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We develop the rough path counterpart of It\^o stochastic integration and - differential equations driven by general semimartingales. This significantly enlarges the classes of (It\^o / forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed.

In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system. In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of L\'epingle's BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased via Kurtz-Protter's uniformly-controlled-variations (UCV) condition. A number of examples illustrate the scope of our results., Comment: Added details to several sections. Accepted version to appear in Annals of Probability

We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the "rough" regime of Hurst parameter $H < 1/2$. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation estimates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approximation formulae from CLT type log-moneyness deviations of order $t^{1/2}$ (recent works of Al\`{o}s, Le\'{o}n & Vives and Fukasawa) to the wider moderate deviations regime.

We develop the algebraic theory of rough path translation. Particular attention is given to the case of branched rough paths, whose underlying algebraic structure (Connes-Kreimer, Grossman-Larson) makes it a useful model case of a regularity structure in the sense of Hairer. Pre-Lie structures are seen to play a fundamental rule which allow a direct understanding of the translated (i.e. renormalized) equation under consideration. This construction is also novel with regard to the algebraic renormalization theory for regularity structures due to Bruned--Hairer--Zambotti (2016), the links with which are discussed in detail., Comment: Final version to appear in Journal of Functional Analysis

We demonstrate two examples of stochastic processes whose lifts to geometric rough paths require a renormalisation procedure to obtain convergence in rough path topologies. Our first example involves a physical Brownian motion subject to a magnetic force which dominates over the friction forces in the small mass limit. Our second example involves a lead-lag process of discretised fractional Brownian motion with Hurst parameter $H \in (1/4,1/2)$, in which the stochastic area captures the quadratic variation of the process. In both examples, a renormalisation of the second iterated integral is needed to ensure convergence of the processes, and we comment on how this procedure mimics negative renormalisation arising in the study of singular SPDEs and regularity structures., Comment: 13 pages

We revisit regularity of SLE trace, for all $\kappa \neq 8$, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia--Rodemich--Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index $\min (1 + \kappa / 8, 2) $, improving on previous works of Werness, and also (optimal) H\"older regularity \`a la Johansson Viklund and Lawler., Comment: 16 pages, 4 figures. Comments are welcome

It is known, since the seminal work [T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998)], that the solution map associated to a controlled differential equation is locally Lipschitz continuous in $q$-variation resp. $1/q$-H\"{o}lder type metrics on the space of rough paths, for any regularity $1/q \in (0,1]$. We extend this to a new class of Besov-Nikolskii-type metrics, with arbitrary regularity $1/q\in (0,1]$ and integrability $p\in [ q,\infty ]$, where the case $p\in \{ q,\infty \} $ corresponds to the known cases. Interestingly, the result is obtained as consequence of known $q$-variation rough path estimates., Comment: To appear in Trans. Amer. Math. Soc., 29 pages

We consider call option prices in diffusion models close to expiry, in an asymptotic regime ("moderately out of the money") that interpolates between the well-studied cases of at-the-money options and out-of-the-money fixed-strike options. First and higher order small-time moderate deviation estimates of call prices and implied volatility are obtained. The expansions involve only simple expressions of the model parameters, and we show in detail how to calculate them for generic local and stochastic volatility models. Some numerical examples for the Heston model illustrate the accuracy of our results.

We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations for the ($k$-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn implies a propagation of chaos result in rough path spaces and allows for a robust subsequent analysis of the particle system and its McKean-Vlasov type limit, as shown in two corollaries., Comment: 42 pages

We study the existence and uniqueness of the stochastic viscosity solutions of fully nonlinear, possibly degenerate, second order stochastic pde with quadratic Hamiltonians associated to a Riemannian geometry. The results are new and extend the class of equations studied so far by the last two authors.

Malliavin calculus is implemented in the context of [M. Hairer, A theory of regularity structures, Invent. Math. 2014]. This involves some constructions of independent interest, notably an extension of the structure which accomodates a robust, and purely deterministic, translation operator, in $L^2$-directions, between "models". In the concrete context of the generalized parabolic Anderson model in 2D - one of the singular SPDEs discussed in the afore-mentioned article - we establish existence of a density at positive times., Comment: Minor revision of [v1]. This version published in Journal of Functional Analysis, Volume 272, Issue 1, 1 January 2017, Pages 363-419

Existence of Loewner trace is revisited. We identify finite energy paths (the "skeleton of Wiener measure") as natural class of regular drivers for which we find simple and natural estimates in terms of their (Cameron--Martin) norm. Secondly, now dealing with potentially rough drivers, a representation of the derivative of the (inverse of the) Loewner flow is given in terms of a rough- and then pathwise F\"ollmer integral. Assuming the driver within a class of It\^o-processes, an exponential martingale argument implies existence of trace. In contrast to classical (exact) SLE computations, our arguments are well adapted to perturbations, such as non-constant $\kappa$ (assuming $<2$ for technical reasons) and additional finite-energy drift terms.