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

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 Finite Horizon Principal-Agent Problem 1 Principal-Agent Problem with Uncertain Parameter 2 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 Finite Horizon Principal-Agent Problem 1 Principal-Agent Problem with Uncertain Parameter 2 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 ( X t ) 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 output process. He chooses his optimal control by solving his own optimization problem: � T α E α [ U A ( ξ − V A ( ξ ) = max c t ( α t ) d t )] . (1) 0 • In order to maximise his expected return, the Principal chooses the optimal contract by solving the following 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

Environment Finite Horizon Principal-Agent Problem The Agent’s Optimization Problem Principal-Agent Problem with Uncertain Parameter 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 Kaitong HU Principal-Agent Problem and FBSDEs

Environment Finite Horizon Principal-Agent Problem The Agent’s Optimization Problem Principal-Agent Problem with Uncertain Parameter 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 Consider ( P α,β , B ) as a weak solution F t := σ { W s , O s , µ, s ≤ t } . of the following controlled system � d X t = ( f ( t ) X t + α t ) d t + σ ( t ) d W t µ 0 = µ , (3) d O t = ( h ( t ) X t + β t ) d t + d B t O 0 = 0. (4) Here, ( X t , F t ), 0 ≤ t ≤ T , is the unobservable process while O is the observation of the system. Kaitong HU Principal-Agent Problem and FBSDEs

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

Environment Finite Horizon Principal-Agent Problem The Agent’s Optimization Problem Principal-Agent Problem with Uncertain Parameter 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 X t however is unknown to neither the Principal nor the Agent. Kaitong HU Principal-Agent Problem and FBSDEs

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

Environment Finite Horizon Principal-Agent Problem The Agent’s Optimization Problem Principal-Agent Problem with Uncertain Parameter The Principal’s Optimization Problem Prior Analysis Proposition Let ˆ X t = E α,β [ X t |F O t ] . We have the following control filter: d ˆ X t = ( f ( t ) ˆ  X t + α t ) d t + h ( t ) V ( t ) d I t µ 0 = m 0 , (5) ˆ   d O t = ( h ( t ) ˆ  X 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 h ( s )( X s − ˆ I t := B t + X s ) d s , (8) 0 which is a F O -adapted P α,β -Brownian motion. Kaitong HU Principal-Agent Problem and FBSDEs

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

Environment Finite Horizon Principal-Agent Problem The Agent’s Optimization Problem Principal-Agent Problem with Uncertain Parameter The Principal’s Optimization Problem Necessary Condition For a contract ξ ∈ Ξ , let ν ∗ ∈ M ∗ ( ξ ). Then the following FBSDE:  t ) − Z t ( h ( t ) ˆ d Y t = ( c t ( ν ∗ X t + β ∗ t )) d t + Z t d O t    �  h ( t ) Z t − ( f ( t ) − V ( t ) h 2 ( t )) P t  d P t =   � − Q t ( h ( t ) ˆ X t + β ∗ t ) d t + Q t d O t ,      d ˆ X t = ( f ( t ) ˆ t + h ( t ) V ( t )( h ( t ) ˆ  X t + α ∗ X t + β ∗ t )) d t + h ( t ) V ( t ) d O t ,  has a solution, denoted ( Y ∗ , Z ∗ , P ∗ , Q ∗ , ˆ X ∗ ), with P T = 0 and Y T = Γ A T ξ . Besides, for all t ∈ [0 , T ], for all ( α, β ) ∈ A × B , the optimal control ν ∗ = ( α ∗ , β ∗ ) must verify � ( P ∗ t + ∂ α c t ( α ∗ t , β ∗ t ))( α − α ∗ t ) ≥ 0 , (10) ( Z ∗ t + V ( t ) h ( t ) P ∗ t − ∂ β c t ( α ∗ t , β ∗ t ))( β − β ∗ t ) ≤ 0 . (11) Kaitong HU Principal-Agent Problem and FBSDEs

Environment Finite Horizon Principal-Agent Problem The Agent’s Optimization Problem Principal-Agent Problem with Uncertain Parameter 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:  t ) − Z t ( h ( t ) ˆ d Y t = ( c t ( ν ∗ X t + β ∗ t )) d t + Z t d O t    �  h ( t ) Z t − ( f ( t ) − V ( t ) h 2 ( t )) P t d P t =    � − Q t ( h ( t ) ˆ X t + β ∗ t ) d t + Q t d O t ,       d ˆ X t = ( f ( t ) ˆ X t + α ∗ t + h ( t ) V ( t )( h ( t ) ˆ X t + β ∗ t )) d t + h ( t ) V ( t ) d O t ,  where ν ∗ = ( α ∗ , β ∗ ) satisfies the condition � ( P ∗ t + ∂ α c t ( α ∗ t , β ∗ t ))( α − α ∗ t ) ≥ 0 , ( Z ∗ t + V ( t ) h ( t ) P ∗ t − ∂ β c t ( 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

Environment Finite Horizon Principal-Agent Problem The Agent’s Optimization Problem Principal-Agent Problem with Uncertain Parameter The Principal’s Optimization Problem Sufficient Condition Define for all ( α, β ) ∈ A × B and x ∈ R , t x 2 + Q ∗ G t ( α, β, x ) := c t ( α, β ) + h ( t ) Q ∗ t x β + P ∗ t α t ˆ − ( Z ∗ t + V ( t ) h ( t ) P ∗ t + Q ∗ X ∗ t ) β. Then ν ∗ = ( α ∗ , β ∗ ) is an optimal control if for all admissible control ν = ( α t , β t ) t ∈ [0 , T ] , we have E ν � � T t , ˆ G t ( α t , β t , X t ) − G t ( α ∗ t , β ∗ X ∗ t ) 0 � t , ˆ t )( X t − ˆ − ∂ X G t ( α ∗ t , β ∗ X ∗ X ∗ t ) d t ≥ 0 . Kaitong HU Principal-Agent Problem and FBSDEs

Environment Finite Horizon Principal-Agent Problem The Agent’s Optimization Problem Principal-Agent Problem with Uncertain Parameter The Principal’s Optimization Problem Sufficient Condition Proposition If ( α, β ) ∈ A × B → c t ( 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 ∂ 2 Q ∗ � � � � t ∈ 0 , 2 h ( t ) · inf β c t ( α, β ) , (12) ( α,β ) ∈ A × B for all t ∈ [0 , T ] . Kaitong HU Principal-Agent Problem and FBSDEs