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On a class of exponential changes of measure for stochastic PDEs

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  • Pieper-Sethmacher, Thorben
  • van der Meulen, Frank
  • van der Vaart, Aad

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 Ph 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 Ph 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 Ph 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.

Suggested Citation

  • Pieper-Sethmacher, Thorben & van der Meulen, Frank & van der Vaart, Aad, 2025. "On a class of exponential changes of measure for stochastic PDEs," Stochastic Processes and their Applications, Elsevier, vol. 185(C).
    • Handle: RePEc:eee:spapps:v:185:y:2025:i:c:s0304414925000717
    DOI: 10.1016/j.spa.2025.104630
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    References listed on IDEAS

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    1. di Nunno, Giulia & Ortiz–Latorre, Salvador & Petersson, Andreas, 2023. "SPDE bridges with observation noise and their spatial approximation," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 170-207.
    2. Goldys, B. & Maslowski, B., 2008. "The Ornstein-Uhlenbeck bridge and applications to Markov semigroups," Stochastic Processes and their Applications, Elsevier, vol. 118(10), pages 1738-1767, October.
    3. Delyon, Bernard & Hu, Ying, 2006. "Simulation of conditioned diffusion and application to parameter estimation," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1660-1675, November.
    4. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
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