Rough Paths Research Articles

Optimal convergence rate of modified Milstein scheme for SDEs with rough fractional diffusions

We develop a new framework for error analysis on stochastic numerical schemes, with the rough path theory and stochastic backward error analysis. Based on our approach, we prove that the almost sure convergence rate of the modified Milstein scheme for stochastic differential equations driven by multiplicative multidimensional fractional Brownian motion with Hurst parameter H∈(14,12) is (2H−12)− for sufficiently smooth coefficients, which is optimal in the sense that it is consistent with the best probable convergence rate of implementable approximations of the Lévy area of fractional Brownian motion. Our result gives a positive answer to the conjecture proposed in [12] for the case H∈(13,12), and reveals for the first time that numerical schemes constructed by a second-order Taylor expansion converge for the case H∈(14,13].

Precise Local Estimates for Differential Equations driven by Fractional Brownian Motion: Elliptic Case

This article is concerned with stochastic differential equations driven by a d-dimensional fractional Brownian motion with Hurst parameter \(H>1/4\), understood in the rough paths sense. Whenever the coefficients of the equation satisfy a uniform ellipticity condition, we establish a sharp local estimate on the associated control distance function and a sharp local lower estimate on the density of the solution.

Construction of Rough Graph through Rough Membership Function

Rough membership function defines the degree of relationship between conditional and decision attributes of an information system. It is defined by <img src=image/13429145_01.gif> where <img src=image/13429145_07.gif> is the subset of <img src=image/13429145_08.gif> under the relation <img src=image/13429145_09.gif> where <img src=image/13429145_08.gif> is the universe of discourse. It can be expressed in different forms like cardinality form, probabilistic form etc. In cardinality form, it is expressed as <img src=image/13429145_02.gif> where as in probabilistic form it can be denoted as <img src=image/13429145_03.gif> where <img src=image/13429145_04.gif> is the equivalence class of <img src=image/13429145_10.gif> with respect to <img src=image/13429145_09.gif>. This membership function is used to measure the value of uncertainty. In this paper we have introduced the concept of graphical representation of rough sets. Rough graph was introduced by He Tong in 2006. In this paper, we propose a novel method for the construction of rough graph through rough membership function <img src=image/13429145_05.gif>. We propose that there is an edge between vertices if <img src=image/13429145_06.gif>. The rough graph is being constructed for an information system; here objects are considered as vertices. Rough path, rough cycle, rough ladder graph are introduced in this paper. We develop the operations on rough graph and also extend the properties of rough graph.

A new definition of rough paths on manifolds

Smooth manifolds are not the suitable context for trying to generalize the concept of rough paths on a manifold. Indeed, when one is working with smooth maps instead of Lipschitz maps and trying to solve a rough differential equation, one loses the quantitative estimates controlling the convergence of the Picard sequence. Moreover, even with a definition of rough paths in smooth manifolds, ordinary and rough differential equations can only be solved locally in such case. In this paper, we first recall the foundations of the Lipschitz geometry, introduced in Rough Paths on Manifolds (Cass, T., Litterer, C. & Lyons, T.), along with the main findings that encompass the classical theory of rough paths in Banach spaces. Then we give what we believe to be a minimal framework for defining rough paths on a manifold that is both less rigid than the classical one and emphasized on the local behaviour of rough paths. We end by explaining how this same idea can be used to define any notion of coloured paths on a manifold.

A Version of Hörmander’s Theorem for Markovian Rough Paths

We consider a rough differential equation of the form dYt = ∑ iVi(Yt)dXti + V0(Yt)dt, where Xt is a Markovian rough path. We demonstrate that if the vector fields (Vi)0≤i≤d satisfy the parabolic Hörmander’s condition, then Yt admits a smooth density with a Gaussian type upper bound, given that the generator of Xt satisfy certain non-degenerate conditions. The main new ingredient of this paper is the study of a non-degenerate property of the Jacobian process of Xt.

A first order description of a nonlinear SPDE in the spirit of rough paths

We consider a nonlinear stochastic partial differential equation (SPDE) in divergence form where the forcing term is a Gaussian noise, that is white in time and colored in space such that the gradient of the solution is Hölder-continuous, but not differentiable. Then, we prove a generalized Taylor expansion of the difference between the solution to the SPDE and the solution to its linearization around a fixed basepoint. The result is reminiscent of the theory of (controlled) rough paths and agrees with the general observation, that, in settings with a rough driver, subtracting the solution to the linearized equation yields a more regular object.

