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Randomisation with moral hazard: a path to existence of optimal contracts

Daniel Kršek, Dylan Possamaï

TL;DR

The paper addresses the existence of optimal contracts in a continuous-time principal–agent setting with moral hazard by introducing a relaxed, measure-valued control framework for the agent. It characterizes the agent’s continuation value via a BSDE driven by a martingale measure and reformulates the principal’s problem as a weak control problem, enabling existence proofs under broad contract constraints through compactness arguments. This approach avoids the regularity pitfalls of fully nonlinear degenerate PDEs and provides implementability via finite observation techniques and three practical application templates. The framework also discusses measurability issues, the role of randomness, and extensions to random horizons, discounted factors, inter-temporal payments, and volatility control, offering a versatile toolkit for robust contract design. Overall, it demonstrates that randomisation can restore existence in complex continuous-time principal–agent models and yields concrete pathways to implementable contracts under realistic constraints.

Abstract

We study a generic principal-agent problem in continuous time on a finite time horizon. We introduce a framework in which the agent is allowed to employ measure-valued controls and characterise the continuation utility as a solution to a specific form of a backward stochastic differential equation driven by a martingale measure. We leverage this characterisation to prove that, under appropriate conditions, an optimal solution to the principal's problem exists, even when constraints on the contract are imposed. In doing so, we employ compactification techniques and, as a result, circumvent the typical challenge of showing well-posedness for a degenerate partial differential equation with potential boundary conditions, where regularity problems often arise.

Randomisation with moral hazard: a path to existence of optimal contracts

TL;DR

The paper addresses the existence of optimal contracts in a continuous-time principal–agent setting with moral hazard by introducing a relaxed, measure-valued control framework for the agent. It characterizes the agent’s continuation value via a BSDE driven by a martingale measure and reformulates the principal’s problem as a weak control problem, enabling existence proofs under broad contract constraints through compactness arguments. This approach avoids the regularity pitfalls of fully nonlinear degenerate PDEs and provides implementability via finite observation techniques and three practical application templates. The framework also discusses measurability issues, the role of randomness, and extensions to random horizons, discounted factors, inter-temporal payments, and volatility control, offering a versatile toolkit for robust contract design. Overall, it demonstrates that randomisation can restore existence in complex continuous-time principal–agent models and yields concrete pathways to implementable contracts under realistic constraints.

Abstract

We study a generic principal-agent problem in continuous time on a finite time horizon. We introduce a framework in which the agent is allowed to employ measure-valued controls and characterise the continuation utility as a solution to a specific form of a backward stochastic differential equation driven by a martingale measure. We leverage this characterisation to prove that, under appropriate conditions, an optimal solution to the principal's problem exists, even when constraints on the contract are imposed. In doing so, we employ compactification techniques and, as a result, circumvent the typical challenge of showing well-posedness for a degenerate partial differential equation with potential boundary conditions, where regularity problems often arise.
Paper Structure (31 sections, 29 theorems, 236 equations)

This paper contains 31 sections, 29 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relaxed agent's problem
  4. The model
  5. The agent's problem and incentive compatibility
  6. Relaxed principal's problem: different points of view
  7. Weak principal's problem
  8. Comparison of the two formulations
  9. Feedback contracts
  10. Additional remarks, extensions and examples
  11. Random time horizon and retirement
  12. Controlled discount factor
  13. Exponential utility does not help
  14. Non-relaxed controls
  15. Bounds on moments are in a certain sense necessary
  16. ...and 16 more sections

Key Result

Lemma 2.4

Let $M$ be a continuous $(\mathbb{F},\mathbb{P})$--martingale measure. Then there is a measurable map $m : \Omega \longrightarrow {\cal M}_f(E \times [0,T])$ such that Moreover, $m$ disintegrates as $m(\mathrm{d}e,\mathrm{d}s)=m_s(\mathrm{d}e)\mathrm{d}s$ for some $($regular$)$$\mathbb{F}$-predictable kernel $m : \Omega \times [0,T] \longrightarrow {\cal M}_f(E).$

Theorems & Definitions (99)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Example 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Lemma 2.9
  • ...and 89 more