Integral representation of generalized grey Brownian motion

ABSTRACT

In this paper, we investigate the representation of a class of non-Gaussian processes, namely generalized grey Brownian motion, in terms of a weighted integral of a stochastic process which is a solution of a certain stochastic differential equation. In particular, the underlying process can be seen as a non-Gaussian extension of the Ornstein–Uhlenbeck process, hence generalizing the representation results of Muravlev, Russian Math. Surveys 66 (2), 2011 as well as Harms and Stefanovits, Stochastic Process. Appl. 129, 2019 to the non-Gaussian case.     

Bridge representation and modal-path approximation

The article shows a bridge representation for the joint density of a system of stochastic processes consisting of a Brownian motion with drift coupled with a correlated fractional Brownian motion with drift. As a result, a small time approximation of the joint density is readily obtained by substituting the conditional expectation under the bridge measure by a single path: the modal-path from the initial point to the terminal point.     

Excursions of a Gaussian process with variable variance above a barrier increasing to infinity

For a family of real-valued Gaussian processes ξ _u (t), t ∈ [0, T], we obtain an exact asymptotics of the probability of crossing a level u as u → ∞ under certain conditions on the variance and correlation. This result is applied to the investigation of excursions of a stationary zero-mean process above a barrier increasing to infinity.     

An Lp-view of the Bahadur-Kiefer Theorem

Summary Let ]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>\alpha_n$ and $\beta_n$ be respectively the uniform empirical and quantile processes, and define $R_n = \alpha_n + \beta_n$, which usually is referred to as the Bahadur--Kiefer process. The well-known Bahadur-Kiefer theorem confirms the following remarkable equivalence: $\|R_n\| /\sqrt{\| \alpha_n \| }\, \sim \, n^{-1/4} (\log n)^{1/2}$ almost surely, as $n$ goes to infinity, where $\| f\| =\sup_{0\le t\le 1} |f(t)|$ is the $L^\infty$-norm. We prove that $\|R_n\|_2 /\sqrt{\| \alpha_n \|_1}\, \sim \, n^{-1/4}$ almost surely, where $\| \, \cdot \, \|_p$ is the $L^p$-norm. It is interesting to note that there is no longer any logarithmic term in the normalizing function. More generally, we show that $n^{1/4} \|R_n\|_p /\sqrt{\| \alpha_n \|_{(p/2)}}$ converges almost surely to a finite positive constant whose value is explicitly known.     

A problem of Földes and Puri on the Wiener process

Let be a real-valued Wiener process starting from 0, and be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of as tends to infinity, i.e. they ask: how long does stay in a tube before ``crossing very much" a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for .

     

Some properties of the sub-fractional Brownian motion

We study several properties of the sub-fractional Brownian motion (fBm) introduced by Bojdecki et al. related to those of the fBm. This process is a self-similar Gaussian process depending on a parameter H ∈ (0, 2) with non stationary increments and is a generalization of the Brownian motion (Bm).

The strong variation of the indefinite stochastic integral with respect to sub-fBm is also discussed.     

Gaussian Approximation to the Partial Sum Processes of Moving Averages

The authors study approximation to the partial sum processes which is based on the stationary sequences of random variables having the structure of the so-called moving averages of independent identically distributed observations. In particular, the rates of convergence both in Donsker's and Strassen's invariance principles are obtained in the case when the limit Gaussian process is a fractional Brownian motion with an arbitrary Hurst parameter.     

布朗运动和泊松过程共同驱动下的欧式期权定价

针对布朗运动和泊松过程共同驱动下股票价格的随机微分方程,利用It0公式和随机积分的方法,得到了该形式下欧式期权定价的模型,并给出了模型的求解.     