Two-dimensional signature of images and texture classification

We introduce a proper notion of two-dimensional signature for images. This object is inspired by the so-called rough paths theory, and it captures many essential features of a two-dimensional object such as an image. It thus serves as a low-dimensional feature for pattern classification. Here, we implement a simple procedure for texture classification. In this context, we show that a low-dimensional set of features based on signatures produces an excellent accuracy.

Geometric rough paths on infinite dimensional spaces

Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For α∈(1/3,1/2), we give criteria for when we can approximate Banach space-valued weakly geometric α-rough paths by signatures of curves of bounded variation, given some tuning of the Hölder parameter. We show that these criteria are satisfied for weakly geometric rough paths on Hilbert spaces. As an application, we obtain Wong-Zakai type result for function space valued martingales using the notion of (unbounded) rough drivers.

Besov rough path analysis

Rough path analysis is developed in the full Besov scale. This extends, and essentially concludes, an investigation started by Prömel and Trabs (2016) [49], further studied in a series of papers by Liu, Prömel and Teichmann. A new Besov sewing lemma, a real-analysis result of interest in its own right, plays a key role, and the flexibility in the choice of Besov parameters allows for the treatment of equations not available in the Hölder or variation settings. Important classes of stochastic processes fit in the present framework.

An Asymptotic Radius of Convergence for the Loewner Equation and Simulation of $$SLE_{\kappa }$$ Traces via Splitting

In this paper, we shall study the convergence of Taylor approximations for the backward Loewner differential equation (driven by Brownian motion) near the origin. More concretely, whenever the initial condition of the backward Loewner equation (which lies in the upper half plane) is small and has the form $Z_{0} = \varepsilon i$, we show these approximations exhibit an $O(\varepsilon)$ error provided the time horizon is $\varepsilon^{2+\delta}$ for $\delta > 0$. Statements of this theorem will be given using both rough path and $L^{2}(\mathbb{P})$ estimates. Furthermore, over the time horizon of $\varepsilon^{2-\delta}$, we shall see that "higher degree" terms within the Taylor expansion become larger than "lower degree" terms for small $\varepsilon$. In this sense, the time horizon on which approximations are accurate scales like $\varepsilon^{2}$. This scaling comes naturally from the Loewner equation when growing vector field derivatives are balanced against decaying iterated integrals of the Brownian motion. As well as being of theoretical interest, this scaling may be used as a guiding principle for developing adaptive step size strategies which perform efficiently near the origin. In addition, this result highlights the limitations of using stochastic Taylor methods (such as the Euler-Maruyama and Milstein methods) for approximating $SLE_{\kappa}$ traces. Due to the analytically tractable vector fields of the Loewner equation, we will show Ninomiya-Victoir (or Strang) splitting is particularly well suited for SLE simulation. As the singularity at the origin can lead to large numerical errors, we shall employ the adaptive step size proposed by Tom Kennedy to discretize $SLE_{\kappa}$ traces using this splitting. We believe that the Ninomiya-Victoir scheme is the first high order numerical method that has been successfully applied to $SLE_{\kappa}$ traces.

The valuation of barrier options under a threshold rough Heston model

In this paper, we propose a novel model for pricing double barrier options, where the asset price is modeled as a threshold geometric Brownian motion time changed by an integrated activity rate process, which is driven by the convolution of a fractional kernel with the CIR process. The new model both captures the leverage effect and produces rough paths for the volatility process. The model also nests the threshold diffusion, Heston and rough Heston models. We can derive analytical formulas for the double barrier option prices based on the eigenfunction expansion method. We also implement the model and numerically investigate the sensitivities of option prices with respect to the parameters of the model.

On exterior differential systems involving differentials of Hölder functions

We study the validity of an extension of Frobenius theorem on integral manifolds for some classes of Pfaff-type systems of partial differential equations involving multidimensional “rough” signals, i.e. “differentials” of given Hölder continuous functions interpreted in a suitable way, similarly to Young Differential Equations in Rough Paths theory. This can be seen as a tool to study solvability of exterior differential systems involving rough differential forms, i.e. the forms involving weak (distributional) derivatives of highly irregular (e.g. Hölder continuous) functions; the solutions (integral manifolds) being also some very weakly regular geometric structures.

