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In previous works Avellaneda et al. pioneered the pricing and hedging of index options - products highly sensitive to implied volatility and correlation assumptions - with large deviations methods, assuming local volatility dynamics for all components of the index. We here present an extension applicable to non-Markovian dynamics and in particular the case of rough volatility dynamics., 22 pages, This version to appear in Mathematical Finance

We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons extension theorem, the renormalization of rough paths in the spirit of [Bruned, Chevyrev, Friz, Prei{\ss}, A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019] as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting., Comment: 47 pages

Several asymptotic results for the implied volatility generated by a rough volatility model have been obtained in recent years (notably in the small-maturity regime), providing a better understanding of the shapes of the volatility surface induced by rough volatility models, and supporting their calibration power to S&P500 option data. Rough volatility models also generate a local volatility surface, via the so-called Markovian projection of the stochastic volatility. We complement the existing results on the implied volatility by studying the asymptotic behavior of the local volatility surface generated by a class of rough stochastic volatility models, encompassing the rough Bergomi model. Notably, we observe that the celebrated "1/2 skew rule" linking the short-term at-the-money skew of the implied volatility to the short-term at-the-money skew of the local volatility, a consequence of the celebrated "harmonic mean formula" of [Berestycki, Busca, and Florent, QF 2002], is replaced by a new rule: the ratio of the at-the-money implied and local volatility skews tends to the constant 1/(H + 3/2) (as opposed to the constant 1/2), where H is the regularity index of the underlying instantaneous volatility process.

We present a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small noise formulae for option prices. Our main tool is the theory of regularity structures, which we use in the form of Bayer et al. (Math. Finance 30 (2020) 782–832) In essence, we implement a Laplace method on the space of models (in the sense of Hairer), which generalizes classical works of Azencott and Ben Arous on path space and then Aida, Inahama–Kawabi on rough path space. When applied to rough volatility models, for example, in the setting of Bayer, Friz and Gatheral (Quant. Finance 16 (2016) 887–904) and Forde–Zhang (SIAM J. Financial Math. 8 (2017) 114–145), one obtains precise asymptotics for European options which refine known large deviation asymptotics.

We take a pathwise approach to classical McKean–Vlasov stochastic differential equations with additive noise, as for example, exposed in Sznitmann (In École D’Été de Probabilités de Saint-Flour XIX—1989 (1991) 165–251, Springer). Our study was prompted by some concrete problems in battery modelling (Contin. Mech. Thermodyn. 30 (2018) 593–628), and also by recent progrss on rough-pathwise McKean–Vlasov theory, notably Cass–Lyons (Proc. Lond. Math. Soc. (3) 110 (2015) 83–107), and then Bailleul, Catellier and Delarue (Bailleul, Catellier and Delarue (2018)). Such a “pathwise McKean–Vlasov theory” can be traced back to Tanaka (In Stochastic Analysis (Katata/Kyoto, 1982) (1984) 469–488, North-Holland). This paper can be seen as an attempt to advertize the ideas, power and simplicity of the pathwise appproach, not so easily extracted from (Bailleul, Catellier and Delarue (2018); Proc. Lond. Math. Soc. (3) 110 (2015) 83–107; In Stochastic Analysis (Katata/Kyoto, 1982) (1984) 469–488, North-Holland), together with a number of novel applications. These include mean field convergence without a priori independence and exchangeability assumption; common noise, càdlàg noise, and reflecting boundaries. Last not least, we generalize Dawson–Gärtner large deviations and the central limit theorem to a non-Brownian noise setting.

Existence and uniqueness for rough flows, transport and continuity equations driven by general geometric rough paths are established., 20 pages

This work is concerned with forest and cumulant type expansions of general random variables on a filtered probability spaces. We establish a "broken exponential martingale" expansion that generalizes and unifies the exponentiation result of Al{\`o}s, Gatheral, and Radoi\v{c}i\'c and the cumulant recursion formula of Lacoin, Rhodes, and Vargas. Specifically, we exhibit the two previous results as lower dimensional projections of the same generalized forest expansion, subsequently related by forest reordering. Our approach also leads to sharp integrability conditions for validity of the cumulant formula, as required by many of our examples, including iterated stochastic integrals, L\'evy area, Bessel processes, KPZ with smooth noise, Wiener-It\^o chaos and "rough" stochastic (forward) variance models., Comment: 25 pages. Additional applications. Analytic proof of forest reordering. Relation to Al\`os et al. (SSRN'17, Quant. Finance '20) and Lacoin et al. (arXiv'19) discussed in detail

