The thesis explores stochastic calculus for fractional Brownian motion. Our approach builds upon a novel technique called stochastic sewing, originally introduced by Lê [Electron. J. Probab. 25:1-55, 2020]. The stochastic sewing has been effectively used to obtain sharp estimates on stochastic Riemann sums.
The main result of the thesis is an extension of Lê’s stochastic sewing, which we refer to as the shifted stochastic sewing. This extension takes advantage of asymptotic decorrelation in stochastic Riemann sums and can be seen as a combination of Lê’s stochastic sewing and the asymptotic independence formulated by Picard [Ann. Probab. 36(6): 2235-2279, 2008].
As applications of the shifted stochastic sewing, we address two important problems in fractional stochastic calculus. Firstly, we characterize the local time of the fractional Brownian motion via level crossings, extending the classical work of Lévy to the fractional setting. Secondly, we establish the pathwise uniqueness of Young and rough differential equations driven by fractional Brownian motion. This result optimizes the regularity of the noise coefficient, which is consistent with the Brownian setting.
Additionally, we demonstrate strong regularization by fractional noise for differential equations with integrable drifts. This result can be viewed as a fractional analogue of the celebrated work by Krylov and Röckner [Probab. Theory Relat. Fields 131: 154–196, 2005].
