Brownian Motion and Stochastic Calculus
Brownian Motion and Stochastic Calculus
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DOI：
10.2307/2289894
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发表时间：
1987
期刊：
影响因子：
--
通讯作者：
I. Karatzas
中科院分区：
文献类型：
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作者：
I. Karatzas
1鞅，停止时间和滤镜。- 1.1。随机过程与场。- 1.2。停止时间。- 1.3。连续时间鞅。——基本不平等。- B.收敛结果。C.可选抽样定理。- 1.4。Doob-Meyer分解。- 1.5。连续的，平方可积的鞅。- 1.6。选定问题的解决方案。- 1.7。笔记。- 2布朗运动。- 2.1。介绍。- 2.2。布朗运动的第一次构造。- A.一致性定理。柯尔莫哥洛夫号？entsov定理。- 2.3。布朗运动的第二种构造。- 2.4。SpaceC[0, ?]、弱收敛性和Wiener测度。- A.收敛性弱。- B.紧。有限维分布的收敛性。D.不变性原理和维纳测度。- 2.5。马尔可夫性质。- A.多维布朗运动。- B.马尔科夫过程和马尔科夫族。C.马尔可夫性质的等价公式。- 2.6。强马尔可夫性质与反射原理。- A.反射原理。- B.强马尔可夫过程和家族。布朗运动的强马尔可夫性质。- 2.7。布朗过滤。强马尔可夫过程增广滤波的右连续性。- B.“通用”过滤。——C.布鲁门撒尔零一定律。- 2.8。基于通过时间的计算。布朗运动及其运行最大值。半线上的布朗运动。——C.有限区间上的布朗运动。- D.涉及最后退出时间的发行版。- 2.9。布朗样本路径。- A.基本属性。B.零集和二次变分。- C.局部最大值和增加点。D.无处可微性。- E.迭代对数定律。- F.连续模。- 2.10。选定问题的解决方案。- 2.11。笔记。- 3随机积分。- 3.1。介绍。- 3.2。随机积分的构造。- A.简单的过程和近似。- B.积分的构造和基本性质。- C.积分的表征。关于连续的局部鞅的积分。- 3.3。变量变换公式。- A.伊藤规则。布朗运动的鞅表征。贝塞尔过程，递归问题。- D.鞅矩不等式。E.补充练习。- 3.4。布朗运动下连续鞅的表示。关于布朗运动作为随机积分的连续局部鞅。作为时变布朗运动的连续局部鞅。——C. F. B. Knight的一个定理布朗鞅作为随机积分。布朗泛函作为随机积分。- 3.5。格萨诺夫定理。- A.基本结果。- B.证明和后果。——带漂移的布朗运动。D.诺维科夫条件。- 3.6。布朗运动的局部时间和广义伊藤规则。- A.当地时间的定义和田中公式。——Trotter存在性定理。C.反映布朗运动和斯科罗hod方程。凸函数的广义伊藤规则。——E.恩格尔伯特-施密特零定律。- 3.7。连续半鞅的本地时间。- 3.8。选定问题的解决方案。- 3.9。笔记。布朗运动与偏微分方程。- 4.1。介绍。- 4.2。调和函数与狄利克雷问题。- A.均值属性。——狄利克雷问题。- C.规则条件。- D.泊松积分公式。E.补充练习。- 4.3。一维热方程。- A. Tychonoff唯一性定理。B.热方程的非负解。- C.布朗运动的边界穿越概率。混合初值/边值问题。- 4.4。费曼和卡茨的公式。- A.多维公式。B.一维公式。- 4.5。选定问题的解决方案。- 4.6。笔记。- 5随机微分方程。- 5.1。介绍。- 5.2。强大的解决方案。- A.定义。——伊藤理论。- C.比较结果和其他改进。D.随机微分方程的近似。E.补充练习。- 5.3。弱的解决方案。- A.独特性的两个概念。- B.利用Girsanov定理的弱解。C.关于正则条件概率的题外话。- Yamada和Watanabe关于弱解和强解的结果。- 5.4。Stroock和Varadhan的鞅问题。A.一些基本的鞅。B.弱解和鞅问题。- C.适位性和强马尔可夫性。D.存在的问题。E.关于独特性的问题。- F.补充练习。- 5.5。一维情况的研究。- A.时间变化的方法。- B.消除漂移的方法。——费勒的爆炸试验。D.补充练习。- 5.6。线性方程。A.高斯-马尔科夫过程。——布朗桥。一般的，一维的，线性方程。D.补充练习。- 5.7。偏微分方程的联系。-狄利克雷问题。柯西问题和费曼-卡茨表示。- C.补充练习。- 5.8。经济学应用。- A.组合和消费过程。B.期权定价。C.最优消费和投资（一般理论）。最优消费和投资（常系数）。- 5.9。选定问题的解决方案。- 5.10。笔记。莱维的布朗地方时理论。- 6.1。介绍。- 6.2。布朗地方时的交替表示。- A.时间流逝的过程。——泊松随机测度。- C.下属。- D.时间流逝的过程。当地时间的偏移和下行表示。- 6.3。两个独立的反射布朗运动。布朗运动的正负两部分。D. Williams的第一个公式。(W(t)， L(t), ？+ (t))。- 6.4。弹性布朗运动。弹性布朗运动的费曼-卡茨公式。B. Ray-Knight对当地时间的描述。D. Williams的第二个公式。- 6.5。应用：具有二值漂移的布朗运动的跃迁概率。- 6.6。选定问题的解决方案。- 6.7。笔记。
1 Martingales, Stopping Times, and Filtrations.- 1.1. Stochastic Processes and ?-Fields.- 1.2. Stopping Times.- 1.3. Continuous-Time Martingales.- A. Fundamental inequalities.- B. Convergence results.- C. The optional sampling theorem.- 1.4. The Doob-Meyer Decomposition.- 1.5. Continuous, Square-Integrable Martingales.- 1.6. Solutions to Selected Problems.- 1.7. Notes.- 2 Brownian Motion.- 2.1. Introduction.- 2.2. First Construction of Brownian Motion.- A. The consistency theorem.