隨機微分方程

內(nèi)容概要

隨機微分方程在數(shù)學(xué)以外的許多領(lǐng)域有著廣泛的應(yīng)用，它對數(shù)學(xué)領(lǐng)域中的許多分支起著有效的聯(lián)結(jié)作用。本書是《Universitext》叢書之一，是一部理想的研究生教材。我們曾影印出版了第2版和第4版，第6版與第4版相比，內(nèi)容做了較大的修改和補充，增加了90頁的篇幅(近1／3內(nèi)容)，包括鞅表示論、變分不等式和隨機控制等內(nèi)容，書后附有部分習(xí)題解答和提示。

書籍目錄

1  Introduction  1.1 Stochastic Analogs of Classical Differential Equations  1.2 Filtering Problems  1.3 Stochastic Approach to Deterministic Boundary Value Problems  1.4 Optimal Stopping  1.5 Stochastic Control  1.6 Mathematical Finance2  Some Mathematical Preliminaries  2.1 Probability Spaces, Random Variables and Stochastic Processes  2.2 An Important Example: Brownian Motion  Exercises3  Ito Integrals  3.1 Construction of the It5 Integral  3.2 Some properties of the It5 integral  3.3 Extensions of the Ito integral  Exercises4  The Ito Formula and the Martingale Representation  Theorem  4.1 The 1-dimensional It5 formula  4.2 The Multi-dimensional It5 Formula  4.3 The Martingale Representation Theorem  Exercises5  Stochastic Differential Equations  5.1 Examples and Some Solution Methods   5.2 An Existence and Uniqueness Result  5.3 Weak and Strong Solutions   Exercises6  The Filtering Problem  6.1 Introduction  6.2 The 1-Dimensional Linear Filtering Problem  6.3 The Multidimensional Linear Filtering Problem  Exercises7  Diffusions: Basic Properties  7.1 The Markov Property   7.2 The Strong Markov Property  7.3 The Generator of an It5 Diffusion  7.4 The Dynkin Formula  7.5 The Characteristic Operator  Exercises8  Other Topics in Diffusion Theory  8.1 Kolmogorov's Backward Equation. The Resolvent  8.2 The Feynman-Kac Formula. Killing  8.3 The Martingale Problem  8.4 When is an It5 Process a Diffusion?  8.5 Random Time Change  8.6 The Girsanov Theorem  Exercises9  Applications to Boundary Value Problems  9.1 The Combined Dirichlet-Poisson Problem. Uniqueness  9.2 The Dirichlet Problem. Regular Points  9.3 The Poisson Problem  Exercises10  Application to Optimal Stopping   10.1 The Time-Homogeneous Case  10.2 The Time-Inhomogeneous Case  10.3 Optimal Stopping Problems Involving an Integral  10.4 Connection with Variational Inequalities  Exercises11  Application to Stochastic Control   11.1 Statement of the Problem  11.2 The Ha.milton-Jacobi-Bellman Equation  11.3 Stochastic control problems with terminal conditions  Exercises12  Application to Mathematical Finance  12.1 Market, portfolio and arbitrage  12.2 Attainability and Completeness  12.3 Option Pricing  ExercisesAppendix A: Normal Random VariablesAppendix B: Conditional ExpectationAppendix C: Uniform Integrability and MartingaleConvergenceAppendix D: An Approximation ResultSolutions and Additional Hints to Some of the Exercises..ReferencesList of Frequently Used Notation and SymbolsIndex

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　　《隨機微分方程》(第6版)為全英文版，適合數(shù)學(xué)專業(yè)研究生閱讀參考。

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