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Register a new userVolterra equations driven by rough signals 2: higher order expansions
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We extend the recently developed rough path theory for Volterra equations from (Harang and Tindel, 2019) to the case of more rough noise and/or more singular Volterra kernels. It was already observed in (Harang and Tindel, 2019) 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 (Gubinelli, 2010) and (Hairer and Kelly, 2015) 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 in driven by more irregular signals.
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We develop the rough path counterpart of It^o stochastic integration and - differential equations driven by general semimartingales. This significantly enlarges the classes of (It^o / forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed.
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 article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter $frac13<H<frac12$. This is a first-order time-discrete numerical approximation scheme, and has been recently introduced by Hu, Liu and Nualart in order to generalize the classical Euler scheme for It^o SDEs to the case $H>frac12$. The current contribution generalizes the modified Euler scheme to the rough case $frac13<H<frac12$. Namely, we show a convergence rate of order $n^{frac12-2H}$ for the scheme, and we argue that this rate is exact. We also derive a central limit theorem for the renormalized error of the scheme, thanks to some new techniques for asymptotics of weighted random sums. Our main idea is based on the following observation: the triple of processes obtained by considering the fBm, the scheme process and the normalized error process, can be lifted to a new rough path. In addition, the Holder norm of this new rough path has an estimate which is independent of the step-size of the scheme.
This paper deals with linear stochastic partial differential equations with variable coefficients driven by L{e}vy white noise. We first derive an existence theorem for integral transforms of L{e}vy white noise and prove the existence of generalized and mild solutions of second order elliptic partial differential equations. Furthermore, we discuss the generalized electric Schrodinger operator for different potential functions $V$.
The aim of the present paper is to introduce a new numerical method for solving nonlinear Volterra integro-differential equations involving delay. We apply trapezium rule to the integral involved in the equation. Further, Daftardar-Gejji and Jafari method (DGJ) is employed to solve the implicit equation. Existence-uniqueness theorem is derived for solutions of such equations and the error and convergence analysis of the proposed method is presented. We illustrate efficacy of the newly proposed method by constructing examples.
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