CLT for an iterated integral with respect to fBm with H > 1/2

Empirical Quantile CLTs For Some Self-Similar Processes

In a paper of Jason Swanson, a CLT for the sample median of independent Brownian motions with value 0 at 0 was proved. Here we extend this result in two ways. We prove such a result for a collection of self-similar processes which include the fractional Brownian motions, all stationary, independent increment symmetric stable processes tied down at 0 as well as iterated and integrated Brownian motions. Second, our results hold uniformly over all quantiles in a compact sub-interval of (0,1). We also examine sample function properties connected with these CLTs.

Stochastic calculus for fractional Brownian motion with Hurst exponent $H>1/4$: A rough path method by analytic extension

The $d$-dimensional fractional Brownian motion (FBM for short) $B_t=((B_t^{(1)},...,B_t^{(d)}),t\in\mathbb{R})$ with Hurst exponent $α$, $α\in(0,1)$, is a $d$-dimensional centered, self-similar Gaussian process with covariance ${\mathbb{E}}[B_s^{(i)}B _t^{(j)}]={1/2}δ_{i,j}(|s|^{2α}+|t|^{2α}-|t-s|^{2 α}).$ The long-standing problem of defining a stochastic integration with respect to FBM (and the related problem of solving stochastic differential equations driven by FBM) has been addressed successfully by several different methods, although in each case with a restriction on the range of eithe…

Fractional Brownian motions ruled by nonlinear equations

In this note we consider generalized diffusion equations in which the diffusivity coefficient is not necessarily constant in time, but instead it solves a nonlinear fractional differential equation involving fractional Riemann-Liouville time-derivative. Our main contribution is to highlight the link between these generalised equations and fractional Brownian motion (fBm). In particular, we investigate the governing equation of fBm and show that its diffusion coefficient must satisfy an additive evolutive fractional equation. We derive in a similar way the governing equation of the iterated fra…

From constructive field theory to fractional stochastic calculus. (I) The Lévy area of fractional Brownian motion with Hurst index $α\in (1/8,1/4)$

Let $B=(B_1(t),\ldots,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $α<1/4$. Defining properly iterated integrals of $B$ is a difficult task because of the low Hölder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We show in this paper how to obtain second-order iterated integrals as the limit when the ultra-violet cut-off goes to infinity of iterated integrals of weakly interacting fields defined using the tools of construc…

缺项系统中重对数律的丢番图条件

Diophantine conditions in the law of the iterated logarithm for lacunary systems

本文研究缺项和 $\sum f(n_k x)$ 满足重对数律所需的丢番图条件，证明节省对数阶因子是最优条件，并揭示重对数律比中心极限定理要求更强的算术条件。

无穷渗流团上布朗运动与随机游走的耦合

Coupling between Brownian motion and random walks on the infinite percolation cluster

本研究针对超临界伯努利边渗流构建无穷团随机游走与布朗运动的耦合，证明路径最大距离均值阶为$T^{1/3+o(1)}$，部分解决公开问题并恢复重对数律、确定极限常数。