Continuous-time Principal-Agent Problem in Partially Observed System - PowerPoint PPT Presentation

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 Finite Horizon Principal-Agent Problem 1 Principal-Agent Problem with Uncertain Parameter 2 Environment The Agent’s Optimization Problem The Principal’s Optimization Problem Path-dependent FBSDEs 3 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 Finite Horizon Principal-Agent Problem 1 Principal-Agent Problem with Uncertain Parameter 2 Environment The Agent’s Optimization Problem The Principal’s Optimization Problem Path-dependent FBSDEs 3 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, one 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 ( X t ) 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 optimal control by solving his optimization problem: � T α E α [ U A ( ξ − V A ( ξ ) = max c t ( α t ) d t )] . (1) 0 • The Principal chooses the optimal contract by solving the non-zero sum Stackelberg game: E α ∗ ( ξ ) [ U P ( X T − ξ )] . V P = max (2) ξ 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 om & Milgrom 1987 . . Holstr¨ . 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 Environment Principal-Agent Problem with Uncertain Parameter The Agent’s Optimization Problem Path-dependent FBSDEs The Principal’s Optimization Problem Outline Finite Horizon Principal-Agent Problem 1 Principal-Agent Problem with Uncertain Parameter 2 Environment The Agent’s Optimization Problem The Principal’s Optimization Problem Path-dependent FBSDEs 3 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 Environment Principal-Agent Problem with Uncertain Parameter The Agent’s Optimization Problem Path-dependent FBSDEs 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 Consider ( P α,β , B ) as a weak solution F t := σ { X s , W s , µ, s ≤ t } . of the following controlled system � d µ t = ( f ( t ) µ t + α t ) d t + σ ( t ) d W t µ 0 = µ , (3) d X t = ( h ( t ) µ t + β t ) d t + d B t X 0 = 0. (4) Here, ( µ t , F t ), 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 Environment Principal-Agent Problem with Uncertain Parameter The Agent’s Optimization Problem Path-dependent FBSDEs The Principal’s Optimization Problem Assumptions (i) The prior distribution of µ is normal of mean and variance m 0 and σ 0 respectively, which is known to both the Agent and the Principal. Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Environment Principal-Agent Problem with Uncertain Parameter The Agent’s Optimization Problem Path-dependent FBSDEs The Principal’s Optimization Problem Assumptions (i) The prior distribution of µ is normal of mean and variance m 0 and σ 0 respectively, which is known to both the Agent and the Principal. (ii) The unobservable process µ t however is unknown to neither the Principal nor the Agent. Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Environment Principal-Agent Problem with Uncertain Parameter The Agent’s Optimization Problem Path-dependent FBSDEs The Principal’s Optimization Problem Assumptions (i) The prior distribution of µ is normal of mean and variance m 0 and σ 0 respectively, which is known to both the Agent and the Principal. (ii) The unobservable process µ t 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 Environment Principal-Agent Problem with Uncertain Parameter The Agent’s Optimization Problem Path-dependent FBSDEs The Principal’s Optimization Problem Prior Analysis Proposition µ t = E α,β [ µ t |F X Let ˆ t ] . We have the following control filter: d ˆ µ t = ( f ( t )ˆ µ t + α t ) d t + h ( t ) V ( t ) d I t µ 0 = m 0 , (5) ˆ    d X t = ( h ( t )ˆ µ t + β t ) d t + d I t X 0 = 0 , (6)  d V ( t ) = 2 f ( t ) V ( t ) − h ( t ) 2 V ( t ) 2 + σ ( t )  V 0 = σ 0 . (7)   d t Here I is the Innovation process define by � t I t := B t + h ( s )( µ s − ˆ µ s ) d s , (8) 0 which is a F X -adapted P α,β -Brownian motion. Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Environment Principal-Agent Problem with Uncertain Parameter The Agent’s Optimization Problem Path-dependent FBSDEs 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 c t the agent’s cost function at time t . The agent’s optimization problem can be then written as follow � T � T 0 ρ A E α,β [ e − t d t ξ − V A = sup c t ( α t , β t ) d t ] . (9) α,β 0 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 Environment Principal-Agent Problem with Uncertain Parameter The Agent’s Optimization Problem Path-dependent FBSDEs 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 P α ∗ ,β ∗ , the agent’s value function satisfies the following FBSDE: d Y t = c t ( α ∗ t , β ∗ t ) d t + Z t d I ∗ Y T = Γ A  T ξ , t   d P t = ( Z t − f ( t ) P t + V ( t ) h 2 ( t ) P t ) d t + Q t d I ∗  P T = 0 ,  t    µ t + β ∗ t ) d t + d I ∗  d X t = ( h ( t )ˆ X 0 = 0 ,  t µ t + α ∗ t ) d t + h ( t ) V ( t ) d I ∗ d ˆ µ t = ( f ( t )ˆ ˆ µ 0 = m 0 , t    ∂ α c t ( α ∗ , β ∗  t ) + P t = 0 ,     ∂ β c t ( α ∗ , β ∗  t ) − Z t − V ( t ) h ( t ) P t = 0 . Kaitong HU Principal-Agent Problem and FBSDEs

Finite Horizon Principal-Agent Problem Environment Principal-Agent Problem with Uncertain Parameter The Agent’s Optimization Problem Path-dependent FBSDEs The Principal’s Optimization Problem Necessary Condition Given a contract of the following form (if such a solution exists)  d Y t = c t ( α ∗ t , β ∗ µ t + β ∗ t ) d t + Z t d ( d X t − ( h ( t )ˆ t ) d t ) ,   d P t = ( Z t − ( f ( t ) − V ( t ) h 2 ( t )) P t ) d t + Q t ( d X t − ( h ( t )ˆ µ t + β ∗ t ) d t ) ,  µ t + α ∗ µ t + β ∗ d ˆ µ t = ( f ( t )ˆ t ) d t + h ( t ) V ( t )( d X t − ( h ( t )ˆ t ) d t ) ,  with terminal condition Y T = Γ A T ξ ( X ) and P T = 0, where � ∂ α c t ( α ∗ , β ∗ t ) + P t = 0 , ∂ β c t ( α ∗ , β ∗ t ) − Z t − V ( t ) h ( t ) P t = 0 . We can show that if the optimal control exists, necessarily the optimal control is ( α ∗ , β ∗ ). Kaitong HU Principal-Agent Problem and FBSDEs