On a class of exponential changes of measure for stochastic PDEs
Thorben Pieper-Sethmacher (TU Delft - Statistics)
FH van der Meulen (TU Delft - Statistics, Vrije Universiteit Amsterdam)
Aad W. VAN DER VAART (TU Delft - Statistics)
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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.
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