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II: 07, 123-139, LNM 51 (1968)
Sur les moments spectraux d'ordre supérieur (Second order processes)
The essential result of the paper (Shiryaev, Th. Prob. Appl., 5, 1960; Sinai, Th. Prob. Appl., 8, 1963) is the definition of multiple stochastic integrals with respect to a second order process whose covariance satisfies suitable spectral properties
Keywords: Spectral representation, Multiple stochastic integrals
Nature: Exposition
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IV: 11, 132-132, LNM 124 (1970)
Théorème de Stone et espérances conditionnelles (Ergodic theory)
It is shown that the spectral projections of the unitary group arising from a group of measure preserving transformations must be complex operators, and in particular cannot be conditional expectations
Comment: This remark arose from the work on flows in Sam Lazaro-Meyer, Z. für W-theorie, 18, 1971
Keywords: Flows, Spectral representation
Nature: Original
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XI: 23, 365-375, LNM 581 (1977)
Changements de temps et intégrales stochastiques (Martingale theory)
A probability space $(\Omega, {\cal F}, P)$ such that $L^1(P)$ is separable (a condition which is often fulfilled) is endowed with a filtration $({\cal F}_t)$ satisfying the usual conditions. Then (extending ideas of Yan, see 925) it is shown that there exists a right continuous strictly increasing process $(O_t)$ such that every optional process is indistinguishable from a deterministic function $f(0_t)$, every previsible process from a deterministic function of $(0_{t-})$. Using the change of time associated with this process, previsible processes of the original filtration are time changed into deterministic processes, and the theory of stochastic integration is reduced to spectral integrals (as Stieltjes integration on the line can be reduced to Lebesgue's). A bounded previsible process $(u_t)$ define a bounded operator $U$ on $L^2$ as follows: starting from $h\in L^2$, construct the closed martingale $E[h|{\cal F}_t] =H_t$, and then $Uh=\int_0^\infty u_s dH_s$. Using the preceding results it is shown that the von Neumann algebra generated by the conditional expectation operators $E[\sc |{\cal F}_T]$ where $T$ is a stopping time consists exactly of these stochastic integral operators. On this point see also 1135
Comment: The last section states an interesting open problem
Keywords: Changes of time, Spectral representation
Nature: Original
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