[Paper Review] Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions

Rainer Buckdahn, Christian Keller+2 more|arXiv (Cornell University)|Jan 28, 2015

Stochastic processes and financial applications3 citations

TL;DR

This paper establishes a unified framework for fully nonlinear stochastic and rough PDEs (SPDEs/RPDEs) by introducing a notion of viscosity solutions via smooth test functions, proving consistency, stability, and a partial comparison principle. When the diffusion coefficient is semi-linear, it achieves a complete theory including global existence and a full comparison principle using the method of characteristics and rough path analysis.

ABSTRACT

We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We develop first a local theory of classical solutions and define then viscosity solutions through smooth test functions. Our notion of viscosity solutions is equivalent to the alternative one using semi-jets. Next, we prove basic properties such as consistency, stability, and a partial comparison principle in the general setting. When the diffusion coefficient is semi-linear (but the drift can be fully nonlinear), we establish a complete theory, including global existence and comparison principle. Our methodology relies heavily on the method of characteristics.

Motivation & Objective

• To develop a consistent theory of viscosity solutions for fully nonlinear second-order forward SPDEs and path-dependent PDEs (PPDEs) in a rough path setting.
• To unify the treatment of stochastic PDEs, path-dependent PDEs, and rough PDEs under a single analytical framework.
• To establish the existence and comparison principle for viscosity solutions when the diffusion coefficient is semi-linear.
• To prove key properties such as consistency, stability, and a partial comparison principle in the general case.
• To extend classical solution theory to local and global well-posedness using the method of characteristics and rough Taylor expansions.

Proposed method

• Formulate the SPDE (1.1) in Stratonovich form and reinterpret it as a forward PPDE (1.2) using Dupire’s path derivatives.
• Re-express the SPDE as a rough PDE (1.3) using Gubinelli’s derivative and rough path theory, linking it to controlled rough paths.
• Define classical solutions via the method of characteristics, deriving local well-posedness for the associated characteristic equations.
• Introduce viscosity solutions through smooth test functions and prove equivalence to the semi-jet-based definition.
• Establish a change of variables formula and stability properties for viscosity solutions using rough path techniques.
• Prove a partial comparison principle in the general case and a full comparison principle when the diffusion coefficient is semi-linear, enabling global existence.

Experimental results

Research questions

• RQ1 How can viscosity solutions be consistently defined for fully nonlinear rough PDEs in a path-dependent setting?
• RQ2 What conditions ensure the stability and consistency of viscosity solutions in the context of rough PDEs?
• RQ3 Under what conditions does a full comparison principle hold for viscosity solutions of rough PDEs?
• RQ4 Can global existence of viscosity solutions be established when the diffusion coefficient is semi-linear?
• RQ5 How does the method of characteristics extend to the rough PDE setting to ensure well-posedness?

Key findings

• The notion of viscosity solutions via smooth test functions is equivalent to the semi-jet-based definition, ensuring consistency in the solution framework.
• The theory proves stability of viscosity solutions under uniform convergence, a key property for numerical and approximation schemes.
• A partial comparison principle is established in the general case, providing a foundation for uniqueness in the absence of full regularity.
• When the diffusion coefficient is semi-linear, a full comparison principle holds, enabling the proof of uniqueness and global existence of viscosity solutions.
• Global existence of viscosity solutions is established under semi-linear diffusion, with solutions in the space $ C^{k, ext{loc}}_{eta,eta}(bR_T) $, ensuring Hölder regularity in space and time.
• The method of characteristics, combined with rough Taylor expansion and rough path integration, yields local well-posedness for classical solutions in $ C^{k+2}_{eta,eta}([0,T]) $ with explicit estimates depending on $ f $, $ g $, and initial data.