Distance between closed sets and the solutions to stochastic partial differential equations

TL;DR

This paper investigates conditions under which solutions to stochastic partial differential equations remain close to specific subsets of the state space, with applications to interest rate curve modeling in finance.

Contribution

It provides new criteria for the stability of solutions near subsets, including finite dimensional submanifolds, extending to deterministic PDEs and finance applications.

Findings

Solutions stay close to subsets under certain conditions

Results include deterministic PDE cases

Applications to interest rate curve evolution

Abstract

The goal of this paper is to clarify when the solutions to stochastic partial differential equations stay close to a given subset of the state space for starting points which are close as well. This includes results for deterministic partial differential equations. As an example, we will consider the situation where the subset is a finite dimensional submanifold with boundary. We also discuss applications to mathematical finance, namely the modeling of the evolution of interest rate curves.