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Results on local times of a Class of Multiparameter Gaussian Processes

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Acta Mathematicae Applicatae Sinica 2006年第1期22卷 81-90页

作者： Zong-mao Cheng Xiu-yun Wang Zheng-yan Lin Department of Mathematics Zhejiang University Hangzhou 310028 China Hangzhou Dianzi University Hangzhou 310018 China

In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t） = （x1 （t）,..., xd（t））,where X1 （t）,…, Xd（t） are independent copies of Y（t）, At first we show the existence and join continuity of the local times of X = {X（t）, t ∈ R＋^N}, then we consider the HSlder conditions for the local times.

关键词： Bi-fractional Brownian motiom local time Gaussian random field

local times of N-parameter Gaussian processes

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Progress in Natural Science:Materials International 2005年第2期 13-19页

作者： LIN Zhengyan1* and CHENG Zongmao 1,2(1. Department of Mathematics, Zhejiang University, Hangzhou 310028, China 2. Hangzhou Institute of Electronics and Engineering, Hangzhou 310012, China)

An N-parameter Gaussian stationary process X=｛X(t): t∈RN+｝ is introduced and the existence and joint continuity of its local times is presented. And the moments of local times are estimated. Furthermore moduli of c... 详细信息

关键词： N-parameter Gaussian process local time moduli of continuity large increments.

The local times and Hausdorff measure for level sets of a Wiener sheet

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Science China Mathematics 2001年第6期44卷 696-708页

作者：林火南 Fujian Teachers Univ Dept Math Fuzhou 350007 Peoples R China

In this paper, we discuss the relation between the local times and the level sets of a Wiener sheet, and give moment estimates for the local times of a Wiener sheet. We show that the correct Hausdorff measure function for level sets of N-parameter Rσ-valued Wiener sheet is φ ( r) = rN-d/2(loglog1/r)d/2 if 2N> d. This solves the problem posed by S. J. Taylor.

关键词： Wiener sheet level set local time Hausdorff measure.

The Fractal Dimensions of the Level Sets of the Generalized Iterated Brownian Motion

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Acta Mathematicae Applicatae Sinica 2013年第3期29卷 597-602页

作者： Chang-qing TONG Zheng-yan LIN Jing ZHENG Institute of applied mathematics Hangzhou Dianzi University Department of Mathematics Zhejiang University

Let{W1（t）, t∈R＋} and {W2（t）, t∈R＋} be two independent Brownian motions with W1（0） = W2（0） = 0. {H （t） = W1（｜W2（t）｜）, t ∈R＋} is called a generalized iterated Brownian motion. In this paper, the Ha... 详细信息

关键词： Hausdorff dimension packing dimension local time generalized iterated Brownian motion

Optimal Selling time in Stock Market over a Finite time Horizon

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Acta Mathematicae Applicatae Sinica 2012年第3期28卷 557-570页

作者： S.C.P.YAM S.P.YUNG W.ZHOU The Chinese University of Hong Kong China The University of Hong Kong China

In this paper, we examine the best time to sell a stock at a price being as close as possible to its highest price over a finite time horizon [0, T], where the stock price is modelled by a geometric Brownian motion and the ＇closeness＇ is measured by the relative error of the stock price to its highest price over [0, T]. More precisely, we want to optimize the expression：V^＊=sup 0≤τ≤T E[Vτ/MT],where （Vt）t≥0 is a geometric Drownian motion with constant drift α and constant volatility σ 〉 0, Mt = max Vs 0≤a〈t is the running maximum of the stock price, and the supremum is taken over all possible stopping times 0 ≤ τ≤T adapted to the natural filtration （Ft）t≥0 of the stock price. The above problem has been considered by Shiryacv, Xu and Zhou （2008） and Du Toit and Peskir （2009）. In this paper we provide an independent proof that when α=1/ 2 σ^2 , a selling strategy is optimal if and only if it sells the stock either at the terminal time T or at the 1 2 moment when the stock price hits its maximum price so far. Besides, when α 〉1/2σ^2 , selling the stock at the terminal time T is the unique optimal selling strategy. Our approach to the problem is purely probabilistic and has been inspired by relating the notion of dominant stopping pr of a stopping time τ to the optimal stopping strategy arisen in the classical ＂Secretary Problem＂.

关键词： optimal Stopping stock selling buy and hold local time

Strong approximation for the general Kesten-Spitzer random walk in independent random scenery

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Science China Mathematics 2001年第5期44卷 619-630页

作者：张立新 1. Department of Mathematics Zhejiang University 310028 Hangzhou China

This paper is to prove that, if a one-dimensional random walk can be approximated by a Brownian motion, then the related random walk in a general independent scenery can be approximated by a Brownian motion in Brownia... 详细信息

关键词： local time random walk in random scenery Brownian motion in Brownian scenery strong approximation the law of iterated logarithm.

Smooth-pasting property on reflected Lévy processes and its applications in credit risk modeling

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Science China Mathematics 2014年第6期57卷 1237-1256页

作者： BO LiJun YANG XueWei Department of Mathematics Xidian University School of Management and Engineering Nanjing University

We study the smooth-pasting property for a class of conditional expectations with reflected Levy process as underlying state process. A relationship between local times and regulators for the doubly reflected Levy process is established. As applications, we derive the analytic pricing formula for a zero-coupon defaultable bond when the default intensity （resp. the stochastic loss rate） is modeled as one-sided （resp. double-sided） reflected Levy processes. Finally, some numerical illustrations are provided.

关键词： smooth-pasting property reflected Levy process local time credit risk default intensity stochas-tic loss rate defaultable bond

The Exact Hausdorff Measure Function of the Level Sets of Multi-parameter Symmetric Stable Process

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Acta Mathematica Sinica,English Series 2005年第5期21卷 1137-1148页

作者： Shui Cao ZHENG Huo Nan LIN Di He HU School of Mathematics Sciences Peking University Beijing 100871 P. R. China School of Sciences Jimei University Xiamen 361021 P. R. China Department of Mathematics Fujian Normal University Fuzhou 350007 P. R. China College of Mathematics and Statistics Wuhan University Wuhan 430072 P. R. China

In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.

关键词： Stable process Level sets local time Hausdorit measure

DIMENSION RESULTS FORORNSTEIN-UHLENBECK TYPE MARKOVPROCESS

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Acta Mathematica Scientia 1999年第4期19卷 417-424页

作者：邓爱姣刘禄勤 Wuhan Univ Dept Math Wuhan 430072 Peoples R China

Let {X-t, t greater than or equal to 0} be an Ornstein-Uhlenbeck type Markov process with Levy process A(t), the authors consider the fractal properties of its ranges, give the upper and lower bounds of the Hausdorff dimensions of the ranges and the estimate of the dimensions of the level sets for the process. The existence of local times and occupation times of X-t are considered in some special situations.

关键词： Levy process Ornstein-Uhlenbeck type Markov process range level set Hausdorff dimension local time occuption time