Multiscaling and rough volatility: An empirical investigation

Pricing derivatives goes back to the acclaimed Black and Scholes model. However, such a modelling approach is known not to be able to reproduce some of the financial stylised facts, including the dynamics of volatility. In the mathematical finance community, it has therefore emerged a new paradigm, named rough volatility modelling, that represents the volatility dynamics of financial assets as a fractional Brownian motion with Hurst exponent very small, which indeed produces rough paths. At the same time, prices’ time series have been shown to be multiscaling, characterised by different Hurst scaling exponents. This paper assesses the interplay, if present, between price multiscaling and volatility roughness, defined as the (low) Hurst exponent of the volatility process. In particular, we perform extensive simulation experiments by using one of the leading rough volatility models present in the literature, the rough Bergomi model. A real data analysis is also conducted to test if the rough volatility model reproduces the same relationship. We find that the model can reproduce multiscaling features of the prices’ time series when a low value of the Hurst exponent is used, but it fails to reproduce what the real data says. Indeed, we find that the dependency between prices’ multiscaling and the Hurst exponent of the volatility process is diametrically opposite to what we find in real data, namely a negative interplay between the two.

Global well-posedness of the 3D Navier–Stokes equations perturbed by a deterministic vector field

We are concerned with the problem of global well-posedness of the 3D Navier–Stokes equations on the torus with unitary viscosity. While a full answer to this question seems to be out of reach of the current techniques, we establish a regularization by a deterministic vector field. More precisely, we consider the vorticity form of the system perturbed by an additional transport type term. Such a perturbation conserves the enstrophy and therefore a priori it does not imply any smoothing. Our main result is a construction of a deterministic vector field v=v(t,x) which provides the desired regularization of the system and yields global well-posedness for large initial data outside arbitrary small sets. The proof relies on probabilistic arguments developed by Flandoli and Luo, tools from rough path theory by Hofmanová, Leahy and Nilssen and a new Wong–Zakai approximation result, which itself combines probabilistic and rough path techniques.

Deterministic homogenization under optimal moment assumptions for fast–slow systems. Part 2

We consider deterministic homogenization for discrete-time fast-slow systems of the form $$ X_{k+1} = X_k + n^{-1}a_n(X_k,Y_k) + n^{-1/2}b_n(X_k,Y_k)\;, \quad Y_{k+1} = T_nY_k\;$$ and give conditions under which the dynamics of the slow equations converge weakly to an It\^o diffusion $X$ as $n\to\infty$. The drift and diffusion coefficients of the limiting stochastic differential equation satisfied by $X$ are given explicitly. This extends the results of [Kelly-Melbourne, J. Funct. Anal. 272 (2017) 4063--4102] from the continuous-time case to the discrete-time case. Moreover, our methods (c\`adl\`ag $p$-variation rough paths) work under optimal moment assumptions. Combined with parallel developments on martingale approximations for families of nonuniformly expanding maps in Part 1 by Korepanov, Kosloff & Melbourne, we obtain optimal homogenization results when $T_n$ is such a family of maps.

Lung Cancer Organoids: The Rough Path to Personalized Medicine.

Simple SummaryLung cancer is very difficult to cure, especially when it has spread to other parts of the body. One of the main reasons that delay the identification of effective therapies is the complexity of lung cancer cells, which can be very different among individual patients. Organoids are small aggregates of tumor cells that are generated from lung tumors and used in cancer research laboratories to study the features of tumor cells. Organoids have unique properties, as they reproduce many characteristics of the tumor derived from each specific patient. Due to their capacity to reproduce individual tumor features in the laboratory setting, organoids are an excellent system to study lung cancer and to identify functional therapies. This review summarizes the challenges encountered by researchers in the field of lung cancer organoids and describes how the advances in organoid technology may allow the future development of individualized therapies for lung cancer patients.Lung cancer is the leading cause of cancer death worldwide. Despite significant advances in research and therapy, a dismal 5-year survival rate of only 10–20% urges the development of reliable preclinical models and effective therapeutic tools. Lung cancer is characterized by a high degree of heterogeneity in its histology, a genomic landscape, and response to therapies that has been traditionally difficult to reproduce in preclinical models. However, the advent of three-dimensional culture technologies has opened new perspectives to recapitulate in vitro individualized tumor features and to anticipate treatment efficacy. The generation of lung cancer organoids (LCOs) has encountered greater challenges as compared to organoids derived from other tumors. In the last two years, many efforts have been dedicated to optimizing LCO-based platforms, resulting in improved rates of LCO production, purity, culture timing, and long-term expansion. However, due to the complexity of lung cancer, further advances are required in order to meet clinical needs. Here, we discuss the evolution of LCO technology and the use of LCOs in basic and translational lung cancer research. Although the field of LCOs is still in its infancy, its prospective development will likely lead to new strategies for drug testing and biomarker identification, thus allowing a more personalized therapeutic approach for lung cancer patients.

