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研究生:趙宗明
研究生(外文):Chao Tsung Ming
論文名稱:Azema鞅及可預期隨機積分方程
論文名稱(外文):Azema martingale and anticipative stochastic integral equation
指導教授:周青松
指導教授(外文):Chou Ching Sung
學位類別:博士
校院名稱:國立中央大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:1999
畢業學年度:87
語文別:中文
論文頁數:99
中文關鍵詞:正規鞅Azema鞅局部時構造方程擴大濾波可預期隨機積分Girsanov轉換柯西問題
外文關鍵詞:Normal martingalesAzema martingaleLocal timeStructure equationEnlargement of filtrationAnticipative stochastic integralsGirsanov transformationCauchy problem
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Azema鞅及可預期隨機積分方程
中文摘要:
本篇論文主要在建立Azema鞅的局部時不等式,並探討由不連續半鞅所支配的可預期隨機積分方程解的存在性與唯一性。本文共分為七章,其主要結果分述如下。
在第一章中,我們主要探討的是正規鞅的多重隨機積分不等式。P. Protter [MPM]等人利用具有混沌表示性質的正規鞅來建構可預期隨機積分理論。以此為基礎,我們建立了正規鞅多重隨機積分不等式,利用此結果,我們可用來討論其指數泛函的可積性問題。
在第二章,我們考慮Azema鞅及期其局部時。Barlow和Yor建立了對於連續鞅的局部時不等式,它是一用途很廣的不等式,受他們結果的啟發,我們建立了Azema鞅的局部時不等式,它是一右連續左極限存在的鞅。
延續第二章的工作,我們針對滿足構造方程(3.1)的鞅研究其軌道及局部時的性質。當β=-1,(3.1)式的解是Azema鞅。Barlow和Yor[BYb]對滿足假設A的半鞅將Ito公式做了推廣,不需要此假設的情形下,我們也可對滿足構造方程(3.1)的鞅做Ito公式的推廣。
第四章的主要技巧是利用擴大濾波的理論,它是解決隨機積分中非可適性被積分過程的主要方法之一。Yor[Yo6]曾經對連續鞅的B-D-G不等式推廣至停止於任意的隨機時間,受其啟示,我們也可推廣由不連續半鞅所之支配的非可適性隨機積分不等式。
第五章主要是建立關於隨機參數的可預期隨機積分,我們主要是利用黎曼和的逼近法,並且我們給了一個關於隨機積分存在的充分條件。黎曼和的收斂主要是利用Garsia-Rodemich-Rumsey引理,此結果可用來解非可適性起始值的線性隨機微分方程。
在第六章中,我們利用一滿足條件BX的隨機變數來擴大濾波,對此擴大的濾波,我們證明了半鞅仍保持其不變性。如果G1是一滿足[Ja]中條件A的隨機變數,G2是一滿足條件BX的隨機變數,由G1、G2來擴大濾波,我們可得到半鞅仍保持其不變性。更進一步,我們可得到關於Poisson鞅在經過擴大濾波與變換機率下的鞅不變性。
根句第五章所建立的隨機積分,在第七章中,我們主要是討論由不連續半鞅所支配的可預期隨機積分方程中,具非可適性起始值的柯西問題解的存在性。另外,在某些適當假設下,我們可得到半線性隨機方程解的唯一性。
Abstract.
This thesis is divided into seven chapters. Its organizationis stated as below.
In Chap. 1, we mainly discuss the multiple stochastic integrals (M.S.I.) for normal martingales. J. Ma et al. [MPM] have a deeply development of the construction of anticipative stochastic integral based on the chaotic representation property for normal martingales. Taking this as foundation, we establish inequalities of M.S.I. for normal martingales. Applying this result, we also obtain the integrability property of exponential funtional of M.S.I. for normal martingales.
In Chap. 2, we treat with Azema martingales and it''s local time. Barlow and Yor [BYa1] (1981) had a nice result about maximal local time inequality for continuous martingales. Inspired by their works, we also establish the maximal local time inequality for Azema martingales which is a family of discontinuous martingale with explicit expressions.
