PaperlayerView exact revision on arXiv

Source-linked AI summary

Dynamic programming approach to principal-agent problems

Jakša Cvitanić, Dylan Possamaï, Nizar Touzi

arXiv:1510.07111v3math.OCecon.GNmath.PRq-fin.EC

TL;DR

The paper addresses how to solve finite-horizon continuous-time Principal-Agent problems with terminal lump-sum payment, including settings where effort affects output volatility. It uses contracts that give the agent’s value a dynamic-programming representation and proves the restriction loses no generality under technical conditions, reducing the game to a standard stochastic control problem.

  • Problem

    With arbitrary contracts, the agent’s stochastic control problem and the principal’s optimization can be difficult, especially when the agent’s optimal control depends nonlinearly on the contract.

    Source: §1 · p. 2

  • Method

    The authors optimize over contracts whose agent-value process has a dynamic-programming representation, characterize the agent’s effort through the Hamiltonian, and use second-order BSDE results.

    Source: §1 · p. 3

  • Results

    The restricted contract family attains the same supremum as all feasible contracts under technical conditions, reducing the game to a finite-dimensional maximization for a standard stochastic control problem.

    Source: §1 · p. 3, §3.2 · p. 12

  • Takeaways & Limitations

    The method provides a systematic approach for most continuous-time Brownian-motion models with a final lump-sum payment, including agent control of drift and volatility.

    Source: §1 · p. 2, §7 · p. 27

    The approach remains to be extended to the infinite-horizon model with possibly continuous payments.

    Source: §1 · p. 2

Abstract

from arXiv · show

We consider a general formulation of the Principal-Agent problem with a lump-sum payment on a finite horizon, providing a systematic method for solving such problems. Our approach is the following: we first find the contract that is optimal among those for which the agent's value process allows a dynamic programming representation, for which the agent's optimal effort is straightforward to find. We then show that the optimization over the restricted family of contracts represents no loss of generality. As a consequence, we have reduced this non-zero sum stochastic differential game to a stochastic control problem which may be addressed by the standard tools of control theory. Our proofs rely on the backward stochastic differential equations approach to non-Markovian stochastic control, and more specifically, on the recent extensions to the second order case.

1 Introduction

The paper develops a systematic method for finite-horizon continuous-time Principal-Agent problems with terminal lump-sum payment, including cases where agent effort controls output volatility. It uses contracts with a dynamic-programming representation of agent value and shows this restriction loses no generality under technical conditions.

1 Introduction

  • The approach is intended for finite-horizon contracts paid at the terminal time; analysis of Sannikov’s infinite-horizon model with continuous payments is left for future work.Source: §1 · pp. 1–2
  • The method covers multidimensional models with arbitrary utility and agent effort affecting both output drift and volatility, without requiring Markovian assumptions.It also includes earlier cases as special cases, subject to technical considerations.Source: §1 · p. 2
  • This approach addresses the difficulty that arbitrary contracts can make the agent’s control problem and the principal’s optimization highly nonlinear and hard to solve.The literature has also approached the general problem using stochastic versions of the Pontryagin maximum principle.Source: §1 · p. 2
  • The authors restrict contracts to those giving the agent’s value process a dynamic-programming representation, making incentive-compatible effort identifiable through a Hamiltonian.Under relatively mild technical conditions, optimizing over this family achieves the same supremum as optimizing over all feasible contracts.Source: §1 · p. 3

2 Principal-Agent problem

The model represents output on a continuous-path space and lets the agent influence its distribution through effort. The principal offers an output-contingent terminal contract, while the paper frames both parties’ problems so they can be reduced to dynamic programming.

2.1 The canonical space of continuous paths

  • Output is represented by the canonical process on continuous paths, with quadratic variation defined consistently across semimartingale measures.Source: §2.1 · p. 4

2.2 Controlled state equation

  • The agent’s optional effort is a pair of controls acting through bounded drift and diffusion coefficients; distinct diffusion choices can induce mutually singular laws.Source: §2.2 · pp. 4–5

2.3 Agent’s problem

  • The agent chooses effort given a terminal compensation contract, incurs effort costs, and accepts only when expected utility meets reservation utility R.The framework also accommodates risk-averse utility, including exponential utilities under modified assumptions [7].Source: §2.3 · pp. 6–7

2.4 Principal’s problem

  • The principal values output over the contract period and pays compensation at its end, but optimization over potentially non-Markovian contracts is not initially a standard dynamic-programming control problem.The model assumes bounded controls to set aside singular-control difficulties; when diffusion may degenerate, the controlled drift is restricted to the diffusion matrix’s range.Source: §2.4 · pp. 7–8
  • When the agent has multiple optimal responses to a contract, the model assumes he selects the response best for the principal.Source: §2.4 · p. 7

3 Reduction to a standard stochastic control problem

The section represents selected contracts through the agent’s value process, making the agent’s optimal action identifiable and the principal’s problem reducible to standard stochastic control. Under stated conditions, restricting contracts preserves the principal’s value; the reduced problem can be characterized by an HJB equation.

