We consider controlled differential equations and give new estimates for higher order Euler schemes. Our proofs are inspired by recent work of A. M. Davie who considers first and second order schemes. In order to implement the general case we make systematic use of geodesic approximations in the free nilpotent group. As application, we can control moments of solutions to rough path differential equations (RDEs) driven by random rough paths with sufficient integrability and have a criteria for L^q - convergence in the Universal Limit Theorem. We also obtain Azencott type estimates and asymptotic expansions for random RDE solution. When specialized to RDEs driven by Enhanced Brownian motion, we (mildly) improve classic estimates for diffusions in the small time limit.
We give a dimension-free Euler estimation of solution of rough differential equations in term of the driving rough path. In the meanwhile, we prove that, the solution of rough differential equation is close to the exponential of a Lie series, with a concrete error bound.
We establish the existence of solutions to path-dependent rough differential equations with non-anticipative coefficients. Regularity assumptions on the coefficients are formulated in terms of horizontal and vertical derivatives.
Based on two isomorphisms of Hopf algebras, we provide a bound in the optimal order on the remainder of the truncated Taylor expansion for controlled differential equations driven by branched rough paths.
We provide a necessary and sufficient condition for a rough control driving a differential equation to be reconstructable, to some order, from observing the resulting controlled evolution. Physical examples and applications in stochastic filtering and statistics demonstrate the practical relevance of our result.
We show in this work how the machinery of C^1-approximate flows introduced in our previous work "Flows driven by rough paths", provides a very efficient tool for proving well-posedness results for path-dependent rough differential equations on flows of the form dϕ= V h(dt) + F X(dt), for smooth enough path-dependent vector fields V,F = (V_1,...,V_\ell), any Holder weak geometric p-rough path X and any a-Holder path h, with a+1/p>1.
These are lecture notes for a Master 2 course on rough differential equations driven by weak geometric Holder p-rough paths, for any p>2. They provide a short, self-contained and pedagogical account of the theory, with an emphasis on flows. The theory is illustrated by some now classical applications to stochastic analysis, such as the basics of Freidlin-Wentzel theory of large deviations for diffusions, or Stroock and Varadhan support theorem.
We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type $$ dX_t = V(X_t,\mathcal{L}(X_t))dt + F(X_t,\mathcal{L}(X_t))dW_t $$ where $W$ is a random rough path and $\mathcal{L}(X_t)$ is the law of $X_t$. We prove propagation of chaos, and provide also an explicit optimal convergence rate. The analysis is based upon the tools we developed in our companion paper [1] for solving mean field rough differential equations and in particular upon a corresponding version of the Itô-Lyons continuity theorem. Th…
A theory of differential equations driven by a non-differentiable path has recently been developed by Lyons. We develop an alternative approach to this theory, using (modified Euler approximations), and investigate its applicability to stochastic differential equations driven by Brownian motion. We also give some other examples showing that the main results are reasonably sharp.
The non-linear sewing lemma constructs flows of rough differential equations from a braod class of approximations called almost flows. We consider a class of almost flows that could be approximated by solutions of ordinary differential equations, in the spirit of the backward error analysis. Mixing algebra and analysis, a Taylor formula with remainder and a composition formula are central in the expansion analysis. With a suitable algebraic structure on the non-smooth vector fields to be integrated, we recover in a single framework several results regarding high-order expansions for various ki…
Rough differential equations and reduced rough paths: a Lie bracket characterization
School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）；Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics（甘肃省数学与统计基础学科研究中心）
We provide in this work a robust solution theory for random rough differential equations of mean field type $$ dX_t = V(X_t,\mathcal{L}(X_t))dt + F(X_t,\mathcal{L}(X_t))dW_t, $$ where $W$ is a random rough path and $\mathcal{L}(X_t)$ stands for the law of $X_t$, with mean field interaction in both the drift and diffusivity. The analysis requires the introduction of a new rough path-like setting and an associated notion of controlled path. We use crucially Lions' approach to differential calculus on Wasserstein space along the way.
N. Katz introduced the notion of the middle convolution on local systems. This can be seen as a generalization of the Euler transform of Fuchsian differential equations. In this paper, we consider the generalization of the Euler transform, the twisted Euler transform, and apply this to differential equations with irregular singular points. In particular, for differential equations with an irregular singular point of irregular rank 2 at $x=\infty$, we describe explicitly changes of local datum caused by twisted Euler transforms. Also we attach these differential equations to Kac-Moody Lie algeb…
By using an explicit ordinary differential equation to approximate the exponential solution flow, we extend the universal limit theorem to rough differential equation in Banach space driven by weak geometric rough path, and give the quantitative dependence of solution in term of the initial value, vector field and driving rough path.
We give in this note a simple treatment of the non-explosion problem for rough differential equations driven by unbounded vector fields and weak geometric rough paths of arbitrary roughness.
