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In this paper, we study two variations of the time discrete Taylor schemes for rough differential equations and for stochastic differential equations driven by fractional Brownian motions. One is the incomplete Taylor scheme which excludes some terms of an Taylor scheme in its recursive computation so as to reduce the computation time. The other one is to add some deterministic terms to an incomplete Taylor scheme to improve the mean rate of convergence. Almost sure rate of convergence and $L_p$-rate of convergence are obtained for the incomplete Taylor schemes. Almost sure rate is expressed in terms of the Holder exponents of the driving signals and the $L_p$-rate is expressed by the Hurst parameters. Both rates involves with the incomplete Taylor scheme in a very explicit way and then provide us with the best incomplete schemes, depending on that one needs the almost sure convergence or one needs $L_p$-convergence. As in the smooth case, general Taylor schemes are always complicated to deal with. The incomplete Taylor scheme is even more sophisticated to analyze. A new feature of our approach is the explicit expression of the error functions which will be easier to study. Estimates for multiple integrals and formulas for the iterated vector fields are obtained to analyze the error functions and then to obtain the rates of convergence.
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For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter $H>frac{1}{2}$, it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $Hrightarrowfrac{1}{2}$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for It^{o} SDEs for $H=frac{1}{2}$, the convergence rate of the naive Euler scheme deteriorates for $Hrightarrowfrac{1}{2}$. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for $H=frac{1}{2}$, and it has the rate of convergence $gamma_n^{-1}$, where $gamma_n=n^{2H-{1}/2}$ when $H<frac{3}{4}$, $gamma_n=n/sqrt{log n}$ when $H=frac{3}{4}$ and $gamma_n=n$ if $H>frac{3}{4}$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if ${X_t,0le tle T}$ is the solution of a SDE driven by a fBm and if ${X_t^n,0le tle T}$ is its approximation obtained by the new modified Euler scheme, then we prove that $gamma_n(X^n-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $Hin(frac{1}{2},frac{3}{4}]$. In the case $H>frac{3}{4}$, we show the $L^p$ convergence of $n(X^n_t-X_t)$, and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.
In this paper we solve real-valued rough differential equations (RDEs) reflected on an irregular boundary. The solution $Y$ is constructed as the limit of a sequence $(Y^n)_{ninmathbb{N}}$ of solutions to RDEs with unbounded drifts $(psi_n)_{ninmathbb{N}}$. The penalisation $psi_n$ increases with $n$. Along the way, we thus also provide an existence theorem and a Doss-Sussmann representation for RDEs with a drift growing at most linearly. In addition, a speed of convergence of the sequence of penalised paths to the reflected solution is obtained. We finally use the penalisation method to prove that the law at time $t>0$ of some reflected Gaussian RDE is absolutely contiuous with respect to the Lebesgue measure.
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 It^o-Lyons continuity theorem. The rate of convergence is obtained by a coupling argument developed first by Sznitman for particle systems with Brownian inputs.
In this note we consider differential equations driven by a signal $x$ which is $gamma$-Holder with $gamma>1/3$, 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 $|xi|^{kappa}$ with $kappain(0,1)$. Two different methods are used in order to construct solutions: (i) In a 1-d setting, we resort to a rough version of Lampertis transform. (ii) For multidimensional situations, we quantify some improved regularity estimates when the solution approaches the origin.
We prove existence of global solutions for differential equations driven by a geometric rough path under the condition that the vector fields have linear growth. We show by an explicit counter-example that the linear growth condition is not sufficient if the driving rough path is not geometric. This settle a long-standing open question in the theory of rough paths. So in the geometric setting we recover the usual sufficient condition for differential equation. The proof rely on a simple mapping of the differential equation from the Euclidean space to a manifold to obtain a rough differential equation with bounded coefficients.
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