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Fractional G-White Noise Theory, Wavelet Decomposition for Fractional G-Brownian Motion, and Bid-Ask Pricing Application to Finance Under Uncertainty

  • Wei Chen
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G-framework is presented by Peng [41] for measure risk under uncertainty. In this paper, we define fractional G-Brownian motion (fGBm). Fractional G-Brownian motion is a centered G-Gaussian process with zero mean and stationary increments in the sense of sub-linearity with Hurst index $H\in (0,1)$. This process has stationary increments, self-similarity, and long rang dependence properties in the sense of sub-linearity. These properties make the fractional G-Brownian motion a suitable driven process in mathematical finance. We construct wavelet decomposition of the fGBm by wavelet with compactly support. We develop fractional G-white noise theory, define G-It\^o-Wick stochastic integral, establish the fractional G-It\^o formula and the fractional G-Clark-Ocone formula, and derive the G-Girsanov's Theorem. For application the G-white noise theory, we consider the financial market modelled by G-Wick-It\^o type of SDE driven by fGBm. The financial asset price modelled by fGBm has volatility uncertainty, using G-Girsanov's Theorem and G-Clark-Ocone Theorem, we derive that sublinear expectation of the discounted European contingent claim is the bid-ask price of the claim.

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File URL: http://arxiv.org/pdf/1306.4070
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Paper provided by arXiv.org in its series Papers with number 1306.4070.

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Date of creation: Jun 2013
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Handle: RePEc:arx:papers:1306.4070
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  1. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  2. T. J. Lyons, 1995. "Uncertain volatility and the risk-free synthesis of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 117-133.
  3. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
  4. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
  5. Shige Peng, 2006. "Modelling Derivatives Pricing Mechanisms with Their Generating Functions," Papers math/0605599, arXiv.org.
  6. Zengjing Chen & Larry Epstein, 2002. "Ambiguity, Risk, and Asset Returns in Continuous Time," Econometrica, Econometric Society, vol. 70(4), pages 1403-1443, July.
  7. Freddy Delbaen & Shige Peng & Emanuela Rosazza Gianin, 2010. "Representation of the penalty term of dynamic concave utilities," Finance and Stochastics, Springer, vol. 14(3), pages 449-472, September.
  8. Jan Ubøe & Bernt Øksendal & Knut Aase & Nicolas Privault, 2000. "White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance," Finance and Stochastics, Springer, vol. 4(4), pages 465-496.
  9. Wei Chen, 2011. "Time Consistent Bid-Ask Dynamic Pricing Mechanisms for Contingent Claims and Its Numerical Simulations Under Uncertainty," Papers 1111.4298, arXiv.org, revised Sep 2013.
  10. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
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