On a class of exponential changes of measure for stochastic PDEs

2024/09/12 by Thorben Pieper-Sethmacher, Frank van der Meulen, Pieper-Sethmacher, Thorben +3 · 3 citations
Economics, Econometrics and Finance · #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2409.08057

openalex publication_date 2024/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a mild solution X to a semilinear stochastic partial differential equation (SPDE), we consider an exponential change of measure based on its infinitesimal generator L, defined in the topology of bounded pointwise convergence. The changed measure ℙ^h depends on the choice of a function h in the domain of L. In our main result, we derive conditions on h for which the change of measure is of Girsanov-type. The process X under ℙ^h is then shown to be a mild solution to another SPDE with an extra additive drift-term. We illustrate how different choices of h impact the law of X under ℙ^h in selected applications. These include the derivation of an infinite-dimensional diffusion bridge as well as the introduction of guided processes for SPDEs, generalizing results known for finite-dimensional diffusion processes to the infinite-dimensional case.

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