Stochastic Calculus for Fractional G-Brownian Motion and its Application to Mathematical Finance

Changhong Guo, Shaomei Fang, Yong He, Yong Zhang

2024, v.30, Issue 4, 477-524

ABSTRACT

Recently, a new concept for some stochastic process called fractional G-Brownian motion (fGBm) was developed. Compared to the standard Brownian motion, fractional Brownian motion and G-Brownian motion, the fGBm can exhibit long-range dependence and feature volatility uncertainty simultaneously. Thus it generalizes the concepts of the former three processes, and can be a better alternative stochastic process in real applications. In this paper, some stochastic calculus for the fGBm is established and
its application to mathematical finance is discussed.
First some stochastic integrals with respect to the fGBm are defined,
including the Wiener integral and fractional G-Wick-It^{o}-Skorohod integral. Then some other important generalizations under the fGBm frame, such as the Quasi-G-conditional expectation, fractional G-Clark-Ocone formula and fractional G-Girsanov theorem are also discussed. As an application, the financial market with a noise process driven by the fGBm is considered, since the financial asset possesses volatility uncertainty, and then the corresponding bid and ask prices for the European contingent claim are derived. These results generalize the existing ones in the frame of nonlinear expectation.

Keywords: Fractional G-Brownian motion; Volatility uncertainty; Stochastic calculus; Fractional G-Girsanov theorem; Financial application

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