Rough Paths Research Articles

Mining massive taxi trajectories for rapid fastest path planning in dynamic multi-level landmark network

This paper presents a speed up algorithm for real-time path planning based on massive trajectory data mining, which comprises three stages: preparatory work, preprocessing stage and the online fastest path query. At the preparatory stage, this algorithm constructs multi-level landmarks and divides the original road network into multiple levels accordingly. At the preprocessing stage, this algorithm first estimates the travel time of all road segments according to the real-time traffic information, then compares all taxi trajectories to extract the experiential fastest paths, and finally makes use of the multi-level landmarks to obtain the rough fastest paths for all landmark pairs. At the online fastest path query stage, the server side first responds by returning a rough fastest path based on the preprocessing result, and then refines it by iteratively introducing the experiential fastest paths. A series of experiments are made to compare the proposed algorithm with the other three algorithms. Experiments indicate that the proposed algorithm has the ability to find more time-saving paths in response to client requests. More importantly, because this algorithm is capable of ensuring the fast completion of pre-computation on the server side, it has an evident advantage in the time cost of the online fastest path query compared with the other three algorithms, which is particularly suitable for the online optimal path query from a larger number of end users.

CONTINUOUS DEPENDENCE ON THE INITIAL DATA OF THE SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH FRACTIONAL BROWNIAN MOTIONS

In the present acticle we consider finite-dimensional stochastic differential equations with fractional Brownian motions having different Hurst indices larger than 1/3 and a drift. These heterogeneous components of the equations are combined into a single process. The solutions of the equations are understood in the integral sense, and the integrals in turn are Gubinelli’s rough path integrals [1] realizing the well-known approach of the rough paths theory [2]. The existence and uniqueness conditions of the solutions of these stochastic differential equations are specified. Such conditions are sufficient to obtain the results related the continuous dependence on the initial data. In this article, we have first proved a continuous dependence on the initial conditions and the right-hand sides of the solutions of the stochastic differential equations under consideration for almost all their trajectories. The result obtained does not depend on the probabilistic properties of fractional Brownian motions, and therefore it can be easily generalized to the case of arbitrary Holder-continuous processes with an exponent greater than 1/3. In this case, the constant arising in the estimates appears to be exponentially dependent on the norms of fractional Brownian motions. Taking into account the last fact and the proved result, an expected logarithmic continuous dependence on the initial conditions and the right-hand sides of the solutions of the stochastic differential equations con - si dered is subsequently derived. This is the major result of this article.

On a fractional stochastic Hodgkin–Huxley model

The model studied in this paper is a stochastic extension of the so-called neuron model introduced by Hodgkin and Huxley. In the sense of rough paths, the model is perturbed by a multiplicative noise driven by a fractional Brownian motion, with a vector field satisfying the viability condition of Coutin and Marie for [Formula: see text]. An application to the modeling of the membrane potential of nerve fibers damaged by a neuropathy is provided.

Rough differential equations with power type nonlinearities

In this note we consider differential equations driven by a signal x which is γ-Hölder with γ>13, and is assumed to possess a lift as a rough path. Our main point is to obtain existence of solutions when the coefficients of the equation behave like power functions of the form |ξ|κ with κ∈(0,1). Two different methods are used in order to construct solutions: (i) In a 1-d setting, we resort to a rough version of Lamperti’s transform. (ii) For multidimensional situations, we quantify some improved regularity estimates when the solution approaches the origin.

Option pricing under fast-varying and rough stochastic volatility

Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it is mean reverting. In this paper it is shown that the price impact of a rapidly mean reverting rough volatility model coincides with that associated with fast mean reverting Markov stochastic volatility models. This reconciles the empirical observation of rough volatility paths with the good fit of the implied volatility surface to models of fast mean reverting Markov volatilities. Moreover, the result conforms with recent numerical results regarding rough stochastic volatility models. It extends the scope of models for which the asymptotic results of fast mean reverting Markov volatilities are valid. The paper concludes with a general discussion of fractional volatility asymptotics and their interrelation. The regimes discussed there include fast and slow volatility factors with strong or small volatility fluctuations and with the limits not commuting in general. The notion of a characteristic term structure exponent is introduced, this exponent governs the implied volatility term structure in the various asymptotic regimes.

