[Paper Review] Spatial rough path lifts of stochastic convolutions

Peter K. Friz, Benjamin Gess+2 more|arXiv (Cornell University)|Oct 31, 2012

Stochastic processes and financial applications11 references3 citations

TL;DR

This paper establishes sufficient conditions for finite controlled rho-variation of the covariance of Gaussian processes with stationary increments, leveraging concavity or convexity of their variance function. These conditions enable the construction of spatial rough path lifts for solutions to fractional stochastic heat equations with additive, possibly colored noise in the space variable, extending recent advances in rough path theory.

ABSTRACT

We present sufficient conditions for finite controlled rho-variation of the covariance of Gaussian processes with stationary increments, based on concavity or convexity of their variance function. The motivation for this type of conditions comes from recent work of Hairer [CPAM,2011]. Our results allow to construct rough paths lifts of solutions to a class of fractional stochastic heat equations with additive, possibly colored Wiener noise with respect to their space variable.

Motivation & Objective

• To identify sufficient conditions on the variance function of Gaussian processes with stationary increments to ensure finite controlled rho-variation of their covariance.
• To extend the applicability of rough path theory to stochastic PDEs with space-time noise.
• To provide a theoretical foundation for constructing rough path lifts of solutions to fractional stochastic heat equations.
• To address the challenge of handling colored Wiener noise in the spatial variable through pathwise integration methods.

Proposed method

• Analyzing the variance function of Gaussian processes with stationary increments to determine its concavity or convexity properties.
• Deriving sufficient conditions on the variance function that guarantee finite controlled rho-variation of the covariance function.
• Applying these conditions to the covariance structure of the solution process of fractional stochastic heat equations.
• Using the resulting rho-variation control to construct a spatial rough path lift of the solution process.
• Leveraging recent theoretical advances in rough path theory, particularly from Hairer (2011), to justify the lift construction.
• Ensuring the method applies even when the noise is colored in space, not just space-time white noise.

Experimental results

Research questions

• RQ1 Under what conditions on the variance function of a Gaussian process with stationary increments is its covariance function of finite controlled rho-variation?
• RQ2 How can the concavity or convexity of the variance function be used to control the rough path properties of the process?
• RQ3 Can rough path lifts be constructed for solutions of fractional stochastic heat equations with additive, possibly colored, space-time noise?
• RQ4 What role does the rho-variation of the covariance play in enabling pathwise solution construction for such SPDEs?
• RQ5 How does this approach extend or generalize prior results in rough path theory for stochastic PDEs?

Key findings

• Finite controlled rho-variation of the covariance function is guaranteed if the variance function is concave or convex, depending on the parameter regime.
• The sufficient conditions on the variance function allow for the construction of a spatial rough path lift of the solution process.
• The method applies to fractional stochastic heat equations with additive noise that is colored in space, not just white in space.
• The results extend Hairer's framework to a broader class of Gaussian noise processes in SPDEs.
• The approach enables pathwise solution construction for SPDEs where traditional Itô or Stratonovich integration may not suffice.