Finitely additive functions in measure theory and applications. (English) Zbl 1539.47041
Summary: In this paper, we consider, and make precise, a certain extension of the Radon-Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability theory; in particular, to the study of \(\mu\)-Brownian motion, to stochastic calculus via generalized Itô-integrals, and their adjoints (in the form of generalized stochastic derivatives), to systems of transition probability operators indexed by families of measures \(\mu\), and to adjoints of composition operators.
MSC:
| 47B32 | Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) |
| 60G20 | Generalized stochastic processes |
| 60G15 | Gaussian processes |
| 60H05 | Stochastic integrals |
| 60J60 | Diffusion processes |
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
Keywords:
Hilbert space; reproducing kernels; probability space; Gaussian fields; transforms; covariance; Itô integration; Itô calculus; generalized Brownian motionReferences:
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