Rough path analysis provides a fresh perspective on Ito's important theory of stochastic differential equations. Key theorems of modern stochastic analysis (existence and limit theorems for stochastic flows, Freidlin-Wentzell theory, the Stroock-Varadhan support description) can be obtained with dramatic simplifications. Classical approximation results and their limitations (Wong-Zakai, McShane's counterexample) receive 'obvious' rough path explanations. Evidence is building that rough paths will play an important role in the future analysis of stochastic partial differential equations and the authors include some first results in this direction. They also emphasize interactions with other parts of mathematics, including Caratheodory geometry, Dirichlet forms and Malliavin calculus. Based on successful courses at the graduate level, this up-to-date introduction presents the theory of rough paths and its applications to stochastic analysis. Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.

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Large classes of multi-dimensional Gaussian processes can be enhanced with stochastic Lévy area(s). In a previous paper, we gave sufficient and essentially necessary conditions, only involving variational properties of the covariance. Following T. Lyons, the resulting lift to a "Gaussian rough path" gives a robust theory of (stochastic) differential equations driven by Gaussian signals with sample path regularity worse than Brownian motion.The purpose of this sequel paper is to establish convergence of Karhunen-Loeve approximations in rough path metrics. Particular care is necessary since martingale arguments are not enough to deal with third iterated integrals. An abstract support criterion for approximately continuous Wiener functionals then gives a description of the support of Gaussian rough paths as the closure of the (canonically lifted) Cameron-Martin space.

We consider risk-neutral returns and show how their tail asymptotics translate directly to asymptotics of the implied volatility smile, thereby sharpening Roger Lee's celebrated moment formula. The theory of regular variation provides the ideal mathematical framework to formulate and prove such results. The practical value of our formulae comes from the vast literature on tail asymptotics and our conditions are often seen to be true by simple inspection of known results.

Abstract. We consider stochastic differential equations dY = V (Y ) dX driven by a multidimensional Gaussian process X in the rough path sense. Using Malliavin Calculus we show that Yt admits a density for t ∈ (0, T ] provided (i) the vector fields V = (V 1 , ..., V d ) satisfy Hörmander's condition and (ii) the Gaussian driving signal X satisfies certain conditions. Examples of driving signals include fractional Brownian motion with Hurst parameter H > 1/4, the Brownian Bridge returning to zero after time T and the Ornstein-Uhlenbeck process.

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