Perron’s method for pathwise viscosity solutions

We use Perron’s method to construct viscosity solutions of fully non-linear degenerate parabolic pathwise (rough) partial differential equations. This provides an intrinsic method for proving the existence of solutions that relies only on a comparison principle, rather than considering equations driven by smooth approximating paths. The result covers the case of multidimensional geometric rough path noise, where the noise coefficients depend non-trivially on space and on the gradient of the solution. Also included in this note is a discussion of the comparison principle and a summary of the pathwise equations for which one has been proved.

Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length

For a path of length L>0, if for all n≥1, we multiply the n-th term of the signature by n!L−n, we say that the resulting signature is ‘normalised’. It has been established (T. J. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths, Springer, 2007) that the norm of the n-th term of the normalised signature of a bounded-variation path is bounded above by 1. In this article, we discuss the super-multiplicativity of the norm of the signature of a path with finite length, and prove by Fekete's lemma the existence of a non-zero limit of the n-th root of the norm of the n-th term in the normalised signature as n approaches infinity.

A Quasi-sure Non-degeneracy Property for the Brownian Rough Path

In the present paper, we are going to show that outside a slim set in the sense of Malliavin (or quasi-surely), the signature path (which consists of iterated path integrals in every degree) of Brownian motion is non-self-intersecting. This property relates closely to a non-degeneracy property for the Brownian rough path arising naturally from the uniqueness of signature problem in rough path theory. As an important consequence we conclude that quasi-surely, the Brownian rough path does not have any tree-like pieces and every sample path of Brownian motion is uniquely determined by its signature up to reparametrization.

Path-dependent convex conservation laws

For convex scalar conservation laws in 1-d, driven by a rough path z(t), in the sense of Lions, Perthame and Souganidis in [32], we show that it is possible to replace z(t) by a piecewise linear path, and still obtain the same solution at a given time. This result is connected to the spatial regularity of solutions. We show that solutions are spatially Lipschitz continuous for an a priori set of times, depending on the path and the initial data. Fine properties of the map z↦u(τ), for a prescribed time τ, are studied. We provide a detailed description of the properties of the rough path z(t) that influence the solution. This description is extracted by a “factorization” of the solution operator (at time τ). In a companion paper [24], we make use of the observations herein to construct computationally efficient numerical methods.

An energy method for rough partial differential equations

We present a well-posedness and stability result for a class of nondegenerate linear parabolic equations driven by geometric rough paths. More precisely, we introduce a notion of weak solution that satisfies an intrinsic formulation of the equation in a suitable Sobolev space of negative order. Weak solutions are then shown to satisfy the corresponding energy estimates which are deduced directly from the equation. Existence is obtained by showing compactness of a suitable sequence of approximate solutions whereas uniqueness relies on a doubling of variables argument and a careful analysis of the passage to the diagonal. Our result is optimal in the sense that the assumptions on the deterministic part of the equation as well as the initial condition are the same as in the classical PDEs theory.

Fracture behaviour of cemented tailing backfill with pre-existing crack and thermal treatment under three-point bending loading: Experimental studies and particle flow code simulation

The three-point bending test was performed on cemented tailing backfill prepared with different offset notches by a loading system with high-speed camera. And the effect of heated temperature on CTB fracture behavior has been investigated. These results indicate that the influence of pre-existing notch and temperature on CTB fracture feature is obvious. The peak load increases linearly with the increasing of the offset ratio when the depth-height ratio of notch is fixed at the same value, while the fracture peak loading decreases sharply with the increase of the notch depth when the offset ratios are the same level. The fracture toughness decreases by 63.5% as the heated temperature rises from 20 °C to 120 °C. The fracture toughness is directly proportional to the porosity of CTB. The increasing of porosity leads to the decrease of the fracture toughness mainly due to the decomposition of hydration products and the coarsening of the pore structure of CTBs. The crack propagates along the interface among tailing particles, resulting in a rough and jagged edge propagation path. The rack propagation pattern of CTB samples contains transgranular fracture and intergranular fracture. The intergranular fracture is clearly predominant during crack propagation. The rack propagation and failure pattern of CTB specimens from numerical modeling are conformed to the failure pattern of samples obtained by experimental testing. PFC is an effective numerical analysis method for exploring the fracture mechanism of micro-cracking in CTB.

Qozloq Route (Astrabad to Shahrud) Impact on Economic Developments of the Region (Safavid Course)

The Qozloq Route was one of the branches of the famous Silk Road in the northeast of Iran, which linked two important and strategic regions of Shahrud and Astrabad. This road constituted rough and smooth paths and was the passage of different nations with different goals. In this context, various cultures have also been published and exchanged in line with the trade of various goods.The presence of different caravansaries around the road indicates its importance and prosperity in the Safavid course, but with all of this, there is little information available on the importance of this route in the existing travel books and historical books. Despite all the inadequacies, in this research, with the descriptive-analytical approach based on the research data, it is concluded that the Qozloq Route has been of great importance in the Safavid course, strategically, and in term of the publication of the culture and prosperity of the economy, and the dynamism of development and awareness.