Schramm–Loewner evolution (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {SLE}_\kappa $$\end{document}SLEκ) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{\kappa }$$\end{document}κ times Brownian motion. This yields a (half-plane) valued random field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = \gamma (t, \kappa ; \omega )$$\end{document}γ=γ(t,κ;ω). (Hölder) regularity of in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (\cdot ,\kappa ;\omega $$\end{document}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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa < 8(2-\sqrt{3})$$\end{document}κ

In this chapter, we consider the most important example of a rough path, which is the one associated to Brownian motion. We discuss the difference, at the level of rough paths, between Ito and Stratonovich Brownian motion. We also provide a natural example of approximation to Brownian motion which converges to neither of them.

Stochastic Volatility Models (SVMs) are ubiquitous in quantitative finance. But is there a Markovian SVM capable of producing extreme (T^(-1/2)) short-dated implied volatility skew? We here propose a modification of a given SVM "backbone", Heston for instance, to achieve just this - without adding jumps or non-Markovian "rough" fractional volatility dynamics. This is achieved via non-smooth leverage function, such as a step function. The resulting Step Stochastic Volatility Model (SSVM) is thus a parametric example of local stochastic volatility model (LSVM). From an IT perspective, SSVM amounts to trivial modifications in the code of existing SVM implementations. From a QF perspective, SSVM offers new flexibility in smile modelling and towards assessing model risk. For comparison, we then exhibit the market-induced leverage function for LSVM, calibrated with the particle method.

We give a short overview of the scopes of both the theory of rough paths and the theory of regularity structures. The main ideas are introduced and we point out some analogies with other branches of mathematics. 1.1 Controlled differential equations Differential equations are omnipresent in modern pure and applied mathematics; many “pure” disciplines in fact originate in attempts to analyse differential equations from various application areas. Classical ordinary differential equations (ODEs) are of the form Ẏt = f(Yt, t); an important sub-class is given by controlled ODEs of the form Ẏt = f0(Yt) + f(Yt)Ẋt , (1.1) where X models the input (taking values in R, say), and Y is the output (in R, say) of some system modelled by nonlinear functions f0 and f , and by the initial state Y0. The need for a non-smooth theory arises naturally when the system is subject to white noise, which can be understood as the scaling limit as h→ 0 of the discrete evolution equation Yi+1 = Yi + hf0(Yi) + √ hf(Yi)ξi+1 , (1.2) where the (ξi) are i.i.d. standard Gaussian random variables. Based on martingale theory, Ito’s stochastic differential equations (SDEs) have provided a rigorous and extremely useful mathematical framework for all this. And yet, stability is lost in the passage to continuous time: while it is trivial to solve (1.2) for a fixed realisation of ξi(ω), after all (ξ1, . . . ξT ;Y0) 7→ Yi is surely a continuous map, the continuity of the solution as a function of the driving noise is lost in the limit. Taking Ẋ = ξ to be white noise in time (which amounts to say that X is a Brownian motion, say B), the solution map S : B 7→ Y to (1.1), known as Ito map, is a measurable map which in general lacks continuity, whatever norm one uses to

Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks in terms of both the input data and the (trained) network weights. As trained network weights are typically very rough when seen as functions of the layer, we propose to derive stability bounds in terms of the total $p$-variation of trained weights for any $p\in[1,3]$. Unlike the $C^1$-theory underlying the neural ODE literature, our estimates remain bounded even in the limiting case of weights behaving like Brownian motions, as suggested in [arXiv:2105.12245]. Mathematically, we interpret residual neural network as solutions to (rough) difference equations, and analyse them based on recent results of discrete time signatures and rough path theory., 21 pages, 2 figures

Second order stochastic partial differential equations are discussed from a rough path point of view. In the linear and finite-dimensional noise case we follow a Feynman–Kac approach which makes good use of concentration of measure results, as those obtained in Sect. 11.2. Alternatively, one can proceed by flow decomposition and this approach also works in a number of non-linear situations. Secondly, now motivated by some semi-linear SPDEs of Burgers’ type with infinite-dimension noise, we study the stochastic heat equation (in space dimension 1) as evolution in Gaussian rough path space relative to the spatial variable, in the sense of Chap. 10.

