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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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