随机分析基础 内容简介

《随机分析基础(英文版)》讲述了：I knew better.At that time.staft members of economics and mathematicsdepartments already discussed the use of the Black and Scholes option pricingformula；courses on stochastic finance were 0fiered at leading institutions suchas ETH Zfirich.Columbia and Stanford；and there Was a general agreementthat not only students and staft members of economics and mathematics de-partments、but also practitioners in financiai institutions should know moreabout this new topic.Soon I realized that there Was not very much literature which could beused for teaching stochastic caiculus at a rather elementary level.I aln fullyaware of the fact that a combination of“elementary”and“stochastic calculus”is a contradiction iU itself Stochastic calculus requires advanced mathematicaitechniques；this theory cannot be fullv understood if one does not know aboutthe basics of measure theory，functional analysis and the theory of stochasticprocesses However.I strongly believe that an interested person who knowsabout elementary probability theory and who can handle the rules of inte-gration and difierentiation is able to understand the main ideas of stochasticcalculus.This is supported by my experience which I gained in courses foreconomics statistics and mathematics students at VUW Wellington and theDepartment of Mathematics in Groningen.I got the same impression as alecturer of crash courses on stochastic calculus at the Summer SchOOl.

随机分析基础 本书特色

Ten years ago I would not have dared to write a book like this: a non-rigorous treatment of a mathematical theory. I admit that I would have been ashamed,and I am afraid that most of my colleagues in mathematics still think like this.However, my experience with students and practitioners convinced me that there is a strong demand for popular mathematics.

随机分析基础 目录

reader guidelines
1 preliminaries
1.1basic concepts fl'om probability theory
1.1.1random variables
1.1.2random vectors
1.1.3independence and dependence
1.2stochastic processes
1.3brownian motion
1.3.1defining properties
1.3.2processes derived from brownian motion
1.3.3simulation of brownian sample paths
1.4conditional expectation
1.4.1conditional expectation under discrete condition
1.4.2about a-fields
1.4.3the general conditional expectation
1.4.4rules for the calculation of conditional expectations
1.4.5the projection property of conditional expectations
1.5martingales
1.5.1defining properties
1.5.2examples
1.5.3the interpretation of a martingale as a fair game
2 the stochastic integral
2.1the riemann and riemann-stieltjes integrals
2.1.1the ordinary riemann integral
2.1.2the riemann-stieltjes integral
2.2the ito integral
2.2.1a motivating example
2.2.2the ito stochastic integral for simple processes
2.2.3the general ito stochastic integral
2.3the ito lemma
2.3.1the classical chain rule of differentiation
2.3.2a simple version of the ito lemma
2.3.3extended versions of the ito lemma
2.4the stratonovich and other integrals
3 stochastic differential equations
3.1deterministic differential equations
3.2ito stochastic differential equations
3.2.1what is a stochastic differential equation?
3.2.2solving ito stochastic differential equations by the itolemma
3.2.3solving ito differential equations via stratonovich calculus
3.3the general linear differential equation
3.3.1linear equations with additive noise
3.3.2homogeneous equations with multiplicative noise
3.3.3the general case
3.3.4the expectation and variance functions of the solution
3.4numerical solution
3.4.1the euler approximation
3.4.2the milstein approximation
4 applications of stochastic calculus in finance
4.1the black-scholes option pricing formula
4.1.1a short excursion into finance
4.1.2what is an option?
4.1.3a mathematical formulation of the option pricing problem
4.1.4the black and scholes formula
4.2a useful technique: change of measure
4.2.1what is a change of the underlying measure?
4.2.2an interpretation of the black-scholes formula by change of measure
appendix
a1modes of convergence
a2inequalities
a3non-differentiability and unbounded variation of brownian sample paths
a4proof of the existence of the general ito stochastic integral
a5the radon-nikodym theorem
aoproof of the existence and uniqueness of the conditional expectation
bibliography
index
list of abbreviations and symbols