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From Dynamic Linear Evaluation Rule to Dynamic CAPM in a Fractional Brownian Motion Environment
School of Science, Beijing University of Posts and Telecommunications, Beijing, China
School of Science, Beijing University of Posts and Telecommunications, Beijing, China
- 1 School of Science, Beijing University of Posts and Telecommunications, Beijing, China
- 2 School of Science, Beijing University of Posts and Telecommunications, Beijing, China
Journal of Mathematical Finance·Volume 02 (2012)·Pages 315–320·Published 19 November 2012·DOI10.4236/jmf.2012.24034
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Abstract
In this paper, we present the fundamental framework of the evaluation problem under which the evaluation operator satisfying some axioms is linear. Based on the dynamic linear evaluation mechanism of contingent claims, studying this evaluation rule in the market driven by fractional Brownian motions has led to a dynamic capital asset pricing model. It is deduced here mainly with the fractional Girsanov theorem and the Clark-Haussmann-Ocone theorem.
KeywordsFractional Brownian MotionClark-Haussmann-Ocone TheoremFractional Girsanov TheoremEvaluation OperatorCapital Asset Pricing Model
- P. H. Dybvig and J. E. Ingersoll, “Mean-Variance Theory in Complete Markets,” Journal of Business, Vol. 55, No. 2, 1982, pp. 233-251. doi:10.1086/296162
- L. P. Hansen and S. F. Richard, “The Role of Conditioning Information in Deducing Testable Restrictions Implied by Dynamic Asset Pricing Models,” Econometrica, Vol. 55, No. 3, 1987, pp. 587-614. doi:10.2307/1913601
- J. H. Cochrane, “Asset Pricing,” Princeton University Press, Princeton, 2001.
- S. Z. Shi, “Ten Presentations of Financial Econometrics,” Shanghai Dometic Publishing, Shanghai, 2004.
- Q. Zhou and W. X. Wu, “From Dynamic Linear Evaluation Rule to Dynamic Capital Asset Pricing Model,” System EngineeringTheory and Practice, Vol. 27, No. 5, 2007, pp. 48-54. doi:10.1016/S1874-8651(08)60033-2
- H. E. Hurst, “Long-Term Storage Capacity in Reservoirs,” Transactions of the American Society of Civil Engineers, Vol. 55, 1951, pp. 400-410.
- H. E. Hurst, “Methods of Using Long-Term Storage in Reservoirs,” Proceedings of the Institution of Civil Engineers Part 1, Vol. 5, 1956, pp. 519-590.
- B. B. Mandelbrot and J. W. Van Ness, “Fractional Brownian Motions, Fractional Noises and Applications,” SIAM Review, Vol. 10, No. 4, 1968, pp. 422-437. doi:10.1137/1010093
- Y. Hu and B. Oksendal, “Fractional White Noise Calculus and Applications to Finance, Infinite Dimensional Analysis,” Quantum Probability and Related Topics, Vol. 6, No. 1, 2003, pp. 1-32.
- Y. Hu, B. ?ksendal and A. Sulem, “Optimal Consumption and Portfolio in a Black-Scholes Market Driven by Fractional Brownian Motion,” Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol. 6, No. 4, 2003, pp. 519-536. doi:10.1142/S0219025703001432
- B. B. Mandelbrot, “Fractals and Scaling in Finance: Discontinuity, Concentration, Risk,” Springer & Verlag, Berlin, 1997.
- A. Shiryaev, “On Arbitrage and Replication for Fractal Models,” In: A. Shiryaev and A. Sulem, Eds., Workshop on Mathematical Finance, INRIA, Paris, 1998.
- Y. Hu, “Integral Transformations and Anticipative Calculus for Fractional Brownian Motions,” American Mathematical Society, Providence, 2005.
- F. Biagini, Y. Hu, B. Oksendal and T. S. Zhang, “Stochastic Calculus for Fractional Brownian Motion and Applications,” Springer, London, 2008. doi:10.1007/978-1-84628-797-8