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The original Donsker theorem says that a standard random walk converges in distribution to a Brownian motion in the space of continuous functions. It has recently been extended to enriched random walks and enriched Brownian motion. We use the Stein-Dirichlet method to precise the rate of this convergence in the topology of fractional Sobolev spaces. [ABSTRACT FROM AUTHOR]

By fractional integration of a square root volatility process, we propose in this paper a long memory extension of the Heston (Rev Financ Stud 6:327-343, 1993) option pricing model. Long memory in the volatility process allows us to explain some option pricing puzzles as steep volatility smiles in long term options and co-movements between implied and realized volatility. Moreover, we take advantage of the analytical tractability of affine diffusion models to clearly disentangle long term components and short term variations in the term structure of volatility smiles. In addition, we provide a recursive algorithm of discretization of fractional integrals in order to be able to implement a method of moments based estimation procedure from the high frequency observation of realized volatilities. [ABSTRACT FROM AUTHOR]

We investigate the sharp large deviation properties of the energy and the maximum likelihood estimator for the Ornstein-Uhlenbeck process driven by a fractional Brownian motion with Hurst index greater than one half. [ABSTRACT FROM AUTHOR]