v(t,Xt)]dt, P − q.s.


Zt = Dv(t,Xt) and ∂tv(t,Xt) +12Tr[σ2tD

2v(t,Xt)]≤ − Ft(v ,Dv , σt)

Finally, the minimality condition on K implies the fully nonlinear PDE

∂tv(t,Xt) + supσ

12Tr[σ2D2v(t,Xt)

]+ Ft(v ,Dv , σ)

= 0















∂tv(t,Xt) + supσ

12Tr[σ2D2v(t,Xt)

]+ Ft(v ,Dv , σ)

= 0















∂tv(t,Xt) + supσ

12Tr[σ2D2v(t,Xt)

]+ Ft(v ,Dv , σ)

= 0















∂tv(t,Xt) + supσ

12Tr[σ2D2v(t,Xt)

]+ Ft(v ,Dv , σ)

= 0





Wellposedness of random horizon 2ndorder backward SDE

Theorem (Soner-NT-Zhang ’12, Possamaï-Tan-Zhou ’18,Lin-Ren-NT-Yang’18)

Let ‖ξ‖Lqρ,τ (PL) <∞, f qρ,τ := EL[( ∫ τ

0

∣∣eρt f 0t

∣∣2ds) q2] 1q <∞, for

some ρ > −µ, q > 1. Then the Random horizon 2BSDE has aunique solution (Y ,Z ) with

Y ∈ Dp

η,τ (PL), Z ∈ Hp

η,τ (PL) for all η ∈ [µ, ρ), p ∈ [1, q)

‖ξ‖pLqρ,τ (P):= EL

[∣∣eρτξ∣∣q]‖Y ‖pDp

η,τ (P):= EL

[supt≤τ

∣∣eηtYt

∣∣p]‖Z‖pHp

η,τ (P):= EL

[( ∫ τ

0

∣∣eηt σTt Zt

∣∣2dt) p2]





Back to Principal-Agent problem


Y Z ,Γt = Y0+

∫ t

0Zs ·dXs+

12

Γs :d〈X 〉s−Hs(X ,Y Z ,Γs ,Zs , Γs)ds, P − q.s.

In the uncontrolled diffusion case, we have :

for all ξ ∈ Lp(P), ∃ (Y0,Z ) s.t. ξ = Y Z ,0T , P− a.s.

(≡ P − q.s.

)If both drift and diffusion are controlled, then :



T −→ ξ”





Back to Principal-Agent problem


Y Z ,Γt = Y0+

∫ t

0Zs ·dXs+

12

Γs :d〈X 〉s−Hs(X ,Y Z ,Γs ,Zs , Γs)ds, P − q.s.

In the uncontrolled diffusion case, we have :

for all ξ ∈ Lp(P), ∃ (Y0,Z ) s.t. ξ = Y Z ,0T , P− a.s.

(≡ P − q.s.

)If both drift and diffusion are controlled, then :



T −→ ξ”





Reduction to second order BSDE

• Ht(ω, y , z , γ) non-decreasing and convex in γ, Then

Ht(ω, y , z , γ) = supσ≥0

12σ2 : γ − H∗t (ω, y , z , σ)

Denote kt := Ht(Yt ,Zt , Γt)− 1

2 σ2t : Γt + H∗t (Yt ,Zt , σt) ≥ 0

Then, required representation ξ = Y Z ,ΓT , P−q.s. is equivalent to

ξ = Y0 +

∫ T

0Zt · dXt + H∗t (Yt ,Zt , σt)dt −

∫ T

0ktdt, P − q.s.

=⇒ 2BSDE up to approximation of nondecreasing process K





Reduction to second order BSDE

• Ht(ω, y , z , γ) non-decreasing and convex in γ, Then

Ht(ω, y , z , γ) = supσ≥0

12σ2 : γ − H∗t (ω, y , z , σ)

Denote kt := Ht(Yt ,Zt , Γt)− 1

2 σ2t : Γt + H∗t (Yt ,Zt , σt) ≥ 0

Then, required representation ξ = Y Z ,ΓT , P−q.s. is equivalent to

ξ = Y0 +

∫ T

0Zt · dXt + H∗t (Yt ,Zt , σt)dt −

∫ T

0ktdt, P − q.s.

=⇒ 2BSDE up to approximation of nondecreasing process K


















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On Continuous Time Contract Theory · 2018-06-07 · The Principal-Agent problem Linear and semilinear representation Fully nonlinear representation OnContinuousTimeContractTheory