吳嬋,陳曄

(1.長(zhǎng)沙理工大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,湖南 長(zhǎng)沙,410114;2.湖南文理學(xué)院 數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,湖南 常德,415000)

帶漂移布朗運(yùn)動(dòng)的一個(gè)局部時(shí)的Laplace變換

吳嬋1,陳曄2

(1.長(zhǎng)沙理工大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,湖南 長(zhǎng)沙,410114;2.湖南文理學(xué)院 數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,湖南 常德,415000)

在Borodin和Salminen(2002)文獻(xiàn)中有關(guān)帶漂移布朗運(yùn)動(dòng)占位時(shí)的Laplace變換結(jié)果的基礎(chǔ)上,運(yùn)用Li等(2014)計(jì)算局部時(shí)的方法,推出了帶漂移布朗運(yùn)動(dòng)在獨(dú)立指數(shù)時(shí)間eq前,及停留在0處的局部時(shí)的Laplace變換表達(dá)式。當(dāng)μ=0時(shí),本文結(jié)果與標(biāo)準(zhǔn)布朗運(yùn)動(dòng)的結(jié)果吻合。

局部時(shí);Laplace變換;帶漂移的布朗運(yùn)動(dòng)

局部時(shí)和占位時(shí)是隨機(jī)過(guò)程理論研究的2個(gè)熱點(diǎn)問(wèn)題,它們?cè)陲L(fēng)險(xiǎn)理論和金融模型中有廣泛應(yīng)用。占位時(shí)是隨機(jī)過(guò)程在一個(gè)特定區(qū)間內(nèi)逗留的時(shí)間總和,而局部時(shí)是其相關(guān)的占位密度。計(jì)算占位時(shí)的Laplace變換的表達(dá)式主要有3種方法,經(jīng)典的方法是通過(guò)Feynman-Kac公式[1-2]得到過(guò)程對(duì)應(yīng)的隨機(jī)微分方程,從而得到相應(yīng)的占位時(shí)的Laplace變換的表達(dá)式[3]。Landriault等[4-5]采用了逼近占位時(shí)的方法并結(jié)合游弋理論,得到了譜負(fù)Lévy過(guò)程的占位時(shí)Laplace變換[6]。為了克服隨機(jī)過(guò)程路徑的無(wú)變差性,Li和Zhou[7]首次運(yùn)用泊松過(guò)程的性質(zhì),將計(jì)算譜負(fù)Lévy過(guò)程的聯(lián)合占位時(shí)的Laplace變換問(wèn)題轉(zhuǎn)化為求某個(gè)隨機(jī)事件的概率問(wèn)題[8]。目前,對(duì)隨機(jī)過(guò)程的局部時(shí)的Laplace變換的研究還比較少。

本文在文獻(xiàn)[9]的有關(guān)帶漂移布朗運(yùn)動(dòng)在隨機(jī)指數(shù)時(shí)間eq之前,停留在區(qū)間(0,a)上的占位時(shí)表達(dá)式的基礎(chǔ)上,運(yùn)用文獻(xiàn)[10]中求局部時(shí)的方法,通過(guò)對(duì)過(guò)程占位時(shí)的 Laplace變換求極限,得到過(guò)程停留在0處的局部時(shí)的Laplace變換表達(dá)式。本文得到了局部時(shí)的Laplace變換表達(dá)式。

1 預(yù)備知識(shí)

設(shè)Xt=μt+Wt是帶漂移布朗運(yùn)動(dòng)[9],其中漂移系數(shù)μ∈ R,Wt是一維標(biāo)準(zhǔn)布朗運(yùn)動(dòng)。帶漂移布朗運(yùn)動(dòng)在獨(dú)立指數(shù)時(shí)間eq之前,停留在區(qū)間(0,a)上占位時(shí)的Laplace變換表達(dá)式[9]為

其中,eq是強(qiáng)度為q的指數(shù)隨機(jī)變量,與過(guò)程X獨(dú)立,且

2 主要結(jié)果

運(yùn)用文獻(xiàn)[10]中求局部時(shí)的方法,通過(guò)對(duì)帶漂移布朗運(yùn)動(dòng)占位時(shí)的 Laplace變換表達(dá)式取極限,得到局部時(shí)的Laplace變換表達(dá)式。

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(責(zé)任編校:劉剛毅)

Laplace transform of one local time on brownian motion with drift

Wu Chan1,Chen Ye2
(1.College of Mathematics and Statistics,Changsha University of Science and Technology,Changsha 410114,China;2.College of Mathematics and Computational Science,Hunan University of Arts and Science,Changde 415000,China)

On the basis of the results in Borodin and Salminen(2002),the approach in Li et al.(2014)is adopted to consider the local time at 0 before independent exponential timeeq,and the Laplace transform of local time on Brownian motion with drift is obtained.The result is.Whenμ=0,the result consists with classical result of Brownian motion.

local time;Laplace transform;brownian motion with drift

O 211.6

A

1672-6146(2017)02-0009-03

吳嬋,1003011369@qq.com。

2017-01-20

國(guó)家自然科學(xué)基金(11571052,11171044);湖南省自然科學(xué)基金(2016JJ4061);湖南省研究生科研創(chuàng)新項(xiàng)目(CX2016B417);湖南文理學(xué)院科學(xué)研究項(xiàng)目(15ZD05)。

10.3969/j.issn.1672-6146.2017.02.003