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Embedding of random vectors into continuous martingales. (English) Zbl 0924.60017

A. V. Skorokhod [“Studies in the theory of random processes” (1965; Zbl 0146.37701)] proved that for any mean zero, square integrable random variable \(X\) there is a Brownian motion \((B_t)_{t\geq 0}\) and a stopping time \(\tau\) such that \(X\) and \(B\) have the same distribution. The Skorokhod embedding is a result for one-dimensional random variables and in general one cannot expect similar results for random vectors with values in \(R^n\) or, even more general, in a Banach space. For this reason the author considers another type of embedding which is not so restricted to the real line and proves the following generalization of the Skorokhod embedding. Let \(E\) be a real, separable Banach space and denote by \(L^0(\Omega,E)\) the space of all \(E\)-valued random vectors defined on the probability space \((\Omega,F,P)\). Then there exists an extension \(\widetilde{\Omega}\) of \(\Omega\), and a filtration \((F_t)_{t\geq 0}\) on \(\widetilde{\Omega}\), such that for every \(X\in L^0(\Omega,E)\) there is an \(E\)-valued, continuous \((\widetilde{F}_t)\)-martingale \((M_t(X))_{t\geq 0}\) in which \(X\) is embedded in the sense that \(X= M_\tau(X)\) a.s. for an a.s. finite stopping time \(\tau\).

MSC:

60G44 Martingales with continuous parameter
60H05 Stochastic integrals
60B11 Probability theory on linear topological spaces

Citations:

Zbl 0146.37701
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