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

摘要

G-framework is presented by Peng [41] for measure risk under uncertainty. Inthis paper, we define fractional G-Brownian motion (fGBm). FractionalG-Brownian motion is a centered G-Gaussian process with zero mean andstationary increments in the sense of sub-linearity with Hurst index $H\in(0,1)$. This process has stationary increments, self-similarity, and long rangdependence properties in the sense of sub-linearity. These properties make thefractional G-Brownian motion a suitable driven process in mathematical finance.We construct wavelet decomposition of the fGBm by wavelet with compactlysupport. We develop fractional G-white noise theory, define G-It\^o-Wickstochastic integral, establish the fractional G-It\^o formula and thefractional G-Clark-Ocone formula, and derive the G-Girsanov's Theorem. Forapplication the G-white noise theory, we consider the financial market modelledby G-Wick-It\^o type of SDE driven by fGBm. The financial asset price modelledby fGBm has volatility uncertainty, using G-Girsanov's Theorem andG-Clark-Ocone Theorem, we derive that sublinear expectation of the discountedEuropean contingent claim is the bid-ask price of the claim.