Continuing the works of Chap. 2. In Chap. 3, we investigate some local time property and the regularity for martingales satisfying the structure equation (3.1). For b =-1, the solution of (3.1) is the Azema martingale. Bouleau and Yor [BYb] have extended Ito''s formula for semimartingale satisfying Hypothesis A (see 3.2.). Without this condition (see Corollary 3.2.2.), we also extend the Ito''s formula to function F, as Bouleau and Yor''s, for martingales satisfying the structure equation (3.1).
The main technique we use in Chap. 4 is the theory of enlargement of filtration (T.E.F.). One of the ways to solve the stochastic integrals with non-adapted integrands is by (T.E.F.). Yor [Yo6] have extended the B-D-G type inequalityfor continuous martingale stopped at any random time. Imkeller [Im] also constructed the inequalities between norms of stochastic integrals of non-adapted integrands for Brownian motion and norms of their quadratic variations. Based on these results, it is worth to establish these inequalities of stochastic integrals with non-adapted integrands for discontinuous martingales.
Chap. 5 is devoted to constructing the anticipative stochastic integrals for processes depending on a random parameter. Here we use the classical Rieman sum approach and give a sufficient condition to ensure the existence of the integrals. The convergence of the Rieman sum is mainly by the aid of Garsia-Rodemich-Rumsey lemma. As a development, we apply these results to stochastic differential equation with non-adapted initial value.
In Chap. 6, let X be a semimartingale,we enlarage the filtration by a random variable satisfying condition BX(see 6.2.) and use Girsanov theorem to prove the invariance of semimartingale property under this larger filtration. In addition, if G1is a random variable that satisfies condition (A) given in [Ja], G2 is a random variable satisfying condition BX, then X is also a semimartingale for the larger filtration enlarged by G1, G2. Moreover, the invariance of martingale property for Poisson martingale under a simultaneous enlargement of filtration and change of equivalent probability measure can be obtained.
Based on the anticipative stochastic integrals constructed in Chap. 5. In Chap. 7, we discuss the existence of the solutions of the anticipative stochastic integrals equation driven by discontinuous semimartingales for the Cauchy problem with the nonadapted initial value. In addition, under some suitable assumptions, the uniqueness of solutions for the semilinear equations
also holds.
Cover
Abstract in Chinese
Acknowledgement
Content
Abstract in English
Chapter 1 On some inequalities of multiple stochastic integrals for normal martingales
1.1. Introduction
1.2. Main inequalities
1.3. The integrability property of exponential functional for (X,Y)
Chapter 2 On some inequalities of local times for Azema martingales
2.1. Introduction
2.2. Basic Notation and some results
2.3. Some Azema Martingale''s Inequalities
Chapter 3 Some remarks on martingales satisfying the structure equation {X,X}=+X-dX
3.1. Introduction
3.2. Some path and local time property
3.3. An extension of Ito''s formula
3.4. Some applications of the extension of Ito''s formula to B-D-G type inequalities
Chapter 4 On some integral''s type inequalities of discontinuous martingales via enlargement of filtration
4.1. Introduction
4.2. Inequalities via enlargement of partition of 
4.3. Inequalities via enlargement by the endpoints of an optional set
Chapter 5 Anticipative stochastic integrals depending on a random parameter
5.1. Introduction and Notations
5.2. Fundamental results
5.3. Forward Stochastic Integration for General Processes
Chapter 6 The theory of enlargement of filtrations via Girsanov Transformation and their application
6.1. Introduction
6.2. Anticipative Girsanov transformation
6.3. Invariance of martingale property for Poisson martingale
Chapter 7 On the Cauchy problem with non-adapted initial value driven by semimartingales
7.1. Introduction
7.2. The existence results for S.D.E.s.
7.3. The uniqueness of solutions for semilinear S.D.E.
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