A continuous Gaussian process gives rise to a Gaussian measure on path-space. Thanks to variation regularity properties of Cameron–Martin paths, powerful tools from the analysis on Gaussian spaces become available. A general Fernique type theorem leads us to integrability properties of rough integrals with Gaussian integrator akin to those of classical stochastic integrals. We then discuss Malliavin calculus for differential equations driven by Gaussian rough paths. As application a version of Hormander’s theorem in this non-Markovian setting is established.

At first sight, the notation \( \int {Y\;dX} \) introduced in Chapter 4 is ambiguous since the resulting controlled rough path depends in general on the choices of both the second-order process \( {\mathbb{X}} \) and the derivative process Y′. Fortunately, this “lack of completeness” in our notations is mitigated by the fact that in virtually all situations of interest, Y is constructed by using a small number of elementary operations described in this chapter. For all of these operations, it turns out to be intuitively rather clear how the corresponding derivative process is constructed.

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.

We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the ‘rough’ regime of Hurst parameter H

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., Minor revision of [v1]. This version published in Journal of Functional Analysis, Volume 272, Issue 1, 1 January 2017, Pages 363-419

International audience; 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 Rie-mannian geometry. The results are new and extend the class of equations studied so far by the last two authors.

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 Follmer integral. Assuming the driver within a class of Ito-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 \(

We consider deterministic fast-slow dynamical systems on $\mathbb{R}^m\times Y$ of the form \[ \begin{cases} x_{k+1}^{(n)} = x_k^{(n)} + n^{-1} a(x_k^{(n)}) + n^{-1/\alpha} b(x_k^{(n)}) v(y_k)\;,\quad y_{k+1} = f(y_k)\;, \end{cases} \] where $\alpha\in(1,2)$. Under certain assumptions we prove convergence of the $m$-dimensional process $X_n(t)= x_{\lfloor nt \rfloor}^{(n)}$ to the solution of the stochastic differential equation \[ \mathop{}\!\mathrm{d} X = a(X)\mathop{}\!\mathrm{d} t + b(X) \diamond \mathop{}\!\mathrm{d} L_\alpha \; , \] where $L_\alpha$ is an $\alpha$-stable L\'evy process and $\diamond$ indicates that the stochastic integral is in the Marcus sense. In addition, we show that our assumptions are satisfied for intermittent maps $f$ of Pomeau-Manneville type., Comment: 36 pages, 3 figures. Minor revision. To appear in Probability Theory and Related Fields

A new paradigm recently emerged in financial modelling: rough (stochastic) volatility, first observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models, also turned out to capture parsimoniously key stylized facts of the entire implied volatility surface, including extreme skews that were thought to be outside the scope of stochastic volatility. On the mathematical side, Markovianity and, partially, semi-martingality are lost. In this paper we show that Hairer's regularity structures, a major extension of rough path theory, which caused a revolution in the field of stochastic partial differential equations, also provides a new and powerful tool to analyze rough volatility models., Dedicated to Professor Jim Gatheral on the occasion of his 60th birthday

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is here examined through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures., Comment: 52 pages, 1 figure, 6 tables; to appear in Forum of Mathematics, Sigma

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

In recent years, substantial progress was made towards understanding convergence of fast-slow deterministic systems to stochastic differential equations. In contrast to more classical approaches, the assumptions on the fast flow are very mild. We survey the origins of this theory and then revisit and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1 (2016), 479-520], taking into account recent progress in $p$-variation and c\`adl\`ag rough path theory., Comment: 27 pages. Minor corrections. To appear in Proceedings of the Conference in Honor of the 75th Birthday of S.R.S. Varadhan

We prove well-posedness and rough path stability of a class of linear and semi-linear rough PDE's on R d using the variational approach. This includes well-posedness of (possibly degenerate) linear rough PDE's in L p ( R d ) , and then – based on a new method – energy estimates for non-degenerate linear rough PDE's. We accomplish this by controlling the energy in a properly chosen weighted L 2 -space, where the weight is given as a solution of an associated backward equation. These estimates then allow us to extend well-posedness for linear rough PDE's to semi-linear perturbations.

