# 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
\$6.5 Probability of Ruins
'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