李軍

(西北師范大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,甘肅 蘭州 730070)

關(guān)于Brown隨機(jī)指數(shù)一致可積性條件的推廣

李軍

(西北師范大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,甘肅 蘭州 730070)

利用Brown運(yùn)動的下函數(shù),推廣了隨機(jī)指數(shù)一致可積性的判定條件,進(jìn)而改進(jìn)了Novikov條件和Kazamaki條件.

隨機(jī)指數(shù);指數(shù)鞅的一致可積性;Brown運(yùn)動的下函數(shù);Novikov條件; Kazamaki條件

1 引言及主要結(jié)果

則結(jié)論依然成立.之后,文獻(xiàn)[4]證明了在(2)式中,取ε=0,即

蘊(yùn)含條件(1).文獻(xiàn)[5]給出了一個更漂亮的條件:

這個條件對于條件(1)是充分的,而且要比條件(2)弱.

對于條件(3)的弱化還可以采取其它方法.在文獻(xiàn)[6-7]中,假設(shè)存在Brown運(yùn)動的一個下函數(shù)φ(關(guān)于下函數(shù)的定義詳見本文第二節(jié)),使得

那么條件(1)成立.文獻(xiàn)[8]也用類似于(5)式的條件對Kazamaki條件(4)進(jìn)行了弱化.文獻(xiàn)[9]也給出了(1)式的充分條件,即

上述兩個條件(6)和(7)在改進(jìn)了文獻(xiàn)[6-8]的結(jié)果的同時,特別地進(jìn)一步對Novikov條件(3)和Kazamaki條件(4)式作了改進(jìn).

本文的主要工作是借助于 Brown的下函數(shù)對文獻(xiàn) [9]中的結(jié)果進(jìn)行推廣,將條件 (6)和(7)統(tǒng)一起來,進(jìn)而得出條件(1)的更一般的條件,從而更進(jìn)一步改進(jìn)了Novikov條件(3)和Kazamaki條件(4).下面給出本文的主要定理.

定理1.1 令φ是Brown運(yùn)動的一個下函數(shù),則下述條件對于(1)式是充分的,即

其中α∈R且α?=1,同時,上確界是取遍所有停時τ而得到的.

2 預(yù)備知識

首先給出 Brown運(yùn)動的下函數(shù)的定義[9-10].設(shè) (Bt)t≥0是標(biāo)準(zhǔn) Brown運(yùn)動且 φ是定義在 R+上的實(shí)值連續(xù)函數(shù).集合 A={ω:?t=t(ω),?s≥t,Bs(ω)＜φ(s)}屬于 σ-代數(shù)X=∩t＞0σ(Bs;s≥t).由Blumenthal 0-1律可得P(A)=0或1.

注 集合A也可以表示成:A={ω:Bt＜φ(t),t→∞}.

定義2.1 若P(A)=0,則稱φ是Brown運(yùn)動的下函數(shù);若P(A)=1,則稱φ是Brown運(yùn)動的上函數(shù).

3 主要結(jié)果的證明

參考文獻(xiàn)

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The generalization of criteria for the un iform in tegrability of B row n ian stochastic exponentials

Li Jun
(College of Mathematics and Statistics,Northwest Normal University,Lanzhou 730070,China)

In this paper,we app ly lower function of Brownian motion to generalize criteria for the uniform integrability of Brownian stochastic exponentials,and im p rove the Novikov condition and Kazam aki condition.

stochastic exponentials,the uniform integrability of exponentialm artingales, lower functions of Brownian motion,Novikov condition,Kazamaki condition

O211.6

A

1008-5513(2012)06-0839-06

2012-06-09.

國家自然科學(xué)基金(11061032).

李軍(1986-),碩士,研究方向:隨機(jī)分析及其應(yīng)用.

2010 M SC:60J65