Related papers: Rough Volterra equations 2: convolutional generali…
The purpose of this paper is to establish the convergence in distribution of the normalized error in the Euler approximation scheme for stochastic Volterra equations driven by a standard Brownian motion, with a kernel of the form…
We consider fractional Brownian motion with the Hurst parameters from (1/2,1). We found that the increment of a fractional Brownian motion can be represented as the sum of a two independent Gaussian processes one of which is smooth in the…
We introduce a canonical way of performing the joint lift of a Brownian motion $W$ and a low-regularity adapted stochastic rough path $\mathbf{X}$, extending [Diehl, Oberhauser and Riedel (2015). A L\'evy area between Brownian motion and…
We study different possibilities to apply the principles of rough paths theory in a non-commutative probability setting. First, we extend previous results obtained by Capitaine, Donati-Martin and Victoir in Lyons' original formulation of…
In this paper, we study reflected differential equations driven by continuous paths with finite $p$-variation ($1\le p<2$) and $p$-rough paths ($2\le p<3$) on domains in Euclidean spaces whose boundaries may not be smooth. We define…
We construct in this article an explicit geometric rough path over arbitrary $d$-dimensional paths with finite $1/\alpha$-variation for any $\alpha\in(0,1)$. The method may be coined as 'Fourier normal ordering', since it consists in a…
The fractional Brownian motion (fBm) is parameterized by the Hurst exponent $H\in(0,1)$, which determines the dependence structure and regularity of sample paths. Empirical findings suggest that the Hurst exponent may be non-constant in…
We apply the averaging method to a coupled system consisting of two evolution equations which has a slow component driven by fractional Brownian motion (FBM) with the Hurst parameter $H_1> \frac12$ and a fast component driven by additive…
Calculus via regularizations and rough paths are two methods to approach stochastic integration and calculus close to pathwise calculus. The origin of rough paths theory is purely deterministic, calculus via regularization is based on…
This paper establishes the averaging method to a coupled system consisting of two stochastic differential equations which has a slow component driven by fractional Brownian motion (FBM) with less regularity $1/3< H \leq 1/2$ and a fast…
We consider additive functionals of stationary Markov processes and show that under Kipnis-Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Levy area that can be…
We reexamine the problem of having nonconservative equations of motion arise from the use of a variational principle. In particular, a formalism is developed that allows the inclusion of fractional derivatives. This is done within the…
The paper considers the integral Volterra equations of the first kind which are related to the inverse boundary-value heat conduction problem. The algorithms have been developed to numerically solve the respective integral equations, which…
In this paper we consider the drift estimation problem for a general differential equation driven by an additive multidimensional fractional Brownian motion, under ergodic assumptions on the drift coefficient. Our estimation procedure is…
This article investigates several properties related to densities of solutions X to differential equations driven by a fractional Brownian motion with Hurst parameter H>1/4. We first determine conditions for strict positivity of the density…
We show well-posedness for McKean--Vlasov equations with rough common noise and progressively measurable coefficients. Our results are valid under natural regularity assumptions on the coefficients, in agreement with the respective…
In this article we extend the framework of rough paths to processes of variable H\"older exponent or variable order paths. We show how a class of multiple discrete delay differential equations driven by signals of variable order are…
We consider slow / fast systems where the slow system is driven by fractional Brownian motion with Hurst parameter $H>{1\over 2}$. We show that unlike in the case $H={1\over 2}$, convergence to the averaged solution takes place in…
{Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha<1/4$, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of $B$ is a…
In Gatheral et al. 2018, first posted in 2014, volatility is characterized by fractional behavior with a Hurst exponent $H < 0.5$, challenging traditional views of volatility dynamics. Gatheral et al. demonstrated this using realized…