[Paper Review] MARTINGALE SOLUTIONS FOR STOCHASTIC EQUATION OF REACTION DIFFUSION TYPE DRIVEN BY L ´ EVY NOISE OR POISSON RANDOM MEASURE

Erika Hausenblas|arXiv (Cornell University)|Oct 28, 2010

Stochastic processes and financial applications48 references16 citations

TL;DR

This paper establishes the existence of martingale solutions for stochastic reaction-diffusion equations driven by Lévy noise or Poisson random measures, using a novel approach that first constructs solutions for SPDEs with bounded and continuous coefficients before extending to the general Lévy case. The key contribution is a positive resolution to a long-standing open problem regarding martingale solutions driven by genuine Lévy processes.

ABSTRACT

In this paper we are interested in a reaction diusion equa- tion driven by Poissonian noise respective Levy noise. For this aim we �rst show existence of a martingale solution for an SPDE of parabolic type driven by a Poisson random measure with only continuous and bounded coecients. This result is transferred to an parabo lic SPDE driven by Levy noise. In a second step, we show existence of a martin- gale solution of reaction diusion type, also driven by Poissonian noise respective Levy noise. Our results answer positively a long standing open question about existence of martingale solutions driven by genuine Levy processes.

Motivation & Objective

• To address the longstanding open problem of proving existence of martingale solutions for reaction-diffusion SPDEs driven by genuine Lévy processes.
• To establish the existence of martingale solutions for parabolic SPDEs driven by Poisson random measures with only continuous and bounded coefficients.
• To extend the existence result from Poisson noise to general Lévy noise via a limiting or approximation procedure.
• To provide a rigorous framework for handling SPDEs with jump noise in the context of reaction-diffusion type equations.

Proposed method

• First, construct a martingale solution for a parabolic SPDE driven by a Poisson random measure using a fixed-point argument under continuity and boundedness assumptions on coefficients.
• Apply a time-changed Lévy process representation to connect the Poisson random measure to Lévy noise, enabling the transfer of results to Lévy-driven SPDEs.
• Use a priori estimates and tightness arguments to handle the convergence of approximating sequences in the space of probability measures on path space.
• Apply the Yamada-Watanabe type argument to ensure the existence of a martingale solution in the weak sense for the reaction-diffusion SPDE.
• Utilize the canonical representation of Lévy processes via Poisson random measures to reduce the general Lévy case to the Poisson case.
• Leverage the structure of the reaction-diffusion equation to ensure the necessary regularity and integrability for the solution to be a local martingale.

Experimental results

Research questions

• RQ1 Does a martingale solution exist for a parabolic SPDE driven by a Poisson random measure with only continuous and bounded coefficients?
• RQ2 Can the existence result for Poisson-driven SPDEs be extended to SPDEs driven by general Lévy noise?
• RQ3 Is it possible to construct a martingale solution for a reaction-diffusion SPDE driven by Lévy noise, even when the noise is purely jump-type?
• RQ4 What conditions on the coefficients ensure the existence of a weak (martingale) solution in the presence of jump noise?
• RQ5 How can the classical Yamada-Watanabe approach be adapted to SPDEs with Lévy noise and non-Lipschitz coefficients?

Key findings

• A martingale solution exists for a parabolic SPDE driven by a Poisson random measure when the coefficients are continuous and bounded.
• The existence result for Poisson-driven SPDEs is successfully extended to SPDEs driven by general Lévy noise via a representation of Lévy processes in terms of Poisson random measures.
• The paper establishes the existence of a martingale solution for a reaction-diffusion SPDE driven by Lévy noise, resolving a long-standing open problem.
• The solution is constructed in the weak (martingale) sense, without requiring Lipschitz continuity of the coefficients.
• The method relies on tightness and weak convergence techniques in the space of càdlàg paths, ensuring the existence of a solution in law.
• The framework applies to both finite and infinite activity Lévy measures, covering a broad class of jump processes.