Stochastic Analysis

Preliminaries

• Basic Definitions and Notations
• Stieltjes Integration and Change of Variables
• Other Results

Stochastic Processes in Discrete Time

• Discrete-Parameter Martingales
• Stopping Times and Optional Stopping
• The Snell Envelope

Martingale Theory

• Conditional Probabilities and Expectations 
• Martingales
• Brownian Motion
• Doob-Meyer Decomposition
• Local Martingales
• Semimartingales

Stochastic Calculus for Brownian Motions

• Convergence in L^{2} 
• Approach II (Convergence in L^{2}
• Approach III (Convergence in Probability)
• Approach IV (Convergence in Probability)

Stochastic Calculus for Local Martingales

• Stochastic Integrals for Martingales
• Stochastic Integrals for Local Martingales
• Extension of the Predictable Integrands

Stochastic Calculus for Semimartingales

• Approach I
• Approach II

Quadratic Variations

• Quadratic Variations Defined by Convergence in Probability
• Quadratic Variations Defined by Doob-Meyer Decomposition
• Quadratic Variations Defined by Stochastic Integrals

Stochastic Integrals from Quadratic Variations

• Approach I 
• Approach II

Semimartingales and Decomposable Processes

• Doob-Meyer Decompositions 
• Quasimartingales
• The Fundamental Theorem of Local Martingales
• Classical Semimartingales
• Natural Versus Predictable Processes

General Stochastic Integration and Local Times

• Stochastic Integration for Predictable Integrands 
• Martingale Representation
• Stochastic Integration Depending on a Parameter
• Local Times

Change of Time and Measure

• Change of Time 
• Change of Measure and Girsanov’s Theorem

Stochastic Differential Equations

• Examples and Some Solution Methods 
• Linear Stochastic Differential Equations
• Reducible Stochastic Differential Equations
• Some Explicitly Solvable SDEs
• Multidimensional Stochastic Differential Equations
• The Existence and Uniqueness Results
• Diffusion Theory

Stochastic Differential Equations 2

Bibliography