随机积分导论(第2版)(影印版)
2014-3
世界图书出版公司
K. L. Chung,R. J. Williams
这是一部可读性很强的讲述随机积分和随机微分方程的入门教程。将基本理论和应用巧妙结合，非常适合学习过概率论知识的研究生，学习随机积分。运用现代方法，随机积分的定义是为了可料被积函数和局部鞅，紧接着是连续鞅的变分公式ito变化。《随机积分导论（第2版）》包括在布朗运动的描述、鞅的hermite多项式、feynman-kac泛函和schrodinger方程。这是第二版，讨论了cameron-martin-giranov变换，并且在最后一章引入随机微分方程和一些学生用的练习。
目次：基础；随机积分的定义；可料被积函数的扩展；二次变分过程；ito公式；ito公式的应用；局部时间和tanaka公式；反射布朗运动；推广的ito公式，时间和测度的变化；随机微分方程。
读者对象：数学专业、概论论、随机统计等学科的研究生和科研人员。
preface
preface to the first edition
abbreviations and symbols
1. preliminaries
1.1 notations and conventions
1.2 measurability, lv spaces and monotone class theorems
1.3 functions of bounded variation and stieltjes integrals
1.4 probability space, random variables, filtration
1.5 convergence, conditioning
1.6 stochastic processes
1.7 optional times
1.8 two canonical processes
1.9 martingales
1.10 local martingales
1.11 exercises
2. definition of the stochastic integral
2.1 introduction
2.2 predictable sets and processes
2.3 stochastic intervals
.2.4 measure on the predictable sets
2.5 definition of the stochastic integral
2.6 extension to local integrators and integrands
2.7 substitution formula
2.8 a sufficient condition for extendability of hz
2.9 exercises
3. extension of the predictable integrands
3.1 introduction
3.2 relationship between p, o, and adapted processes
3.3 extension of the integrands
3.4 a historical note
3.5 exercises
4. quadratic variation process
4.1 introduction
4.2 definition and characterization of quadratic variation
4.3 properties of quadratic variation for an l2-wartingale
4.4 direct definition of μm
4.5 decomposition of (m)2
4.6 a limit theorem
4.7 exercises
5. the ito formula
5.1 introduction
5.2 one-dimensional it5 formula
5.3 mutual variation process
5.4 multi-dimensional it5 formula
5.5 exercises
6. applications of the ito foi~mula
6.1 characterization of brownian motion
6.2 exponential processes
6.3 a family of martingales generated by m
6.4 feynman-kac functional and.the schrsdinger equation
6.5 exercises
7. local time and tanaka's formula
7.1 introduction
7.2 local time
7.3 tanaka's formula
7.4 proof of lemma 7.2
7.5 exercises
8. reflected brownian motions
8.1 introduction
8.2 brownian motion reflected at zero
8.3 analytical theory of z via the it5 formula
8.4 approximations in storage theory
8.5 reflected brownian motions in a wedge
8.6 alternative derivation of equation (8.7)
8.7 exercises
9. generalized fro formula,change of time and measure
9.1 introduction
9.2 generalized it5 formula
9.3 change of time
9.4 change of measure
9.5 exercises
10. stochastic differential equations
10.1 introduction
10.2 existence and uniqueness for lipschitz coefficients
10.3 strong markov property of the solution
10.4 strong and weak solutions
10.5 examples
10.6 exercises
references
index
