da Maia, L. (2024). An Itô-Wentzell formula for the fractional Brownian motion (arXiv:2402.06328v2). arXiv.
This paper aims to develop an Itô-Wentzell formula for the fractional Brownian motion (fBm) with Hurst parameter H > 1/2 and apply it to study the existence and uniqueness of solutions for a class of stochastic differential equations driven by fBm.
The author utilizes the framework of stochastic calculus for fractional Brownian motion, specifically employing the stochastic integral defined by Duncan and Hu (2000) based on the Wick product. The Itô-Wentzell formula is derived by leveraging this integral definition and properties of the φ-derivative. The existence and uniqueness of solutions for the considered class of SDEs are then established by applying the derived Itô-Wentzell formula and analyzing the associated flow.
The paper provides a valuable contribution to the field of stochastic calculus for fractional Brownian motion by establishing a new Itô-Wentzell formula and demonstrating its applicability in analyzing SDEs driven by fBm. The results offer a theoretical foundation for studying a broader range of stochastic models involving fBm, particularly in fields like finance and physics.
The development of an Itô-Wentzell formula for fBm with H > 1/2 significantly expands the toolkit for analyzing stochastic processes with long-range dependence, which are prevalent in various real-world phenomena. The application to SDEs further highlights the formula's utility in understanding the behavior and solutions of these equations in the context of fBm.
The paper focuses on fBm with H > 1/2 and a specific class of SDEs. Exploring similar results for the case of 0 < H < 1/2 and more general SDEs with potentially non-Lipschitz coefficients could be promising avenues for future research. Additionally, investigating potential applications of the derived formula in other areas like stochastic control and filtering would be of interest.
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