關于Brown隨機指數一致可積性條件的推廣

李軍

(西北師范大學數學與統計學院,甘肅 蘭州 730070)

關于Brown隨機指數一致可積性條件的推廣

李軍

(西北師范大學數學與統計學院,甘肅 蘭州 730070)

利用Brown運動的下函數,推廣了隨機指數一致可積性的判定條件,進而改進了Novikov條件和Kazamaki條件.

隨機指數;指數鞅的一致可積性;Brown運動的下函數;Novikov條件; Kazamaki條件

1 引言及主要結果

則結論依然成立.之后,文獻[4]證明了在(2)式中,取ε=0,即

蘊含條件(1).文獻[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[6-7]中,假設存在Brown運動的一個下函數φ(關于下函數的定義詳見本文第二節),使得

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

上述兩個條件(6)和(7)在改進了文獻[6-8]的結果的同時,特別地進一步對Novikov條件(3)和Kazamaki條件(4)式作了改進.

本文的主要工作是借助于 Brown的下函數對文獻 [9]中的結果進行推廣,將條件 (6)和(7)統一起來,進而得出條件(1)的更一般的條件,從而更進一步改進了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運動的一個下函數,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 預備知識

首先給出 Brown運動的下函數的定義[9-10].設 (Bt)t≥0是標準 Brown運動且 φ是定義在 R+上的實值連續函數.集合 A={ω:?t=t(ω),?s≥t,Bs(ω)＜φ(s)}屬于 σ-代數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運動的下函數;若P(A)=1,則稱φ是Brown運動的上函數.

3 主要結果的證明

參考文獻

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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.

國家自然科學基金(11061032).

李軍(1986-),碩士,研究方向:隨機分析及其應用.

2010 M SC:60J65