[Paper Review] Jump-Diffusions in Hilbert Spaces: Existence, Stability and Numerics

Damir Filipović, Stefan Tappe+1 more|arXiv (Cornell University)|Oct 28, 2008

Stochastic processes and financial applicationsEconomics, Econometrics and Finance9 references3 citations

TL;DR

This paper introduces the 'method of the moving frame' to establish existence, uniqueness, and stability of mild and weak solutions for stochastic partial differential equations (SPDEs) with path-dependent coefficients driven by infinite-dimensional Wiener processes and compensated Poisson random measures. By applying a time-dependent coordinate transform that eliminates the drift term, the SPDE is reduced to a simpler stochastic differential equation (SDE), enabling systematic analysis and numerical approximation under general conditions, including non-Markovian and jump-diffusion settings.

ABSTRACT

By means of an original approach, called "method of the moving frame", we establish existence, uniqueness and stability results for mild and weak solutions of stochastic partial differential equations (SPDEs) with path dependent coefficients driven by an infinite dimensional Wiener process and a compensated Poisson random measure. Our approach is based on a time-dependent coordinate transform, which reduces a wide class of SPDEs to a class of simpler SDE problems. We try to present the most general results, which we can obtain in our setting, within a self-contained framework to demonstrate our approach in all details. Also several numerical approaches to SPDEs in the spirit of this setting are presented.

Motivation & Objective

• To address the lack of a unified framework for analyzing SPDEs with path-dependent coefficients and general Lévy noise, particularly jump-diffusions.
• To overcome the analytical challenge of non-invertible drift operators in infinite-dimensional SPDEs by introducing a time-dependent transformation.
• To establish existence, uniqueness, and stability of mild and weak solutions under general conditions, including non-Markovian and non-linear coefficients.
• To provide a self-contained, general-purpose approach that reduces complex SPDEs to solvable SDEs via coordinate transformation.
• To enable the development of high-order numerical schemes and facilitate large deviation and rough path analyses.

Proposed method

• The method employs a time-dependent coordinate transform, $ r_t \mapsto S_{-t} r_t $, to convert the original SPDE into a transformed SDE without a drift term.
• The transformed SDE is analyzed in the 'moving frame' where the dynamics are driven only by volatility and jump components, simplifying solution analysis.
• The approach relies on the Szőkefalvi-Nagy theorem to extend pseudo-contractive semigroups into groups, enabling backward time evolution crucial for the transformation.
• Existence and uniqueness are established via fixed-point arguments and $ L^p $-estimates on the transformed SDE, leveraging the Itô isometry and Fubini-type theorems.
• Stability results are derived through convergence of approximating sequences in $ L_T^2(\mathbb{P} \otimes \lambda) $, using Lebesgue's dominated convergence and Cauchy sequence arguments.
• Numerical schemes are constructed by approximating the transformed SDE using sequences in $ V \subset L_T^2(\mu \otimes \lambda) $, with convergence proven via Hölder’s inequality and the Itô isometry.

Experimental results

Research questions

• RQ1 Can existence, uniqueness, and stability of mild and weak solutions be established for SPDEs with path-dependent coefficients and general Lévy noise, including jumps and infinite-dimensional Wiener processes?
• RQ2 How can the drift term in infinite-dimensional SPDEs be eliminated via a time-dependent transformation without requiring invertibility in negative time?
• RQ3 What conditions ensure that the transformed SDE in the moving frame admits a solution that can be mapped back to yield a mild solution of the original SPDE?
• RQ4 To what extent can high-order numerical schemes be derived from this transformation-based approach for SPDEs with jumps and path dependence?
• RQ5 Can the method support advanced stochastic analysis, such as large deviations or rough path formulations, in the SPDE context?

Key findings

• The method of the moving frame reduces a wide class of SPDEs with path-dependent coefficients to a simpler SDE in the transformed space, enabling existence and uniqueness results.
• The Szőkefalvi-Nagy theorem is used to extend pseudo-contractive semigroups into groups, allowing the time-reversal required for the coordinate transformation.
• The transformed SDE is shown to have a unique solution in $ L_T^2(\mathbb{P} \otimes \lambda) $, with convergence of approximating sequences established via $ L^p $-estimates and Lebesgue's theorem.
• The paper proves that the solution of the original SPDE can be recovered via the inverse transformation $ r_t = S_t f_t $, yielding a mild solution.
• The method supports the construction of high-order numerical schemes by reducing the SPDE to a numerically tractable SDE in the moving frame.
• The framework allows for future extensions to large deviation principles and rough path theory, as the transformed SDE inherits regularity properties suitable for such analyses.