Path properties of Lévy driven mixed moving average processes
School of Applied Mathematics and Informatics, J. J. Strossmayer University of Osijek（奥西耶克约瑟夫·斯特罗斯马耶尔大学应用数学与信息学院）；Bolyai Institute, University of Szeged（塞格德大学博莱伊研究所）；Department of Mathematical Sciences, Aalborg University（奥尔堡大学数学科学系）
We derive general sufficient conditions for the existence of càdlàg and continuous modifications of Lévy-driven mixed moving average processes. The conditions are explicit and easy to verify and applied to supOU, well-balanced supOU, trawl, and power-weighted supOU processes. In these examples, the conditions are shown to be close to optimal.
FunctionaL Regular Variation of Lévy-driven Multivariate Mixed Moving Average Processes
本文研究了$X_t = \int\int f(A, t - s) Λ(d A, d s)$型多元混合移动平均过程的càdlàg函数在$\mathbb{D}$空间中的泛函正则变差.本文给出了MMA过程$(X_t)$有càdlàg样本路的充分条件.作为我们的主要结果，我们证明了在$\mathbb{D}$中$(X_t)$是正则变化的，如果驱动Lévy基是正则变化的，并且核函数$f$满足一定的自然（连续）条件.最后，讨论了供应过程的特殊情况，例如，在金融领域的应用中，我们会详细考虑。
A recently proposed alternative to multifractional Brownian motion (mBm) with random Hurst exponent is studied, which we refer to as Itô-mBm. It is shown that Itô-mBm is locally self-similar. In contrast to mBm, its pathwise regularity is almost unaffected by the roughness of the functional Hurst parameter. The pathwise properties are established via a new polynomial moment condition similar to the Kolmogorov-Chentsov theorem, allowing for random local Hölder exponents. Our results are applicable to a broad class of moving average processes where pathwise regularity and long memory properties …
For $(X_t)$ a two-sided $α$-stable moving average, this paper studies the conditional distribution of future paths given a piece of observed trajectory when the process is far from its central values. Under this framework, vectors of the form $\boldsymbol{X}_t=(X_{t-m},\ldots,X_t,X_{t+1},\ldots,X_{t+h})$, $m\ge0$, $h\ge1$, are multivariate $α$-stable and the dependence between the past and future components is encoded in their spectral measures. A new representation of stable random vectors on unit cylinders -sets $\{\boldsymbol{s}\in\mathbb{R}^{m+h+1}: \hspace{0.3cm} \|\boldsymbol{s}\|=1\}$ f…
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We obtain central limit theorems for stationary random fields employing a novel measure of dependence called $θ$-lex weak dependence. We show that this dependence notion is more general than strong mixing, i.e., it applies to a broader class of models. Moreover, we discuss hereditary properties for $θ$-lex and $η$-weak dependence and illustrate the possible applications of the weak dependence notions to the study of the asymptotic properties of stationary random fields. Our general results apply to mixed moving average fields (MMAF in short) and ambit fields. We show general conditions such th…
On the mixing structure of stationary increment and self-similar symmetric α-stable processes
在平稳对称α稳定（SαS）过程的遍历分解中，出现了混合滑动平均过程.它们对应于产生SαS过程的“确定性”流的耗散部分（Rosinski，1995）。沿着这条路线，我们研究平稳增量和自相似SαS过程。由于平稳增量过程类和自相似过程类可以分别通过Masani变换和Lamperti变换嵌入到平稳过程类中，我们用非奇异流和相关的上循环刻画了这两类SαS过程.我们用Surgailis，Rosinski，Mandrekar and Cambanis，1992中引入的自相似混合移动平均SαS过程的各种例子来说明这种方法。
This paper is devoted to studying the averaging principle for fast-slow system of rough differential equations driven by mixed fractional Brownian rough path. The fast component is driven by Brownian motion, while the slow component is driven by fractional Brownian motion with Hurst index $H ~(1/3 < H\leq 1/2)$. Combining the fractional calculus approach to rough path theory and Khasminskii's classical time discretization method, we prove that the slow component strongly converges to the solution of the corresponding averaged equation in the $L^1$-sense. The averaging principle for a fast-slow…
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We compute analytically the probability $S(t)$ that a set of $N$ Brownian paths do not cross each other and stay below a moving boundary $g(τ)= W \sqrtτ$ up to time $t$. We show that for large $t$ it decays as a power law $S(t) \sim t^{- β(N,W)}$. The decay exponent $β(N,W)$ is obtained as the ground state energy of a quantum system of $N$ non-interacting fermions in a harmonic well in the presence of an infinite hard wall at position $W$. Explicit expressions for $β(N,W)$ are obtained in various limits of $N$ and $W$, in particular for large $N$ and large $W$. We obtain the joint distribution…
