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Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs

Continuous-time Principal-Agent Problem in Partially Observed System and Path-dependent FBSDEs

Kaitong HU

CMAP, Ecole Polytechnique

joint work with Zhenjie REN, Nizar Touzi Mai 2018

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs

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

3

Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs

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

3

Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs

Moral hazard

• Adam Smith (1723-1790): moral hazard is a major risk in

economics: In a situation where an agent may benefit from an action whose cost is supported by others,

• ne should not count on agents’ morality.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs

Principal-Agent Problem

• The Principal delegate the management of the output process

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

• By receiving the salary (and signing the contract), the Agent

devotes his effort and manage the output process. He chooses his

• ptimal control by solving his optimization problem:

VA(ξ) = max

α Eα[UA(ξ −

T ct(αt)dt)]. (1)

• The Principal chooses the optimal contract by solving the

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 Path-dependent FBSDEs

Continuous time Principal-Agent Literature

Holstr¨

• m & Milgrom 1987 .

. . Cvitani´ c & Zhang 2012: book Sannikov 2008 . . . Cvitanic, Possama¨ ı & Touzi 2015 Elie, Mastrolia, Possama¨ ı 2016 (Many Agents) Mastrolia, Ren 2017 (Many Principals)

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs 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

3

Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Kaitong HU Principal-Agent Problem and FBSDEs

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

Introduction

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

• f the following controlled system

dµt = (f (t)µt + αt)dt + σ(t)dWt µ0 = µ, (3) dXt = (h(t)µt + βt)dt + dBt X0 = 0. (4) Here, (µt, Ft), 0 ≤ t ≤ T, is the unobservable process while X is the observation of the system.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs 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 Path-dependent FBSDEs 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 µt 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 Path-dependent FBSDEs 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 µt 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 Path-dependent FBSDEs Environment The Agent’s Optimization Problem The Principal’s Optimization Problem

Prior Analysis

Proposition Let ˆ µt = Eα,β[µt|FX

t ]. We have the following control filter:

       dˆ µt = (f (t)ˆ µt + αt)dt + h(t)V (t)dIt ˆ µ0 = m0, (5) dXt = (h(t)ˆ µt + 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)(µs − ˆ µs)ds, (8) which is a FX-adapted Pα,β-Brownian motion.

Kaitong HU Principal-Agent Problem and FBSDEs

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

Necessary Condition

Given a contract ξ, assume that an optimal control (α∗, β∗) exists for the Agent’s problem, then we have the following result. Theorem (Necessary Condition) Under Pα∗,β∗, the agent’s value function satisfies the following FBSDE:                    dYt = ct(α∗

t , β∗ t )dt + ZtdI ∗ t

YT = ΓA

Tξ,

dPt = (Zt − f (t)Pt + V (t)h2(t)Pt)dt + QtdI ∗

t

PT = 0, dXt = (h(t)ˆ µt + β∗

t )dt + dI ∗ t

X0 = 0, dˆ µt = (f (t)ˆ µt + α∗

t )dt + h(t)V (t)dI ∗ t

ˆ µ0 = m0, ∂αct(α∗, β∗

t ) + Pt = 0,

∂βct(α∗, β∗

t ) − Zt − V (t)h(t)Pt = 0.

Kaitong HU Principal-Agent Problem and FBSDEs

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

Necessary Condition

Given a contract of the following form (if such a solution exists)      dYt = ct(α∗

t , β∗ t )dt + Ztd(dXt − (h(t)ˆ

µt + β∗

t )dt),

dPt = (Zt − (f (t) − V (t)h2(t))Pt)dt + Qt(dXt − (h(t)ˆ µt + β∗

t )dt),

dˆ µt = (f (t)ˆ µt + α∗

t )dt + h(t)V (t)(dXt − (h(t)ˆ

µt + β∗

t )dt),

with terminal condition YT = ΓA

Tξ(X) and PT = 0, where

∂αct(α∗, β∗

t ) + Pt = 0,

∂βct(α∗, β∗

t ) − Zt − V (t)h(t)Pt = 0.

