[Paper Review] Perron's method for viscosity solutions of semilinear path dependent PDEs

Zhenjie Ren|arXiv (Cornell University)|Mar 7, 2015

Stochastic processes and financial applications18 references3 citations

TL;DR

This paper establishes the existence of viscosity solutions to semilinear path-dependent PDEs using Perron's method, showing that the supremum of viscosity subsolutions is a viscosity solution. It introduces a novel regularization technique for semicontinuous viscosity solutions and proves a comparison principle, enabling a new, concise proof of the optimal stopping problem under nonlinear expectations.

ABSTRACT

This paper proves the existence of viscosity solutions of path dependent semilinear PDEs via Perron's method, i.e. via showing that the supremum of viscosity subsolutions is a viscosity solution. We use the notion of viscosity solutions introduced by Ekren, Keller, Touzi and Zhang, in whose work all smooth processes which are tangent in mean are considered as test functions. We also provide a comparison result for semicontinuous viscosity solutions, by using a regularization technique. As an interesting byproduct, we give a new short proof for the optimal stopping problem with semicontinuous obstacles.

Motivation & Objective

• To establish the existence of viscosity solutions for semilinear path-dependent PDEs using PDE-based methods rather than backward SDEs.
• To extend the comparison principle to semicontinuous viscosity solutions, which is essential for Perron's method.
• To develop a new regularization technique that preserves viscosity subsolution properties and ensures continuity.
• To provide a short, self-contained proof of the optimal stopping problem with semicontinuous obstacles under nonlinear expectations.
• To lay groundwork for future analysis of fully nonlinear path-dependent PDEs by introducing a novel regularization approach.

Proposed method

• Applies Perron's method by taking the pointwise supremum of all viscosity subsolutions to construct a candidate solution.
• Introduces a new regularization technique based on backward distance along paths to mollify semicontinuous viscosity subsolutions into continuous ones.
• Uses a regularization that preserves the viscosity subsolution property and ensures convergence to the original function.
• Employs the Skorokhod decomposition's minimum condition to derive a new, simplified proof of the optimal stopping result for semicontinuous obstacles.
• Establishes a comparison result for upper semicontinuous viscosity subsolutions and lower semicontinuous viscosity supersolutions via regularization.
• Relies on the notion of viscosity solutions defined via test functions that are smooth and tangent in mean, as introduced in [9].

Experimental results

Research questions

• RQ1 Can Perron's method be successfully applied to construct viscosity solutions for semilinear path-dependent PDEs?
• RQ2 Does a comparison principle hold for semicontinuous viscosity solutions in the path-dependent setting?
• RQ3 Can a regularization technique be constructed that preserves the viscosity subsolution property while making the function continuous?
• RQ4 Is the optimal stopping problem with semicontinuous obstacles solvable under nonlinear expectations using PDE methods?
• RQ5 Can the regularization technique developed be extended to the fully nonlinear case of path-dependent PDEs?

Key findings

• The supremum of all viscosity subsolutions is a viscosity solution, establishing existence via Perron's method in the path-dependent setting.
• A novel regularization based on backward distance along paths preserves the viscosity subsolution property and ensures continuity and convergence.
• A comparison principle is proven for semicontinuous viscosity solutions using the proposed regularization, enabling the application of Perron's method.
• A new, concise proof is provided for the optimal stopping problem with semicontinuous obstacles under nonlinear expectations, using the minimum condition of Skorokhod decomposition.
• The regularization technique is shown to be continuous and stable under the given nonlinear expectation, with convergence established via the expectation of hitting times.
• The method provides a PDE-based alternative to backward SDEs for constructing viscosity solutions, particularly valuable in the fully nonlinear case where SDEs are not available.