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.

We consider the solutions lying on the global attractor of the two-dimensional Navier-Stokes equations with periodic boundary conditions and analytic forcing. We show that in this case the value of a solution at a finite number of nodes determines elements of the attractor uniquely, proving a conjecture due to Foias and Temam. Our results also hold for the complex Ginzburg-Landau equation, the Kuramoto-Sivashinsky equation, and reaction-diffusion equations with analytic nonlinearities.

We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983) 46-57]. This condition is verified in many examples, even in absence of explicit expressions for the covariance or Volterra kernels. Of special interest are random Fourier series, with covariance given as Fourier series itself, and we formulate conditions directly in terms of the Fourier coefficients. We also establish convergence and rates of convergence in rough path metrics of approximations to such random Fourier series. An application to SPDE is given. Our criterion also leads to an embedding result for Cameron-Martin paths and complementary Young regularity (CYR) of the Cameron-Martin space and Gaussian sample paths. CYR is known to imply Malliavin regularity and also Itô-like probabilistic estimates for stochastic integrals (resp., stochastic differential equations) despite their (rough) pathwise construction. At last, we give an application in the context of non-Markovian Hörmander theory.

We survey recent results on the behavior of the Black-Scholes implied volatility at extreme strikes. There are simple and universal formulae that give quantitative links between tail behavior and moment explosions of the underlying on one hand, and growth of the famous volatility smile on the other hand. Some original results are included as well.

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A hybrid theory of rough stochastic analysis is built. It seamlessly combines the advantages of both Itô's stochastic -and Lyons' rough differential equations. Well-posedness of rough stochastic differential equation is obtained, under natural assumptions and with precise estimates; many examples and applications are mentioned. A major role is played by a new stochastic variant of Gubinelli's controlled rough paths spaces, with norms that reflect some generalized stochastic sewing lemma, and which may prove useful whenever rough paths and Itô integration meet.

Schramm–Loewner evolution ($$\hbox {SLE}_\kappa $$ SLE κ ) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by $$\sqrt{\kappa }$$ κ times Brownian motion. This yields a (half-plane) valued random field $$\gamma = \gamma (t, \kappa ; \omega )$$ γ = γ ( t , κ ; ω ) . (Hölder) regularity of in $$\gamma (\cdot ,\kappa ;\omega $$ γ ( · , κ ; ω ), a.k.a. SLE trace, has been considered by many authors, starting with Rohde and Schramm (Ann Math (2) 161(2):883–924, 2005). Subsequently, Johansson Viklund et al. (Probab Theory Relat Fields 159(3–4):413–433, 2014) showed a.s. Hölder continuity of this random field for $$\kappa < 8(2-\sqrt{3})$$ κ < 8 ( 2 - 3 ) . In this paper, we improve their result to joint Hölder continuity up to $$\kappa < 8/3$$ κ < 8 / 3 . Moreover, we show that the SLE$$_\kappa $$ κ trace $$\gamma (\cdot ,\kappa )$$ γ ( · , κ ) (as a continuous path) is stochastically continuous in $$\kappa $$ κ at all $$\kappa \ne 8$$ κ ≠ 8 . Our proofs rely on a novel variation of the Garsia–Rodemich–Rumsey inequality, which is of independent interest.

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