Advancements in Stochastic Calculus for Fractional Brownian Motion

Introduction to Fractional Brownian Motion Calculus

This paper delves into the intricacies of stochastic calculus when applied to fractional Brownian motion (fBm), specifically for cases where the Hurst index H is greater than 1/2. Unlike classical Brownian motion, fBm exhibits long-range dependence, which significantly alters the properties of stochastic integrals and differentials.

Key Contributions and Concepts

• Generalized Itô's Formula: A fundamental aspect explored is the extension of Itô's formula. For a function f(Xt), the differential can be expressed as df(Xt) = f'(Xt) dXt + 1/2 f''(Xt) (dXt)2. The paper meticulously details how this formula adapts to the non-semimartingale nature of fBm.
• Quadratic Variation: The authors highlight that the quadratic variation, denoted as <BH>t, is non-zero for fractional Brownian motion, a stark contrast to standard Brownian motion. This property is crucial for understanding the integration theory developed in the paper.
• Integration Techniques: Various methods for constructing stochastic integrals with respect to fBm are discussed, providing a robust framework for further research in this domain.

Conclusion

The research presented in this document, originally published as arXiv hal-00920484, provides significant advancements in the field of stochastic calculus, particularly for processes characterized by long-range dependence. It offers essential tools and insights for researchers working with complex financial models, physics, and other areas where fractional dynamics are prevalent.