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Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter

Continuous-time Principal-Agent Problem in Partially Observed System

Kaitong HU

CMAP, Ecole Polytechnique

joint work with Zhenjie REN, Nizar Touzi

International Conference on Control, Games and Stochastic Analysis Hammamet, October 2018

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter

Table of Contents

1

Finite Horizon Principal-Agent Problem

2

Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter

Outline

1

Finite Horizon Principal-Agent Problem

2

Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter

Principal-Agent Problem

• The Principal delegates the management of the output process

(Xt)t ∈ [0, T] to the Agent. He alone can only oversee X and decides the salary of the Agent.

• By receiving the salary (therefore signing the contract), the

Agent devotes his effort and expertise to the management of the

• utput process. He chooses his optimal control by solving his own
• ptimization problem:

VA(ξ) = max

α Eα[UA(ξ −

T ct(αt)dt)]. (1)

• In order to maximise his expected return, the Principal chooses

the optimal contract by solving the following non-zero sum Stackelberg game: VP = max

ξ

Eα∗(ξ)[UP(XT − ξ)]. (2)

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Outline

1

Finite Horizon Principal-Agent Problem

2

Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Introduction

Let (Ω, F, P) be a Probability space. Let (W , O) be a standard 2-dimensional Brownian motion, µ a Gaussian variable. We assume that W , O, µ are mutually independent. Denote Ft := σ{Ws, Os, µ, s ≤ t}. Consider (Pα,β, B) as a weak solution

• f the following controlled system

dXt = (f (t)Xt + αt)dt + σ(t)dWt µ0 = µ, (3) dOt = (h(t)Xt + βt)dt + dBt O0 = 0. (4) Here, (Xt, Ft), 0 ≤ t ≤ T, is the unobservable process while O is the observation of the system.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Assumptions

(i) The prior distribution of µ is normal of mean and variance m0 and σ0 respectively, which is known to both the Agent and the Principal.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Assumptions

(i) The prior distribution of µ is normal of mean and variance m0 and σ0 respectively, which is known to both the Agent and the Principal. (ii) The unobservable process Xt however is unknown to neither the Principal nor the Agent.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Assumptions

(i) The prior distribution of µ is normal of mean and variance m0 and σ0 respectively, which is known to both the Agent and the Principal. (ii) The unobservable process Xt however is unknown to neither the Principal nor the Agent. (iii) The Principal doesn’t observe the Agent’s effort, namely α and β.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Prior Analysis

Proposition Let ˆ Xt = Eα,β[Xt|FO

t ]. We have the following control filter:

       dˆ Xt = (f (t) ˆ Xt + αt)dt + h(t)V (t)dIt ˆ µ0 = m0, (5) dOt = (h(t) ˆ Xt + βt)dt + dIt X0 = 0, (6) dV (t) dt = 2f (t)V (t) − h(t)2V (t)2 + σ(t) V0 = σ0. (7) Here I is the Innovation process define by It := Bt + t h(s)(Xs − ˆ Xs)ds, (8) which is a FO-adapted Pα,β-Brownian motion.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Agent’s Optimization Problem

We assume that the agent will only receive his pay ξ at the end of the contract. Denote ct the agent’s cost function at time t. The agent’s optimization problem can be then written as follow VA = sup

α,β

Eα,β[e−

T

0 ρA t dtξ −

T ct(αt, βt)dt]. (9) Here we only consider implementable contracts such that the Agent has at least one optimal control.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Necessary Condition

For a contract ξ ∈ Ξ, let ν∗ ∈ M∗(ξ). Then the following FBSDE:                dYt = (ct(ν∗

t ) − Zt(h(t) ˆ

Xt + β∗

t ))dt + ZtdOt

dPt =

• h(t)Zt − (f (t) − V (t)h2(t))Pt

−Qt(h(t) ˆ Xt + β∗

t )

• dt + QtdOt,

dˆ Xt = (f (t) ˆ Xt + α∗

t + h(t)V (t)(h(t) ˆ

Xt + β∗

t ))dt + h(t)V (t)dOt,

has a solution, denoted (Y ∗, Z ∗, P∗, Q∗, ˆ X ∗), with PT = 0 and YT = ΓA

Tξ. Besides, for all t ∈ [0, T], for all (α, β) ∈ A × B, the

• ptimal control ν∗ = (α∗, β∗) must verify

(P∗

t + ∂αct(α∗ t , β∗ t ))(α − α∗ t ) ≥ 0,

(10) (Z ∗

t + V (t)h(t)P∗ t − ∂βct(α∗ t , β∗ t ))(β − β∗ t ) ≤ 0.

