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Stochastic analysis based on deterministic Brownian motion

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Abstract

A deterministic version of the Itô calculus is presented. We consider a modelY t=H(N t ,t) with a deterministic Brownian N t and an unknown functionH. We predictY c from the observation {Y t;t ∈ [a, b]}, wherea<b<c. We prove that there exists an estimatorY t based on the observation such thatE[(Ŷ t−Y c)2]=O((c−b)2) asc ↓b.

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References

  1. Teturo Kamae,Linear expansions, strictly ergodic homogeneous cocycles and fractals, Israel Journal of Mathematics106 (1998), 313–337.

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  2. Benoit B. Mandelbrot,A multifractal walk down Wall Street, Scientific American, February, 1999.

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Authors and Affiliations

  1. Department of Mathematics, Osaka City University, 558-8585, Osaka, Japan

    Teturo Kamae

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  1. Teturo Kamae
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Correspondence to Teturo Kamae.

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Kamae, T. Stochastic analysis based on deterministic Brownian motion. Isr. J. Math. 125, 317–346 (2001). https://doi.org/10.1007/BF02773385

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  • DOI: https://doi.org/10.1007/BF02773385

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