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Probability structure preserving and absolute continuity. (English) Zbl 1002.60046
The probability structure preserving mapping (PSPM) in the sense that $$V(F+G)= V(F)+V(G), V(FG)= V(F)V(G), E[F] = {\tilde E}[V(F)]$$ for all elements $$F$$ and $$G$$ of the space $$\varepsilon$$ of finite linear combinations of the exponential functionals defined from an abstract Wiener space together with the corresponding expectation $$E$$ (respectively, $$V(F) \in \tilde {\varepsilon}$$ with $$\tilde E$$) is introduced similarly to a correspondence introduced in a previous article of the author and P. A. Meyer [in: Stochastic processes, 141-147 (1993; Zbl 0798.60057)] but a similar role is played by multiple Itô type integrals. A particular PSPM between Brownian motion and fractional Brownian motion (FBM) with Hurst parameter $$H$$ is established to define stochastic integral for FBM and to prove a Girsanov type theorem on absolute continuity for FBM.
Reviewer: T.N.Pham (Hanoi)

##### MSC:
 60H05 Stochastic integrals 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60B11 Probability theory on linear topological spaces 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 60H07 Stochastic calculus of variations and the Malliavin calculus
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