(11)

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Sufficient Condition

Let ξ be an contract and (Y ∗, Z ∗, P∗, Q∗, ˆ X ∗) be a solution in the reference Probability space of the associated FBSDE:                dYt = (ct(ν∗

t ) − Zt(h(t) ˆ

Xt + β∗

t ))dt + ZtdOt

dPt =

• h(t)Zt − (f (t) − V (t)h2(t))Pt

−Qt(h(t) ˆ Xt + β∗

t )

• dt + QtdOt,

dˆ Xt = (f (t) ˆ Xt + α∗

t + h(t)V (t)(h(t) ˆ

Xt + β∗

t ))dt + h(t)V (t)dOt,

where ν∗ = (α∗, β∗) satisfies the condition (P∗

t + ∂αct(α∗ t , β∗ t ))(α − α∗ t ) ≥ 0,

(Z ∗

t + V (t)h(t)P∗ t − ∂βct(α∗ t , β∗ t ))(β − β∗ t ) ≤ 0,

for all t ∈ [0, T] and for all (α, β) ∈ A × B.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Sufficient Condition

Define for all (α, β) ∈ A × B and x ∈ R, Gt(α, β, x) := ct(α, β) + h(t)Q∗

t x2 + Q∗ t xβ + P∗ t α

− (Z ∗

t + V (t)h(t)P∗ t + Q∗ t ˆ

X ∗

t )β.

Then ν∗ = (α∗, β∗) is an optimal control if for all admissible control ν = (αt, βt)t∈[0,T], we have Eν T Gt(αt, βt, Xt) − Gt(α∗

t , β∗ t , ˆ

X ∗

t )

− ∂XGt(α∗

t , β∗ t , ˆ

X ∗

t )(Xt − ˆ

X ∗

t )dt

• ≥ 0.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Sufficient Condition

Proposition If (α, β) ∈ A × B → ct(α, β) is convex and t → h(t) is non-negative, than v∗ is an optimal control if the functional G is convex, which is equivalent to Q∗

t ∈

• 0, 2h(t) ·

inf

(α,β)∈A×B

• ∂2

βct(α, β)

, (12) for all t ∈ [0, T].

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Principal’s Optimization Problem

The Principal’s expected utility can be formulated as VP = sup

ξ∈Ξ

sup

ν∗∈M∗(ξ)

Eν∗[ΓP

T(OT − ξ)],

where ∀t ∈ [0, T], ΓP

t := e t

0 ρP s ds represents the discount factor for

the Principal.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Principal’s Optimization Problem

Theorem Let ¯ VP := sup

y0≥R

sup

Z∈Z(y0)

Eν∗ ΓP

T

• OT − Y Z

T

ΓA

T

• .

(13) We have VP ≤ ¯ VP, and the equality is attained if there exists an optimizer (y∗

0 , Z ∗) for

the problem (13) such that QZ ∗ satisfies the condition (12).

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

An Example

Consider the following output process X given the Agent’s effort process α: Xt = t (µ + αs)ds + Bt. (14) The time-invariant productivity is denoted by µ whereas αt ∈ U is the effort provided by the Agent. The Agent’s action thus shifts the average output but does not directly affect its volatility. However, µ is unknown at time 0 and the common priors are normal with mean m0 and precision h0.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

An Example

We will be consider the following Principal-Agent problem: VA = sup

α Eα[e− T

0 ρA t dtξ −

T ct(αt)dt]. (15) VP = sup

ξ,VA≥R0

Eα∗(ξ)[e−

T

0 ρP t dt(XT − ξ)]

(16)

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

An Example

Theorem Assume that for all 0 ≤ t ≤ T, the function α → α − (ΓA

T)−1ct(α)

has a global maximum and denote ¯ α∗

t a maximizer. If in addition

α → c′

t(α) is onto and U = R, we have

VP = ΓP

T

T (¯ α∗

t −(ΓA T)−1ct(¯

α∗

t ))dt −ΓP T(ΓA T)−1R0 +(ΓA T)−1m0T.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

An Example

Theorem The optimal contract is given by ξ = R + T (c′(¯ α∗

t ) − Z ∗ t bt(Xt, A∗ t , ¯

α∗

t ))dt +

T Z ∗

t dXt,

where Z ∗

t = c′(¯

α∗

t ) +

T

t

e

s

t dr hr c′(¯

α∗

s)

hs ds, and A∗

t :=

t ¯ α∗

sds, ht = h0 + t.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

References I

Alain Bensoussan. Stochastic Control of Partially Observable Systems. Cambridge University Press, 1992. Jaksa Cvitanic and Jianfeng Zhang. Contract Theory in Continuous-Time Models. Springer, 2012. Detao Zhang Jin Ma, Zhen Wu and Jianfeng Zhang. On well-posedness of forward-backward sdes - a unified approach. The Annals of Applied Probability, Vol.25, No.4, 2168-2214, 2015.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

References II

Magdalena Kobylanski. Backward stochastic differential equations and partial differential equations with quadratic growth. The Annals of Probability, Vol.28, No.2, 558-602, 2000. Julien Prat and Boyan Jovanovic. Dynamic incentive contracts under parameter uncertainty. IZA, DP No. 5323, 2010. Jianfeng Zhang. The wellposedness of fbsdes (ii). arXiv:1708.05785, 2017.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Thanks for your attention!

Kaitong HU Principal-Agent Problem and FBSDEs