On the exponential process associated with a CARMA-type process

We study the correlation decay and the expected maximal increments of the exponential processes determined by continuous-time autoregressive moving average (CARMA)-type processes of order (p, q). We consider two background driving processes, namely fractional Brownian motions and Lévy processes with exponential moments. The results presented in this paper are significant extensions of those very recent works on the Ornstein–Uhlenbeck-type case (p = 1, q = 0), and we develop more refined techniques to meet the general (p, q). In the concluding section, we discuss the perspective role of exponential CARMA-type processes in stochastic modelling of the burst phenomena in telecommunications and the leverage effect in financial econometrics.     

Deterministic equivalent for a continuous-time linear-convex stochastic control problem

We consider a finite-horizon control model with additive input. There are two convex functions which describe the running cost and the terminal cost within the system. The cost of input is proportional to the input and can take both positive and negative values. It is shown that there exists a deterministic control problem whose optimal cost is the same as the one in the stochastic control problem. The optimal policy for the stochastic problem consists of keeping the process as close to the optimal deterministic trajectory as possible.This research is supported by NSERC Grant A4619, MRCO, NSF Grant DMS-86-01510, and AFOSR Grant 87-0278.     

Immigration process in catalytic medium

The longtime behavior of the immigration process associated with a catalytic super-Brownian motion is studied. A large number law is proved in dimension d≤3 and a central limit theorem is proved for dimension d=3.     

带干扰的索赔次数为复合Poisson-Geometric过程的负风险和模型

引进带干扰的索赔次数为复合Poisson-Geometric过程的负风险和模型,给出该模型的破产概率所满足的积分-微分方程及解析式.     

An enlargement of filtration for Brownian motion

Let Bt be an Ft Brownian motion and Gt be an enlargement of filtration of Ft from some Gaussian random variables. We obtain equations for ht such that Bt ht is a Gt-Brownian motion.     

On the Asymptotic Behavior of the Storage Process Fed by a Markov Modulated Brownian Motion

Consider a storage model fed by a Markov modulated Brownian motion. We prove that the stationary distribution of the model exits and that the running maximum of the storage process over the interval [0, t] grows asymptotically like log t as t→∞.     

Testing the goodness of fit of a mixed Poisson process

In the literature on the statistical analysis of point processes certain tests for homogeneous Poisson processes are proposed, which in fact are tests for mixed Poisson processes. Some conclusions from this fact are drawn.     

经济环境下带扩散扰动古典风险过程的破产概率

对于经济环境下带扩散扰动古典风险过程的重要性质进行了讨论,给出了有限时间的破产概率的上界估计.     

An explicit calculation of the mean of the perimeter of the convex hull of a plane random walk

If (X _n ) _n ⁼¹ is a sequence of i.i.d. random variables in the Euclidean plane such that we compute the mean of the perimeter of theconvex hull ofX ₁++X _k; 0kn}.     

A representation for the posterior distribution of the location of a moving target

A target moves according to a continuous stochastic process in Euclidean R_^H. A search is conducted by choosing a search strategy and by observing the state of a detection process. Methods for representing the posterior distributions for target location given no detection are discussed. In particular, methods for parameterizing Gaussian models for target motion and the transition intensity of the detection process which yield tractable representations are introduced. The methodology is discussed in terms of two examples.     

Maximal inequalities for the Ornstein-Uhlenbeck process

Let be the Ornstein-Uhlenbeck velocity process solving

with , where 0$"> and is a standard Brownian motion. Then there exist universal constants 0$">and 0$"> such that

for all stopping times of . In particular, this yields the existence of universal constants 0$"> and 0$"> such that

for all stopping times of . This inequality may be viewed as a stopped law of iterated logarithm. The method of proof relies upon a variant of Lenglart's domination principle and makes use of Itô calculus.

     

Time Difference on the Brownian Curve

Let X _t and Y _t be respectively the locations of the maximum and minimum, over [0, t], of a real-valued Wiener process. We establish limsup and liminf iterated logarithm laws for , the time difference between the maximum and the minimum, as well as for max(X _t, Y _t) and min(X _t, Y _t).