- B. The Kolmogorov-?entsov theorem.- 2.3. Second Construction of Brownian Motion.- 2.4. The SpaceC[0, ?), Weak Convergence, and Wiener Measure.- A. Weak convergence.- B. Tightness.- C. Convergence of finite-dimensional distributions.- D. The invariance principle and the Wiener measure.- 2.5. The Markov Property.- A. Brownian motion in several dimensions.- B. Markov processes and Markov families.- C. Equivalent formulations of the Markov property.- 2.6. The Strong Markov Property and the Reflection Principle.- A. The reflection principle.- B. Strong Markov processes and families.- C. The strong Markov property for Brownian motion.- 2.7. Brownian Filtrations.- A. Right-continuity of the augmented filtration for a strong Markov process.- B. A "universal" filtration.- C. The Blumenthal zero-one law.- 2.8. Computations Based on Passage Times.- A. Brownian motion and its running maximum.- B. Brownian motion on a half-line.- C. Brownian motion on a finite interval.- D. Distributions involving last exit times.- 2.9. The Brownian Sample Paths.- A. Elementary properties.- B. The zero set and the quadratic variation.- C. Local maxima and points of increase.- D. Nowhere differentiability.- E. Law of the iterated logarithm.- F. Modulus of continuity.- 2.10. Solutions to Selected Problems.- 2.11. Notes.- 3 Stochastic Integration.- 3.1. Introduction.- 3.2. Construction of the Stochastic Integral.- A. Simple processes and approximations.- B. Construction and elementary properties of the integral.- C. A characterization of the integral.- D. Integration with respect to continuous, local martingales.- 3.3. The Change-of-Variable Formula.- A. The Ito rule.- B. Martingale characterization of Brownian motion.- C. Bessel processes, questions of recurrence.- D. Martingale moment inequalities.- E. Supplementary exercises.- 3.4. Representations of Continuous Martingales in Terms of Brownian Motion.- A. Continuous local martingales as stochastic integrals with respect to Brownian motion.- B. Continuous local martingales as time-changed Brownian motions.- C. A theorem of F. B. Knight.- D. Brownian martingales as stochastic integrals.- E. Brownian functionals as stochastic integrals.- 3.5. The Girsanov Theorem.- A. The basic result.- B. Proof and ramifications.- C. Brownian motion with drift.- D. The Novikov condition.- 3.6. Local Time and a Generalized Ito Rule for Brownian Motion.- A. Definition of local time and the Tanaka formula.- B. The Trotter existence theorem.- C. Reflected Brownian motion and the Skorohod equation.- D. A generalized Ito rule for convex functions.- E. The Engelbert-Schmidt zero-one law.- 3.7. Local Time for Continuous Semimartingales.- 3.8. Solutions to Selected Problems.- 3.9. Notes.- 4 Brownian Motion and Partial Differential Equations.- 4.1. Introduction.- 4.2. Harmonic Functions and the Dirichlet Problem.