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

TL;DR

This paper develops a new mathematical framework for principal-agent problems with moral hazard, proving the existence of optimal contracts using measure-valued controls and backward stochastic differential equations.

Contribution

It introduces a novel approach employing measure-valued controls and BSDEs to establish the existence of optimal contracts under constraints in continuous-time models.

Findings

Existence of optimal contracts proven under new framework

Use of measure-valued controls simplifies well-posedness issues

Circumvents PDE regularity problems with compactification techniques

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.