The Sewing lemma for 0 < γ ≤ 1

We establish a Sewing lemma in the regime γ∈(0,1], constructing a Sewing map which is neither unique nor canonical, but which is nonetheless continuous with respect to the standard norms. Two immediate corollaries follow, which hold on any commutative graded connected locally finite Hopf algebra: a simple constructive proof of the Lyons-Victoir extension theorem which associates to a Hölder path a rough path, with the additional result that this map can be made continuous; the bicontinuity of a transitive free action of a space of Hölder functions on the set of Rough Paths.

Volterra equations driven by rough signals 2: Higher-order expansions

We extend the recently developed rough path theory for Volterra equations from [F. Harang and S. Tindel, Volterra equations driven by rough signals, Stoch. Process. Appl. 142 (2021) 34–78] to the case of more rough noise and/or more singular Volterra kernels. It was already observed in [F. Harang and S. Tindel, Volterra equations driven by rough signals, Stoch. Process. Appl. 142 (2021) 34–78] that the Volterra rough path introduced there did not satisfy any geometric relation, similar to that observed in classical rough path theory. Thus, an extension of the theory to more irregular driving signals requires a deeper understanding of the specific algebraic structure arising in the Volterra rough path. Inspired by the elements of “non-geometric rough paths” developed in [M. Gubinelli, Ramification of rough paths, J. Differential Equations 248 (2010) 693–721; M. Hairer and D. Kelly, Geometric versus non-geometric rough path, Ann. Inst. Henri Poincaré-Probab. Stat. 51 (2015) 207–251], we provide a simple description of the Volterra rough path and the controlled Volterra process in terms of rooted trees, and with this description we are able to solve rough Volterra equations driven by more irregular signals.

On the uniqueness of higher order Gubinelli derivatives and an analogue of the Doob – Meyer theorem for rough paths of the arbitrary positive Holder index

In this paper, we investigate the features of higher order Gubinelli derivatives of controlled rough paths having an arbitrary positive Holder index. There is used a notion of the (α, β)-rough map on the basis of which the sufficient conditions are given for the higher order Gubinelli derivatives uniqueness. Using the theorem on the uniqueness of higher order Gubinelli derivatives an analogue of the Doob – Meyer theorem for rough paths with an arbitrary positive Holder index is proved. In the final section of the paper, we prove that the law of the local iterated logarithm for fractional Brownian motion allows using all the main results of this paper for integration over the multidimensional fractional Brownian motions of the arbitrary Hurst index. The examples demonstrating the connection between the rough path integrals and the Ito and Stratonovich integrals are represented.

Solution properties of the incompressible Euler system with rough path advection

The present paper aims to establish the local well-posedness of Euler's fluid equations on geometric rough paths. In particular, we consider the Euler equations for the incompressible flow of an ideal fluid whose Lagrangian transport velocity possesses an additional rough-in-time, divergence-free vector field. In recent work, we have demonstrated that this system can be derived from Clebsch and Hamilton-Pontryagin variational principles that possess a perturbative geometric rough path Lie-advection constraint. In this paper, we prove the local well-posedness of the system in L2-Sobolev spaces Hm with integer regularity m≥⌊d/2⌋+2 and establish a Beale-Kato-Majda (BKM) blow-up criterion in terms of the Lt1Lx∞-norm of the vorticity. In dimension two, we show that the Lp-norms of the vorticity are conserved, which yields global well-posedness and a Wong-Zakai approximation theorem for the stochastic version of the equation.