A rough calculus approach to level sets in the Heisenberg group

We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in $\mathbb{R}^2$. These equations can be seen as a sub-Riemannian counterpart to classical ODEs arising from the implicit function theorem. We show that they enjoy all the natural well-posedness properties, thus allowing for a good calculus on nonsmooth level sets. We apply these results to prove an area formula for the intrinsic measure of level sets, along with the corresponding coarea formula.

Symplectic Runge–Kutta methods for Hamiltonian systems driven by Gaussian rough paths

We consider Hamiltonian systems driven by multi-dimensional Gaussian processes in rough path sense, which include fractional Brownian motions with Hurst parameter H∈(1/4,1/2]. We prove that the phase flow preserves the symplectic structure almost surely and this property could be inherited by symplectic Runge–Kutta methods, which are implicit methods in general. If the vector fields satisfy some smoothness and boundedness conditions, we obtain the pathwise convergence rates of Runge–Kutta methods. When vector fields are linear, we get the solvability of the midpoint scheme for skew symmetric cases, and obtain its pathwise convergence rate. Numerical experiments verify our theoretical analysis.

Differential equations driven by rough paths with jumps

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

A support and density theorem for Markovian rough paths

We establish two results concerning a class of geometric rough paths $\mathbf{X}$ which arise as Markov processes associated to uniformly subelliptic Dirichlet forms. The first is a support theorem for $\mathbf{X}$ in $\alpha$-H\"older rough path topology for all $\alpha \in (0,1/2)$, which answers in the positive a conjecture of Friz-Victoir (2010). The second is a H\"ormander-type theorem for the existence of a density of a rough differential equation driven by $\mathbf{X}$, the proof of which is based on analysis of (non-symmetric) Dirichlet forms on manifolds.

Rough Linearization By One-Dimensional Rough Paths

In this paper, we propose a linearization method of nonlinear control systems by using rough path analysis. The linearization is achieved by the addition of rough signals, which are designed by functions with unbounded variations such as a sinusoidal wave having infinitely-large frequency Therefore, the proposed method is considered to be an extension of the analysis of Lie bracket motions such as the approximation algorithms by Liu and Sussmann in 1990’s, or, the transverse function approach by Morin et al. in 2000’s. Note, however, that, the characteristic feature of the method is to derive controllable linear approximate models for nonholonomic systems even if no Lie bracket motion is generated. Instead, another motions appear based on the similar effects of Wong-Zakai correction terms in stochastic analysis, provided that any stochastic signals are never dealt with in this paper. The validity of the method is confirmed by a few simple examples of linearization problems for nonlinear control systems.

Rough Paths Theory and Impulsive Control: A Promising Connection

Abstract This short discussion note is devoted to establishing some connections between the framework of dynamical systems driven by signals of a low regularity (rough differential equations) and the mathematical theory of impulsive control (measure-driven systems), which, as we believe, would essentially enrich one another. More precisely, we attempt to elaborate an impulsive (discontinuous) extension of the theory of rough differential equation for input-affine models with states of unbounded first variation. In the paper, we are focused on a concept of solution for impulsive rough differential equations with vector-valued controls of bounded p-variation, p£ (1,2), and its constructive representation through a (discrete-continuous) Young’s integral equation.

Lévy area with a drift as a renormalization limit of Markov chains on periodic graphs

A careful look at rough path topology applied to Brownian motion reveals new possible properties of the well-known Lévy area, in particular the presence of an intrinsic drift of this area. Using renormalization limit of Markov chains on periodic graphs, we present a construction of such a non-trivial drift and give an explicit formula for it. Several examples with explicit computations are included.

Decay rate of iterated integrals of branched rough paths

Iterated integrals of paths arise frequently in the study of the Taylor's expansion for controlled differential equations. We will prove a factorial decay estimate, conjectured by M. Gubinelli, for the iterated integrals of non-geometric rough paths. We will explain, with a counter example, why the conventional approach of using the neoclassical inequality fails. Our proof involves a concavity estimate for sums over rooted trees and a non-trivial extension of T. Lyons' proof in 1994 for the factorial decay of iterated Young's integrals.