Rough Paths Research Articles

Fatores intervenientes na preceptoria num serviço especializado em HIV/Aids: estudo etnográfico

Aim: to identify the factors impacting on the development of preceptorship as an educational method. Method: this is an ethnographic study with a qualitative approach. Five preceptors of a Multidisciplinary Residency Program took part in the research. Data collection happened between April and June 2014. The data collected were interpreted and analysed using content analysis, while Edgar Morin's thoughts were used as the theoretical framework. Results: it is evident that the lack of training, motivation and good working and ethical conditions represented factors which influenced the teaching and learning process. Discussion: the identified impacting factors are challenges that accompany the preceptors during their journey. Given this, it is recommended that there be an association between the seven facets of essential knowledge as proposed by Morin, and practice. Conclusion: while teaching residents, there is a long and rough path to be followed when seeking quality support for people with HIV/AIDS. In conclusion, it is necessary for preceptors to specialize in this area of education in order to improve the teaching of residents.

A Fractional Version of the Heston Model with Hurst Parameter H (1/2, 1)

We consider a fractional version of the Heston model where the two standard Brownian motions are replaced by two fractional Brownian motions with Hurst parameter H ∈ (1/2, 1). We show that the stochastic differential equation admits a unique positive solution by adapting and generalizing some results of Y. Hu, D. Nualart and X. Song on singular equations driven by rough paths. Moreover, we show that the fractional version of the variance, which is a version of the fractional Cox-Ingersoll-Ross model, is still a mean-reverting process.

Invariance for rough differential equations

In 1990, in Itô’s stochastic calculus framework, Aubin and Da Prato established a necessary and sufficient condition of invariance of a nonempty compact or convex subset C of Rd (d∈N∗) for stochastic differential equations (SDE) driven by a Brownian motion. In Lyons rough paths framework, this paper deals with an extension of Aubin and Da Prato’s results to rough differential equations. A comparison theorem is provided, and the special case of differential equations driven by a fractional Brownian motion is detailed.

Integration of controlled rough paths via fractional calculus

Abstract We develop a fractional calculus approach to rough path analysis, introduced by Y. Hu and D. Nualart [6], and show that our integration can be generalized so that it is consistent with the rough path integration introduced by M. Gubinelli [5].

Pathwise stochastic integrals for model free finance

We present two different approaches to stochastic integration in frictionless model free financial mathematics. The first one is in the spirit of It\^o's integral and based on a certain topology which is induced by the outer measure corresponding to the minimal superhedging price. The second one is based on the controlled rough path integral. We prove that every "typical price path" has a naturally associated It\^o rough path, and justify the application of the controlled rough path integral in finance by showing that it is the limit of non-anticipating Riemann sums, a new result in itself. Compared to the first approach, rough paths have the disadvantage of severely restricting the space of integrands, but the advantage of being a Banach space theory. Both approaches are based entirely on financial arguments and do not require any probabilistic structure.

Characteristic functions of measures on geometric rough paths

We define a characteristic function for probability measures on the signatures of geometric rough paths. We determine sufficient conditions under which a random variable is uniquely determined by its expected signature, thus partially solving the analogue of the moment problem. We furthermore study analyticity properties of the characteristic function and prove a method of moments for weak convergence of random variables. We apply our results to signature arising from L\'evy, Gaussian and Markovian rough paths.

QUASI-SURE EXISTENCE OF GAUSSIAN ROUGH PATHS AND LARGE DEVIATION PRINCIPLES FOR CAPACITIES

We construct a quasi-sure version (in the sense of Malliavin) of geometric rough paths associated with a Gaussian process with long-time memory. As an application we establish a large deviation principle (LDP) for capacities for such Gaussian rough paths. Together with Lyons' universal limit theorem, our results yield immediately the corresponding results for pathwise solutions to stochastic differential equations driven by such Gaussian process in the sense of rough paths. Moreover, our LDP result implies the result of Yoshida on the LDP for capacities over the abstract Wiener space associated with such Gaussian process.

