Stochastic integrals on general topological measurable spaces |
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Authors: | Huang Zhiyuan |
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Institution: | (1) Mathematics Department, Wuhan University, Wuchang, Hubei, The People's Republic of China |
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Abstract: | Summary A general theory of stochastic integral in the abstract topological measurable space is established. The martingale measure is defined as a random set function having some martingale property. All square integrable martingale measures constitute a Hilbert space M
2. For each M
2, a real valued measure on the predictable -algebra is constructed. The stochastic integral of a random function
with respect to is defined and investigated by means of Riesz's theorem and the theory of projections. The stochastic integral operator I
is an isometry from L
2() to a stable subspace of M
2, its inverse is defined as a random Radon-Nikodym derivative. Some basic formulas in stochastic calculus are obtained. The results are extended to the cases of local martingale and semimartingale measures as well. |
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Keywords: | |
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