On a class of exponential changes of measure for stochastic PDEs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912249043484672 |
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| author | Pieper-Sethmacher, Thorben van der Meulen, Frank van der Vaart, Aad |
| author_facet | Pieper-Sethmacher, Thorben van der Meulen, Frank van der Vaart, Aad |
| contents | 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 $\mathbb{P}^h$ 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 $\mathbb{P}^h$ 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 $\mathbb{P}^h$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08057 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a class of exponential changes of measure for stochastic PDEs Pieper-Sethmacher, Thorben van der Meulen, Frank van der Vaart, Aad Probability 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 $\mathbb{P}^h$ 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 $\mathbb{P}^h$ 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 $\mathbb{P}^h$ 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. |
| title | On a class of exponential changes of measure for stochastic PDEs |
| topic | Probability |
| url | https://arxiv.org/abs/2409.08057 |