On the integration of weakly geometric rough paths

We close a gap in the theory of integration for weakly geometric rough paths in the in…nite-dimensional setting. We show that the integral of a weakly geometric rough path against a su¢ ciently regular one form is, once again, a weakly geometric rough path.

Rough differential equations with unbounded drift term

We study controlled differential equations driven by a rough path (in the sense of T. Lyons) with an additional, possibly unbounded drift term. We show that the equation induces a solution flow if the drift grows at most linearly. Furthermore, we show that the semiflow exists assuming only appropriate one-sided growth conditions. We provide bounds for both the flow and the semiflow. Applied to stochastic analysis, our results imply strong completeness and the existence of a stochastic (semi)flow for a large class of stochastic differential equations. If the driving process is Gaussian, we can further deduce (essentially) sharp tail estimates for the (semi)flow and a Freidlin–Wentzell-type large deviation result.

Variational estimates for the bilinear iterated Fourier integral

We prove pointwise variational Lp bounds for a bilinear Fourier integral operator in a large but not necessarily sharp range of exponents. This result is a joint strengthening of the corresponding bounds for the classical Carleson operator, the bilinear Hilbert transform, the variation norm Carleson operator, and the bi-Carleson operator. Terry Lyon's rough path theory allows for extension of our result to multilinear estimates. We consider our result a proof of concept for a wider array of similar estimates with possible applications to ordinary differential equations.

An automated nudged elastic band method.

A robust, efficient, dynamic, and automated nudged elastic band (AutoNEB) algorithm to effectively locate transition states is presented. The strength of the algorithm is its ability to use fewer resources than the nudged elastic band (NEB) method by focusing first on converging a rough path before improving upon the resolution around the transition state. To demonstrate its efficiency, it has been benchmarked using a simple diffusion problem and a dehydrogenation reaction. In both cases, the total number of force evaluations used by the AutoNEB method is significantly less than the NEB method. Furthermore, it is shown that for a fast and robust relaxation to the transition state, a climbing image elastic band method where the full spring force, rather than only the component parallel to the local tangent to the path, is preferred especially for pathways through energy landscapes with multiple local minima. The resulting corner cutting does not affect the accuracy of the transition state as long as this is located with the climbing image method. Finally, a number of pitfalls often encountered while locating the true transition state of a reaction are discussed in terms of systematically exploring the multidimensional energy landscape of a given process.

Stochastic calculus with respect to G-Brownian motion viewed through rough paths

In this paper, we study rough path properties of stochastic integrals of It\^{o}'s type and Stratonovich's type with respect to $G$-Brownian motion. The roughness of $G$-Brownian Motion is estimated and then the pathwise Norris lemma in $G$-framework is obtained.

Pathwise stability of likelihood estimators for diffusions via rough paths

We consider the classical estimation problem of an unknown drift parameter within classes of nondegenerate diffusion processes. Using rough path theory (in the sense of T. Lyons), we analyze the Maximum Likelihood Estimator (MLE) with regard to its pathwise stability properties as well as robustness toward misspecification in volatility and even the very nature of the noise. Two numerical examples demonstrate the practical relevance of our results.

Exact convergence rate of the Wong–Zakai approximation to RDEs driven by Gaussian rough paths

We consider a solution to a stochastic differential equation driven by a Gaussian process in the rough differential equation sense and the Wong–Zakai approximation to the solution. We give an upper bound of the error of the Wong–Zakai approximation. We also show that the upper bound is optimal in a particular case.

Stabilization by unbounded‐variation noises

SummaryIn this paper, we claim the availability of deterministic noises for stabilization of the origins of dynamical systems, provided that the noises have unbounded variations. To achieve the result, we first consider the system representations of rough systems based on rough path analysis; then, we provide the notion of asymptotic stability for rough systems to analyze the stability for the systems. In the procedure, we also confirm that the system representations include stochastic differential equations; we also found that asymptotic stability for rough systems is the same property as uniform almost sure asymptotic stability provided by Bardi and Cesaroni. After the discussion, we confirm that there is a case that deterministic noises are capable of making the origin become asymptotically stable for rough systems while stochastic noises do not achieve the same stabilization results. Copyright © 2016 John Wiley & Sons, Ltd.

