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Study of stochastic differential equations driven by fractional brownian motion

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Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Curs: 2022-2023. Director: David Márquez

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[en] In this thesis we study and develop in detail the research paper Differential equations driven by fractional brownian motion by D. Nualart and A. Rascanu, 7]. It is a landmark paper in which the authors prove the existence and uniqueness of solution to stochastic differential equations driven by fractional Brownian motion of Hurst parameter $H \in(1 / 2,1)$. Moreover, they show that, under additional hypothesis, the solution has finite moments of all orders. They take a path-by-path approach given the Hölder-continuity property of the paths of the fractional Brownian motion.
On our part, after a gentle introduction to the fractional integrals and derivatives and to the generalized Stieltjes integral, we fully develop the results and proofs of this paper. Not only that but we insert our own remarks and comment on the obtained results regarding the measurability of the solution. As a result, this thesis could be considered a companion paper intended to the reader interested in this important result but not versed in the foundations of stochastic differential equations.

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cc by-nc-nd (c) Abel Serrat i Castella, 2023

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Study of stochastic differential equations driven by fractional brownian motion