We can show that if the optimal control exists, necessarily the

• ptimal control is (α∗, β∗).

Kaitong HU Principal-Agent Problem and FBSDEs

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

Sufficient Condition

Define Ht(α, β, µ) := −ct(α, β) − Qtµ2 − Qtµβ − Ptα + (Zt + V (t)h(t)Pt − Qµ∗

t )β.

We have VA(α, β) − VA(α∗, β∗) = Eα,β[ T (Ht(αt, βt, ˆ µt) − Ht(α∗

t , β∗ t , ˆ

µ∗

t )

− ∂µHt(α∗

t , β∗ t , ˆ

µ∗

t )δµ)dt].

The right hand side of the above equality is negative if for all t ∈ [0, T], Hess(H) has only negative Eigenvalues.

Kaitong HU Principal-Agent Problem and FBSDEs

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

Principal’s Optimization Problem

The Principal has to maximize his utility by anticipating Agent’s action and choosing an optimal contract. VP := sup

ξ

Eα∗(ξ),β∗(ξ)[e−

T

0 ρP t dt(XT − ξ)]

:= e−

T

0 ρP t dt

sup

Y0,Z,Q,PT =0

E∗[ T (h(t)ˆ µt + β∗

t )dt − e T

0 ρA t dtY Z

T ]

:= e−

T

0 ρP t dt

m0 T h(t)dt + sup

Y0,Z,Q,PT =0

E∗[ T (β∗

t

− e

T

0 ρA t dtct(α∗

t , β∗ t )dt]

• .

Kaitong HU Principal-Agent Problem and FBSDEs

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

Principal’s Optimization Problem

Theorem Assume that for all α ∈ Uα, the function β → β − e

T

0 ρA t dtct(α, β)

has a maximiser, denoted ¯ βt(α) and the equation ∂αct(α, ¯ βt(α)) + Pt = 0 (10) has at least one solution, where Pt is given by the ODE dPt = (1 − f (t)Pt)dt. (11) Then there exists an optimal contract for the Principal’s problem.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs 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. (12) 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 Path-dependent FBSDEs 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]. (13) VP = sup

ξ,VA≥R0

Eα∗(ξ)[e−

T

0 ρP t dt(XT − ξ)]

(14)

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs 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 Path-dependent FBSDEs 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 Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

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

3

Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Markovian FBSDEs

Let T > 0 be a fixed time horizon We consider the following system of strongly coupled forward-backward stochastic differential equation (abbreviated FBSDE): dXt = bt(Xt, Yt, Zt)dt + σt(Xt, Yt, Zt)dWt X0 = x dYt = −ft(Xt, Yt, Zt)dt + ZtdWt YT = g(XT).

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Markovian FBSDEs Literature

Antonelli 1993 . . . Pardoux & Tang 1999 (Contracting Mapping) Ma, Protter & Yong 1994 (Four Step Scheme) Hu & Peng 1995 . . . Peng & Wu 1999 (Method of Continuation) Ma, Wu, Zhang (Detao) & Zhang (Jianfeng) 2015 (Unified Approach)

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Path-dependent FBSDEs

Let T > 0 be a fixed time horizon We consider the following system of strongly coupled forward-backward stochastic differential equation: dXt = bt(X, Yt, Zt)dt + σt(X, Yt, Zt)dWt X0 = x dYt = −ft(X, Yt, Zt)dt + ZtdWt YT = g(X).

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Assumptions

Assumption

• The coefficients

b : [0, T] × Ω × C([0, T], Rd) × Rn × Mn(R) → Rd f : [0, T] × Ω × C([0, T], Rd) × Rn × Mn(R) → Rn σ : [0, T] × Ω × C([0, T], Rd) × Rn × Mn(R) → Md,n(R) are F-progressively measurable, for fixed (X, y, z) ∈ C([0, T], Rd) × Rn × Mn(R); the function g : C([0, T], Rd) × Ω → Rn is FT-measurable.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Assumptions

Assumption

• The following integrability condition holds:

E T

• b + f
• (t, ¯

0, 0, 0)dt 2 + T σ2(t, ¯ 0, 0, 0)dt + g(¯ 0)