Real-Time Trajectory Planning for Autonomous Urban Driving: Framework, Algorithms, and Verifications

This paper focuses on the real-time trajectory planning problem for autonomous vehicles driving in realistic urban environments. To solve the complex navigation problem, we adopt a hierarchical motion planning framework. First, a rough reference path is extracted from the digital map using commands from the high-level behavioral planner. The conjugate gradient nonlinear optimization algorithm and the cubic B-spline curve are employed to smoothen and interpolate the reference path sequentially. To follow the refined reference path as well as handle both static and moving objects, the trajectory planning task is decoupled into lateral and longitudinal planning problems within the curvilinear coordinate framework. A rich set of kinematically feasible path candidates are generated to deal with the dynamic traffic both deliberatively and reactively. In the meanwhile, the velocity profile generation is performed to improve driving safety and comfort. After that, the generated trajectories are carefully evaluated by an objective function, which combines behavioral decisions by reasoning about the traffic situations. The optimal collision-free, smooth, and dynamically feasible trajectory is selected and transformed into commands executed by the low-level lateral and longitudinal controllers. Field experiments have been carried out with our test autonomous vehicle on the realistic inner-city roads. The experimental results demonstrated capabilities and effectiveness of the proposed trajectory planning framework and algorithms to safely handle a variety of typical driving scenarios, such as static and moving objects avoidance, lane keeping, and vehicle following, while respecting the traffic rules.

Discretely sampled signals and the rough Hoff process

We introduce a canonical method for transforming a discrete sequential data set into an associated rough path made up of lead–lag increments. In particular, by sampling a d-dimensional continuous semimartingale X:[0,1]→Rd at a set of times D={ti}, we construct a piecewise linear, axis-directed process XD:[0,1]→R2d comprised of a past and a future component. We call such an object the Hoff process associated with the discrete data {Xt}ti∈D. The Hoff process can be lifted to its natural rough path enhancement and we consider the question of convergence as the sampling frequency increases. We prove that the Itô integral can be recovered from a sequence of random ODEs driven by the components of XD. This is in contrast to the usual Stratonovich integral limit suggested by the classical Wong–Zakai Theorem (Wong and Zakai, 1965). Such random ODEs have a natural interpretation in the context of mathematical finance.

The signature of a rough path: Uniqueness

In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. We extend Hambly–Lyons' result and their notion of tree-like paths to the setting of weakly geometric rough paths in a Banach space. At the heart of our approach is a new definition for reduced path and a lemma identifying the reduced path group with the space of signatures.

Stochastic control with rough paths

We study a class of controlled differential equations driven by rough paths (or rough path realizations of Brownian motion) in the sense of Lyons. It is shown that the value function satisfies a HJB type equation; we also establish a form of the Pontryagin maximum principle. Deterministic problems of this type arise in the duality theory for controlled diffusion processes and typically involve anticipating stochastic analysis. We make the link to old work of Davis and Burstein (Stoch Stoch Rep 40:203---256, 1992) and then prove a continuous-time generalization of Roger's duality formula [SIAM J Control Optim 46:1116---1132, 2007]. The generic case of controlled volatility is seen to give trivial duality bounds, and explains the focus in Burstein---Davis' (and this) work on controlled drift. Our study of controlled rough differential equations also relates to work of Mazliak and Nourdin (Stoch Dyn 08:23, 2008).

Scalar conservation laws with rough flux and stochastic forcing

In this paper, we study scalar conservation laws where the flux is driven by a geometric Holder p-rough path for some $$p\in (2,3)$$ and the forcing is given by an Ito stochastic integral driven by a Brownian motion. In particular, we derive the corresponding kinetic formulation and define an appropriate notion of kinetic solution. In this context, we are able to establish well-posedness, i.e., existence, uniqueness and the $$L^1$$ -contraction property that leads to continuous dependence on initial condition. Our approach combines tools from rough path analysis, stochastic analysis and theory of kinetic solutions for conservation laws. As an application, this allows to cover the case of flux driven for instance by another (independent) Brownian motion enhanced with Levy’s stochastic area.