The indefinite integral of the homogenized Ornstein-Uhlenbeck process is a well-known model for physical Brownian motion, modelling the behaviour of an object subject to random impulses [L. S. Ornstein, G. E. Uhlenbeck: On the theory of Brownian Motion. In: Physical Review. 36, 1930, 823-841]. One can scale these models by changing the mass of the particle, and in the small mass limit one has almost sure uniform convergence in distribution to the standard idealized model of mathematical Brownian motion. This provides one well-known way of realising the Wiener process. However, this result is less robust than it would appear, and important generic functionals of the trajectories of the physical Brownian motion do not necessarily converge to the same functionals of Brownian motion when one takes the small mass limit. In the presence of a magnetic field the area process associated to the physical process converges - but not to Lévy’s stochastic area. As this area is felt generically in settings where the particle interacts through force fields in a non-linear way, the remark is physically significant and indicates that classical Brownian motion, with its usual stochastic calculus, is not an appropriate model for the limiting behaviour. We compute explicitly the area correction term and establish convergence, in the small mass limit, of the physical Brownian motion in the rough path sense. The small mass limit for the motion of a charged particle in the presence of a magnetic field is, in distribution, an easily calculable, but “non-canonical” rough path lift of Brownian motion. Viewing the trajectory of a charged Brownian particle with small mass as a rough path is informative and allows one to retain information that would be lost if one only considered it as a classical trajectory. We comment on the importance of this point of view.

In the framework of non-equilibrium thermodynamics, we derive a new model for many-particle electrodes. The model is applied to $$\text {LiFePO}_{4}$$ (LFP) electrodes consisting of many LFP particles of nanometer size. The phase transition from a lithium-poor to a lithium-rich phase within LFP electrodes is controlled by both different particle sizes and surface fluctuations leading to a system of stochastic differential equations. An explicit relation between battery voltage and current controlled by the thermodynamic state variables is derived. This voltage–current relation reveals that in thin LFP electrodes lithium intercalation from the particle surfaces into the LFP particles is the principal rate-limiting process. There are only two constant kinetic parameters in the model describing the intercalation rate and the fluctuation strength, respectively. The model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate-limiting capacity. Moreover we study the impact of both the particle size distribution and the active surface area on the voltage–charge characteristics of the electrode. Finally we carefully discuss the phase transition for varying charging/discharging rates.

We consider rough paths with jumps. In particular, the analogue of Lyons’ extension theorem and rough integration are established in a jump setting, offering a pathwise view on stochastic integration against càdlàg processes. A class of Lévy rough paths is introduced and characterized by a sub-ellipticity condition on the left-invariant diffusion vector fields and a certain integrability property of the Carnot–Caratheodory norm with respect to the Lévy measure on the group, using Hunt’s framework of Lie group valued Lévy processes. Examples of Lévy rough paths include a standard multi-dimensional Lévy process enhanced with a stochastic area as constructed by D. Williams, the pure area Poisson process and Brownian motion in a magnetic field. An explicit formula for the expected signature is given.

There are several (mathematical) reasons why Dupire’s formula fails in the non-diffusion setting. And yet, in practice, ad-hoc preconditioning of the option data works reasonably well. In this note, we attempt to explain why. In particular, we propose a regularization procedure of the option data so that Dupire’s local vol diffusion process recreates the correct option prices, even in manifest presence of jumps.

In [17] we discussed density expansions for multidimensional diffusions X 1 ,...,X d � , at fixed time T and projected to their first l coordinates, in the small noise regime. Global conditions were found which replace the well-known ”not-in-cutlocus” condition known from heat-kernel asymptotics. In the present paper we discuss financial applications; these include tail and implied volatility asymptotics in some correlated stochastic volatility models. In particular, we solve a problem left open by A. Gulisashvili and E.M. Stein (2009).

Density expansions for hypoelliptic diffusions (X1,…,Xd) are revisited. We are particularly interested in density expansions of the projection at time T > 0 with l ≤ d. Global conditions are found that replace the well-known “not-in-cut-locus” condition known from heat kernel asymptotics. Our small-noise expansion allows for a “second order” exponential factor. As an application, new light is shed on the Takanobu-Watanabe expansion of Brownian motion and Levy's stochastic area. Further applications include tail and implied volatility asymptotics in some stochastic volatility models, discussed in the companion paper [12].© 2013 Wiley Periodicals, Inc.

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. $1/q$-Holder 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.

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 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 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