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

## Author Info

Listed author(s):
• Wei Chen
Registered author(s):

## Abstract

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.

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

File URL: http://arxiv.org/pdf/1306.4070

## Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1306.4070.

as
in new window

## References

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

as
in new window

1. 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.
2. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters,in: Theory Of Valuation, chapter 8, pages 229-288 World Scientific Publishing Co. Pte. Ltd..
3. 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.
4. Zengjing Chen & Larry Epstein, 2002. "Ambiguity, Risk, and Asset Returns in Continuous Time," Econometrica, Econometric Society, vol. 70(4), pages 1403-1443, July.
5. 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.
6. Shige Peng, 2006. "Modelling Derivatives Pricing Mechanisms with Their Generating Functions," Papers math/0605599, arXiv.org.
7. 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.
8. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
9. 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 637-654, May-June.
10. 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.
Full references (including those not matched with items on IDEAS)

## Lists

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

## Corrections

When requesting a correction, please mention this item's handle: RePEc:arx:papers:1306.4070. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (arXiv administrators)

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.