[Paper Review] Viscosity Solutions for HJB Equations on the Process Space

Jianjun Zhou, Nizar Touzi+1 more|arXiv (Cornell University)|Jan 10, 2024

Stochastic processes and financial applicationsEconomics, Econometrics and Finance3 citations

TL;DR

This paper introduces a novel viscosity solution framework for fully nonlinear, second-order, path-dependent Hamilton-Jacobi-Bellman (HJB) equations on the process space, enabling the treatment of mean field control problems with common noise and degenerate volatility. By incorporating a singular, absolutely continuous component into test functions, the authors establish existence and a comparison principle under merely Lipschitz continuity, ensuring uniqueness of the viscosity solution and well-posedness of the HJB equation.

ABSTRACT

In this paper we investigate a path dependent optimal control problem on the process space with both drift and volatility controls, with possibly degenerate volatility. The dynamic value function is characterized by a fully nonlinear second order path dependent HJB equation on the process space, which is by nature infinite dimensional. In particular, our model covers mean field control problems with common noise as a special case. We shall introduce a new notion of viscosity solutions and establish both the existence and the comparison principle, under merely Lipschitz/Holder continuity assumptions. The main feature of our notion is that, besides the standard smooth part, the test function consists of an extra singular component which allows us to handle the second order derivatives of the smooth test functions without invoking the Crandall-Ishii lemma. We shall use the doubling variable arguments, combined with the Ekeland-Borwein-Preiss variational principle in order to overcome the noncompactness of the state space. A smooth gauge-type function on the path space is crucial for our estimates.

Motivation & Objective

• To address the lack of a robust viscosity solution theory for fully nonlinear, second-order, path-dependent HJB equations arising in stochastic control with path-dependent and degenerate volatility.
• To extend the applicability of viscosity solutions to mean field control problems with common noise, a framework not previously covered under standard HJB theory.
• To establish a comparison principle and uniqueness of viscosity solutions under minimal regularity assumptions—specifically, Lipschitz continuity—without requiring smoothness or compactness.
• To develop a new test function structure that incorporates a singular, absolutely continuous component to bypass technical obstacles in second-order derivative estimation.
• To ensure well-posedness of the HJB equation on the Wasserstein space induced by mean field control with common noise.

Proposed method

• Propose a new notion of viscosity solution where test functions are decomposed into a smooth part and a singular, absolutely continuous-in-time component to handle second-order derivatives without Ishii’s lemma.
• Use the doubling variable method combined with the Ekeland-Borwein-Preiss variational principle to overcome noncompactness of the path space.
• Construct a smooth gauge-type function on the path space to control estimates and ensure uniform bounds in the comparison principle argument.
• Apply a transformation-based idea from the constant volatility case to motivate the construction of the singular component in test functions.
• Tailor the singular component to cancel diffusion terms in the HJB equation, enabling control over second-order derivative terms in the doubling variable argument.
• Leverage the structure of joint law dependence between state and control processes to derive estimates in the path-dependent setting.

Experimental results

Research questions

• RQ1 Can a viscosity solution theory be developed for fully nonlinear, second-order, path-dependent HJB equations on the process space with degenerate volatility?
• RQ2 How can the comparison principle be established under only Lipschitz continuity, without requiring smoothness or compactness?
• RQ3 What is the role of a singular, absolutely continuous component in test functions for handling second-order derivatives in infinite-dimensional HJB equations?
• RQ4 Can this framework cover mean field control problems with common noise as a special case?
• RQ5 Is the dynamic value function the unique viscosity solution of the HJB equation under minimal regularity assumptions?

Key findings

• The authors establish a comparison principle for the HJB equation under only Lipschitz continuity assumptions, a significant weakening of prior conditions in the literature.
• The dynamic value function of the path-dependent optimal control problem is proven to be the unique viscosity solution of the HJB equation.
• The new test function structure—consisting of a smooth part and a singular, absolutely continuous component—enables the bypassing of Ishii’s lemma and controls second-order derivative terms.
• The framework ensures well-posedness of the HJB equation on the Wasserstein space induced by mean field control with common noise.
• The method overcomes noncompactness of the state space via the Ekeland-Borwein-Preiss variational principle and a carefully constructed gauge function on the path space.
• The results generalize existing viscosity solution theories for mean field control, covering both drift and volatility controls with possibly degenerate diffusion.