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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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