This paper revisits the concept of rough paths of inhomogeneous degree of smoothness (geometric Π-rough paths in our terminology) sketched by Lyons ("Differential equations driven by rough signals", Revista Mathematica Iber. Vol 14, Nr. 2,215-310, 1998). Although geometric Π-rough paths can be treated as p-rough paths for a sufficiently large p and the theory of integration of Lip-γone-forms (γ>p-1) along geometric p-rough paths applies, we prove the existence of integrals of one-forms under weaker conditions. Moreover, we consider differential equations driven by geometric Π-rough paths and g…
We provide an account for the existence and uniqueness of solutions to rough differential equations under the framework of controlled rough paths. The case when the driving path is $β$-Hölder continuous, for $β>1/3$, is widely available in the literature. In its extension to the case when $β\leqslant1/3,$ a main challenge and missing ingredient is to show that controlled roughs paths are closed under composition with Lipschitz transformations. Establishing such a property precisely, which has a strong algebraic nature, is a main purpose of the present article.
We show how to use geometric arguments to prove that the terminal solution to a rough differential equation driven by a geometric rough path can be obtained by driving the same equation by a piecewise linear path. For this purpose, we combine some results of the seminal work of Sussmann on orbits of vector fields with the rough calculus on manifolds developed by Cass, Litterer and Lyons.
We give an overview of the recent approach to the integration of rough paths that reduces the problem to classical Young integration. As an application, we extend an argument of Schwartz to rough differential equations, and prove the existence, uniqueness and continuity of the solution, which is applicable when the driving path takes values in nilpotent Lie group or Butcher group.
We give an elementary proof that Davie's definition of a solution to a rough differential equation and the notion of solution given by Bailleul in (Flows driven by rough paths) coincide. This provides an alternative point on view on the deep algebraic insights of Cass and Weidner in their work (Tree algebras over topological vector spaces in rough path theory).
In this article, we present an innovative method to find particular solutions of the non-homogeneous Cauchy-Euler equations in several variables and Sturm-Liouville equations. The method is basically built on the application of the ponderation ring which introduced by Assal and Zeyada [2].
The central aim of this work is to understand rough differential equations on homogeneous spaces. We focus on the formal approach, by giving an explicit expansion of the solution at each point of the real line in terms of decorated planar forests. For this we develop the notion of planarly branched rough paths, following M. Gubinelli's branched rough paths. The definition is similar to the one in the flat case, the main difference being the replacement of the Butcher--Connes--Kreimer Hopf algebra of non-planar rooted forests by the Munthe-Kaas--Wright Hopf algebra of planar rooted forests. We …
We obtain a remainder estimate for the truncated Taylor expansion for differential equations driven by weakly geometric $Π$-rough paths for $Π=\left( p_{1},\cdots ,p_{k}\right) $, $p_{i}\geq 1$. When there exists $ p\geq 1$ such that $p_{i}=pk_{i}^{-1}\ $for some $k_{i}\in \left\{ 1,\dots , \left[ p\right] \right\} $, we obtain a refined Taylor remainder estimate that contains a factorial decay component. The remainder estimates are in the right order as they are comparable to the next term in the Taylor expansion.
It is shown, how to generate infinite sequences of differential equations of the second order based on some standard equations, using Euler-Imshenetsky-Darboux (EID) transformation. For all this, factorizations of differential operators and operational identities are used.Some generalizations of integrable cases of Schrödinger's equations are finded. The example of integrable equation with liouvillian coefficients, that, apparently, can not be solved by Singer and Kovacic's algorithm (and its modifications), was built. The alogorithm and the program for solving conctructed classes of equations…
In this paper, we prove that $ζ$ is not a solution of any non-trivial algebraic differential equation whose coefficients are polynomials in $Γ, Γ^{(n)}, Γ^{(l)}$ over the ring of polynomials in $\mathbb{C}$, $l>n\geq 1$ are positive integers. We extended the result that $ζ$ does not satisfy any non-trivial algebraic differential equation whose coefficients are polynomials in $Γ, Γ', Γ''$ over the field of complex numbers, which is proved by Li and Ye[7].
Recently there has been a renewed interest in asymptotic Euler-MacLaurin formulas, partly due to applications to spectral theory of differential operators. Using elementary means, we recover such formulas for compactly supported smooth functions f on intervals, polygons, and 3-dimensional polytopes, where the coefficients in the asymptotic expansion are sums of differential operators involving only derivatives of f in directions normal to the faces of the polytope. Our formulas apply to wedges of any dimension. This paper builds on, and is motivated by, works of Guillemin, Sternberg, and other…
We review some recent results of the fractional variational calculus. Necessary optimality conditions of Euler-Lagrange type for functionals with a Lagrangian containing left and right Caputo derivatives are given. Several problems are considered: with fixed or free boundary conditions, and in presence of integral constraints that also depend on Caputo derivatives.
