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A representation result for time-space Brownian chaos. (English) Zbl 0990.60081
Summary: Given a Brownian motion $$X$$, we say that a square-integrable functional $$F$$ belongs to the $$n$$th time-space Brownian chaos if $$F$$ is contained in the $$L^2$$-closed vector space $$\overline\Pi_n$$, generated by r.v.’s of the form $$f_1(X_{t_1})\cdots f_n(X_{t_n})$$, and $$F$$ is orthogonal to $$\overline\Pi_{n-1}$$. We show that every element of the $$n$$th time-space Brownian chaos can be represented as a multiple time-space Wiener integral of the $$n$$th order, thus proving a new chaotic representation property for Brownian motion.

##### MSC:
 60J65 Brownian motion 60H05 Stochastic integrals 60G99 Stochastic processes
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