3.1 Family of restricted contracts

  • For restricted contracts, the agent’s value equals the constructed process, and optimal actions maximize the Hamiltonian.The resulting contract form is independent of the agent’s control model.Source: §3.1 · pp. 9–10

3.2 Restricted Principal’s problem and main reduction result

  • Under conditions in Theorem 3.6, restricting contracts does not affect the principal’s value; a general case uses convenient approximations to attain the optimal value.Exact representation is sufficient when volatility is uncontrolled and the weak solution has the predictable representation property.Source: §3.2 · p. 12

3 Reduction to a standard stochastic control problem

  • The restricted problem is standard stochastic control with controls (Z, Γ) and controlled state process (X, Y).The dynamics of X are given in weak formulation, while Y follows the restricted-contract dynamics.Source: §3.3 · p. 12

3.3 HJB characterization of the reduced problem

  • The Markovian reduced problem admits an HJB verification result when the value function and Hamiltonian maximizers satisfy the stated conditions.The result identifies the value and an optimal control.Source: §3.3 · pp. 12–13
  • Principal’s problem generally has both x and y as state variables, although Section 6 considers conditions for reducing their number.Source: §3.3 · p. 13
  • An optimal contract’s existence requires sufficient regularity of the value function for its partial derivatives to serve as feedback in the state equation.The authors relate this existence question to that of a generic stochastic control problem.Source: §3.3 · p. 13

4 Fixed volatility of the output

For drift-only effort, the section characterizes the admissible contracts through the agent’s value process and a backward-SDE representation. Under the stated conditions, this characterization covers the full admissible contract set.

4 Fixed volatility of the output

  • The main result does not require Condition (4.2), although the existing continuous-time Principal-Agent literature falls within that framework.The section also notes that the predictable martingale representation property holds under a Lipschitz condition on the diffusion coefficient.Source: §4 · p. 13
  • Under Condition (4.2) and nonempty V, Theorem 4.2 identifies the admissible contracts Ξ with the family represented by (Y,Z).The proof reduces the result to a backward SDE representation of any ξ ∈ Ξ.Source: §4 · p. 14
  • The representation is established using backward-SDE existence and uniqueness under the stated integrability and Lipschitz conditions.The proof invokes predictable martingale representation and the zero-one law under P′.Source: §4 · p. 14

5 The general case

The general case extends the reduction to settings where the agent controls both drift and volatility. The proof uses second-order backward SDEs to represent the agent’s value and establish the restricted-contract reduction.

5 The general case

  • The general case covers agent control of both the drift and volatility of the output, bypassing Condition (4.2) from the fixed-volatility setting.The volatility-control case requires recent second-order backward-SDE extensions.Source: §5 · p. 14

5.1 From fully nonlinear to semilinear Hamiltonian

  • The fully nonlinear Hamiltonian is recast as semilinear by isolating its partial maximization over squared diffusion.The passage describes this transformation through a convex-conjugate relation.Source: §5.1 · p. 15

5.2 Isolating the control on quadratic variation

  • For fixed quadratic variation, the agent’s value admits a backward-SDE representation; across measures, its dynamic value yields a supermartingale representation with nondecreasing K.When an optimal measure exists, K is zero under that measure; otherwise its minimality is accounted for in the representation.Source: §5.2 · p. 16

5.3 2BSDE characterization of Agent’s problem

  • The 2BSDE has a unique solution, represents the agent’s value as sup over P⁰ of E[Y₀], and characterizes optimal controls through K = 0 and maximization of F.The value representation is stated in Proposition 5.4; the optimality conditions are described in its proof.Source: §5.3 · pp. 18–19

5.4 Proof of Theorem 3.6

  • The proof establishes the reduction by approximating arbitrary admissible contracts with restricted contracts while preserving the principal’s objective.The section identifies the optimization over the restricted family with the unrestricted value.Source: §5.4 · pp. 19–20

6 Special cases and examples

The special cases show how the principal’s problem simplifies under particular utility and model assumptions, and how optimal contracts perform in drift- and volatility-control settings. In the no-cost volatility case, a contract can attain the principal’s corresponding first-best expected utility under the stated conditions.

6.1 Coefficients independent of x

  • With coefficients independent of x, an exponential-utility transformation reduces the problem to two state variables, while linear liquidation reduces it further to an ODE on [0, T].These reductions apply under the stated conditions, including zero effort cost and linear liquidation for the ODE case.Source: §6.1 · p. 21
  • With a risk-neutral principal and zero effort cost, a change of variables reduces the problem to [0, T]×R^d, and the principal offers the agent minimum reservation utility R.The minimum-reservation-utility result follows from the increasing flow in the agent’s initial value.Source: §6.1 · p. 21

6.2 Drift control and quadratic cost: Cvitani´c, Wan and Zhang [8]

  • With the stated uniqueness and HJB conditions, the optimal drift-control contract depends only on terminal output and induces the principal-optimal effort, giving both parties their optimal expected utilities.The contract is described as a moral-hazard modification of the Borch rule: the marginal-utility ratio is linear in the principal’s utility.Source: §6.2 · pp. 22–23

6.3 Volatility control with no cost: Cadenillas, Cvitani´c and Zapatero [5]

  • Under the stated ODE, regularity, and participation conditions, the volatility-control contract lets the principal attain her corresponding first-best expected utility.The agent’s chosen action need not match the action the principal would dictate in the first-best case.Source: §6.3 · p. 25

7 Conclusions

The paper presents a dynamic-programming approach for finite-horizon principal-agent problems with terminal lump-sum payments, restricting attention to contracts that represent the agent’s value dynamically and then establishing no loss of generality. The approach covers most existing continuous-time Brownian models with final lump-sum payment, while continuous-payment models remain future work.

7 Conclusions

  • Restricting contracts to those with a dynamic-programming representation of the agent’s value makes incentive-compatible effort straightforward to identify, and the restriction entails no loss of generality.The proofs rely on recent second-order backward stochastic differential equation results.Source: §7 · p. 27
  • The method encompasses most existing continuous-time Brownian-motion models with only a final lump-sum payment.Source: §7 · p. 27
  • Extending the approach to possibly continuous payments remains future work, with a proposed roadmap of representing the agent’s value dynamically and optimizing the principal’s objective over payments expressed through that value.Source: §7 · p. 27
Loading 1510.07111v3…