An alternative approach to Privault's discrete-time chaotic calculus |
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Authors: | Caishi Wang Yanchun Lu Huifang Chai |
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Affiliation: | School of Mathematics and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, PR China |
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Abstract: | In this paper, we present an alternative approach to Privault's discrete-time chaotic calculus. Let Z be an appropriate stochastic process indexed by N (the set of nonnegative integers) and l2(Γ) the space of square summable functions defined on Γ (the finite power set of N). First we introduce a stochastic integral operator J with respect to Z, which, unlike discrete multiple Wiener integral operators, acts on l2(Γ). And then we show how to define the gradient and divergence by means of the operator J and the combinatorial properties of l2(Γ). We also prove in our setting the main results of the discrete-time chaotic calculus like the Clark formula, the integration by parts formula, etc. Finally we show an application of the gradient and divergence operators to quantum probability. |
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Keywords: | Discrete-time chaotic calculus Gradient and divergence Finite power set Full Wiener integral operator |
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