On the smoothness of value functions and the existence of optimal strategies in diffusion models

Abstract

Studies of dynamic economic models often rely on each agent having a smooth value function and a well-defined optimal strategy. For time-homogeneous optimal control problems with a one-dimensional diffusion, we prove that the corresponding value function must be twice continuously differentiable under Lipschitz, growth, and non-vanishing-volatility conditions. Under similar conditions, the value function of any optimal stopping problem is shown to be (once) continuously differentiable. We also provide sufficient conditions, based on comparative statics and differential methods, for the existence of an optimal control in the sense of strong solutions. The results are applied to growth, experimentation, and dynamic contracting settings.

Introduction to Symposium on Dynamic Contracts and Mechanism Design

“…Under these conditions, Strulovici and Szydlowski (2015) establish that the value function is the unique smooth solution to the HJB equation. In turn, they show that the HJB equation can be used to express a candidate optimal strategy in terms of the value function and its derivatives.…”

“…In their leading example of repeat sales of a good or service, they establish that commonly observed contract features such as flat rates, free consumption units, and two-part tariffs are natural parts of optimal contracts. The contributions by Strulovici and Szydlowski (2015) and Williams (2015) in this Symposium develop some of the associated optimization results and derive explicit solutions to a class of dynamic principal-agent problems, respectively.…”

“…The ninth paper in the Symposium by Strulovici and Szydlowski (2015) considers a family of single-agent dynamic problems where uncertainty is generated by a Brownian motion. The authors consider both stochastic control and optimal stopping problems, such as those analyzed in the agency models of Bergemann and Strack (2015a) and Kruse and Strack (2015) in this Symposium.…”

“…In turn, they show that the HJB equation can be used to express a candidate optimal strategy in terms of the value function and its derivatives. Finally, Strulovici and Szydlowski (2015) provide conditions under which the candidate solution indeed leads to an admissible optimal control.…”

Breakdowns

“…Our paper is thus the first in the economics literature to identify in a fully analytic fashion-and even in closed form when breakdowns are conclusive-a failure of smooth pasting in a continuous-time, non-zero-sum stochastic game and to recognize its replacement by the principle of continuous pasting. 9 Starting with Samuelson (1965), smooth-pasting conditions have been used extensively in (financial) economics to solve problems of optimal stopping or control of one-dimensional diffusions; see Dumas (1991), Dixit (1993), and Dixit and Pindyck (1994) for classic references, and Strulovici and Szydlowski (2014) for a recent contribution. All these papers specify a diffusion coefficient that is bounded away from zero, which (for sufficiently well behaved terminal reward functions) ensures smooth pasting even when the underlying process can jump; see Bayraktar and Xing (2012).…”