Introduction to Stochastic Process

English Translation of Original Edition

project submitted to publishers

$1 Basic Probability

$1.1 Meaning of Probability
$1.2 Definition of Probability
$1.3 Events and Probability
'One –point' Exercise

$2 Random Variable and its Probability Distributions

$2.1 Random Variables
$2.2 Representing Probability Distributions
$2.3 Notion of Expected Value
$2.4 Notion of Variance and its Role
$2.5 Shapes of Distributions
$2.6 Probability of 'less than or equal to'
'One –point' Exercise

$3 Probability Distributions

$3.1 4 Basic Probability Distributions
$3.2 Binomial Distribution
$3.3 Poisson Distribution
$3.4 Exponential Distribution
$3.5 Normal Distribution
$3.6 The Origin of the Central Limit Theorem
$3.7 Use of Moments Generating Function
'One –point' Exercise

$4 Multidimensional Random Variables

$4.1 Set of Random Variables
$4.2 Joint Probability Distribution
$4.3 Marginal Probability Distribution
$4.4 Covariance and Correlation Coefficient
$4.5 Application to Portfolio Selection
$4.6 Illustration of Joint Probability Distribution
'One –point' Exercise

$5 Independent Random Variables and their Application

$5.1 Sums of Independent Random Variables
$5.2 Distribution of Sums
$5.3 Conditioning the Means
$5.4 Operating the Conditional Means
$5.5 Deriving the Bivariate Normal Distribution
$5.6 Example of Application to Stochastic Process
$5.7 Uncorrelatedness and Independence
'One –point' Exercise

$6 Random Walk

$6.1 Simple Random Walk
$6.2 General Random Walk
$6.3 Notion of Martingale
$6.4 Probability of Return to the Origin
$6.5 Probability of Ruins
$6.6 Probability of 'leads'
'One –point' Exercise

$7 Foundation of Limit Theorems

$7.1 Algebra of Events
$7.2 Definition of Probability by Axioms
$7.3 Expression of 'eventually' and 'forever'
$7.4 Sets in completely addictive family
$7.5 Complete list of information
$7.6 The Law of Large Numbers(I)
$7.7 The Central Limit Theorem
$7.8 The Law of Large Numbers(II)
$7.9 Review of convergences
$7.10 Strong Convergence and Weak Convergence
'One –point' Exercise

$8 Brownian Motion

$8.1 Case of Continuous Time
$8.2 Definition of Brownian Motion
$8.3 Continuity of Paths
$8.4 Infinite Length and Finite Quadratic Variation of Paths
$8.5 Past Values
$8.6 Non-predictability for Perfect-Information Investors
'One –point' Exercise

$9 Stochastic Integrals and Ito's Formula

$9.1 Defining the Brownian Motion
$9.2 Review of Differentiation and Integration
$9.3 Stochastic Integrals
$9.4 Ito Integrals
$9.5 Introducing Ito Process
$9.6 Ito's Formula
$9.7 Multidimensional Case
$9.8 Application and Extension
'One –point' Exercise

$10 Application to Financial Mathematics

$10.1 Shifting the Distributions
$10.2 Measure Change and Non-arbitrage
$10.3 Girsanov's Theorem I
$10.4 Girsanov's Theorem II
$10.5 Security Markets
$10.6 Deflators
$10.7 Portfolio, Self-financing and Non-arbitrage
$10.8 Use of Girsanov's Theorem: Condition for Non-arbitrage
$10.9 Illustrative Examples
$10.10 Applications: Claims and Risk-hedge
$10.11 Complete Markets
$10.12 Conditions for Completeness
$10.13 Pricing the Claims
$10.14 The Black-Scholes's Formula
'One –point' Exercise

Japanese Edition