• < ∞,

where ¯ 0 is the function constantly equals to 0 and · is the Euclidean distance.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Assumptions

Assumption

• The coefficients b, σ, f satisfy the following Lipschitz condition

in the spacial variable (X, y, z) ∈ C([0, T], Rd) × Rn × Mn(R): ∃K0 > 0 such that ξ(t, X, y, z)−ξ(t, X ′, y′, z′) ≤ K0

• X−X ′2,t+y−y′+z−z′
• uniformly in ω ∈ Ω, where ξ can be b, σ, f and

X − X ′2

2,t :=

t

0 X(s) − X ′(s)2ds + X(t) − X ′(t)2.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Assumptions

Assumption

• The function g satisfies the following Lipschitz condition:

∃K1 > 0, ∀X, X ′ ∈ C([0, T], Rd) : g(X)−g(X ′)2 ≤ K 2

1

T X −X ′2(t)dt +K 2

1 X(T)−X ′(T)2

uniformly in ω ∈ Ω.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Well-posedness on Small Time Interval

Theorem Under the above Assumptions, if σz <

1 K1 , then ∃δ > 0 such that

∀T < δ, the FBSDE dXt = bt(X, Yt, Zt)dt + σt(X, Yt, Zt)dWt X0 = x dYt = −ft(X, Yt, Zt)dt + ZtdWt YT = g(X) has an unique solution (X, Y , Z).

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Some Examples of Non-solvable FBSDEs

Case: σzK1 < 1 isn’t satisfied: dXt = (σ0 + Zt)dWt X0 = x0 (15) dYt = ZtdWt YT = XT. (16) From (15) we get XT − Xt = T

t (σ0 + Zs)ds. Combing with (16),

we get Yt = Xt + σ0(WT − Wt). (17) In particular, Y0 = x0 + σ0WT, where lays the contradiction.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Some Examples of Non-solvable FBSDEs

Case: No solution on R: dXt = Ytdt X0 = x0 dYt = ZtdWt YT = XT. We can actually solve explicitly the above FBSDE on [t, T]:            Xt = x0 1 − (T − s) 1 − (T − t) Yt = x0 1 − (T − t). Zt = 0. We can see that there is explosion in both forward and backward process when T − t approach 1.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Decoupling Field

Definition An F-progressively measurable random field u : [0, T] × Ω × C([0, T], Rd) → Rn with u(T, X) = g(X) is said to be a decoupling field of FBSDE if there exists a constant δ > 0 such that, for any 0 ≤ t1 < t2 ≤ T with t2 − t1 ≤ δ and any x ∈ L2(Ft1), the FBSDE with initial value x ∈ C([0, T], Rd) and terminal condition u(t2, ·) has an unique solution that satisfies Yt = u(t, X∧t) = u(t, X), t ∈ [0, T], P − a.s.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Well-posedness on Larger Time Interval

Theorem Under the above Assumptions, if σ depends only on X, then there exists Tmax > 0 depending on the Lipschitz constants of the coefficients of the following FBSDE dXt = bt(X, Yt, Zt)dt + σt(X)dWt X0 = x dYt = −ft(X, Yt, Zt)dt + ZtdWt YT = g(X) such that the FBSDE has an unique solution (X, Y , Z) on [0, Tmax]. Furthermore, there exists an unique decoupling field u such that ∀X, X ′, u(t, X) − u(t, X ′)2 ≤ K(t)||X − X ′||2,t, (18) where t → K(t) is a deterministic function.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Estimation of Z

Theorem Assume that all the coefficients of the FBSDE are deterministic and all the assumptions in the above theorem are satisfied, denote (X, Y , Z) the solution of the FBSDE, then |Zt| ≤ K(t) · ||σ||∞, (19) where K is the deterministic function given by the previous theorem.

Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Principal-Agent Problem with Uncertain Parameter Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

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 Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

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 Path-dependent FBSDEs Introduction Well-posedness on Small Time Interval Well-posedness on Larger Time Interval Estimation of Z

Thanks for your attention!

Kaitong HU Principal-Agent Problem and FBSDEs