- A. The mean-value property.- B. The Dirichlet problem.- C. Conditions for regularity.- D. Integral formulas of Poisson.- E. Supplementary exercises.- 4.3. The One-Dimensional Heat Equation.- A. The Tychonoff uniqueness theorem.- B. Nonnegative solutions of the heat equation.- C. Boundary crossing probabilities for Brownian motion.- D. Mixed initial/boundary value problems.- 4.4. The Formulas of Feynman and Kac.- A. The multidimensional formula.- B. The one-dimensional formula.- 4.5. Solutions to selected problems.- 4.6. Notes.- 5 Stochastic Differential Equations.- 5.1. Introduction.- 5.2. Strong Solutions.- A. Definitions.- B. The Ito theory.- C. Comparison results and other refinements.- D. Approximations of stochastic differential equations.- E. Supplementary exercises.- 5.3. Weak Solutions.- A. Two notions of uniqueness.- B. Weak solutions by means of the Girsanov theorem.- C. A digression on regular conditional probabilities.- D. Results of Yamada and Watanabe on weak and strong solutions.- 5.4. The Martingale Problem of Stroock and Varadhan.- A. Some fundamental martingales.- B. Weak solutions and martingale problems.- C. Well-posedness and the strong Markov property.- D. Questions of existence.- E. Questions of uniqueness.- F. Supplementary exercises.- 5.5. A Study of the One-Dimensional Case.- A. The method of time change.- B. The method of removal of drift.- C. Feller's test for explosions.- D. Supplementary exercises.- 5.6. Linear Equations.- A. Gauss-Markov processes.- B. Brownian bridge.- C. The general, one-dimensional, linear equation.- D. Supplementary exercises.- 5.7. Connections with Partial Differential Equations.- A. The Dirichlet problem.- B. The Cauchy problem and a Feynman-Kac representation.- C. Supplementary exercises.- 5.8. Applications to Economics.- A. Portfolio and consumption processes.- B. Option pricing.- C. Optimal consumption and investment (general theory).- D. Optimal consumption and investment (constant coefficients).- 5.9. Solutions to Selected Problems.- 5.10. Notes.- 6 P. Levy's Theory of Brownian Local Time.- 6.1. Introduction.- 6.2. Alternate Representations of Brownian Local Time.- A. The process of passage times.- B. Poisson random measures.- C. Subordinators.- D. The process of passage times revisited.- E. The excursion and downcrossing representations of local time.- 6.3. Two Independent Reflected Brownian Motions.- A. The positive and negative parts of a Brownian motion.- B. The first formula of D. Williams.- C. The joint density of (W(t), L(t), ? +(t)).- 6.4. Elastic Brownian Motion.- A. The Feynman-Kac formulas for elastic Brownian motion.- B. The Ray-Knight description of local time.- C. The second formula of D. Williams.- 6.5. An Application: Transition Probabilities of Brownian Motion with Two-Valued Drift.- 6.6. Solutions to Selected Problems.- 6.7. Notes.