Stochastic Processes

9.

Stochastic Calculus

9.1.

Stochastic Integration

9.1.1.

Limitations of Classical Integration

9.1.2.

Itô Integral Construction

9.1.2.1.

Simple Processes

9.1.2.2.

Extension to General Processes

9.1.2.3.

Isometry Property

9.1.3.

Properties of Itô Integral

9.1.3.1.

9.1.3.2.

Martingale Property

9.1.3.3.

Quadratic Variation

9.2.

9.2.1.

One-Dimensional Case

9.2.1.1.

Statement and Proof

9.2.1.2.

Chain Rule Interpretation

9.2.2.

Multidimensional Case

9.2.3.

9.2.3.1.

Geometric Brownian Motion

9.2.3.2.

Ornstein-Uhlenbeck Process

9.3.

Stochastic Differential Equations

9.3.1.

Definition and Interpretation

9.3.2.

Existence and Uniqueness

9.3.2.1.

Lipschitz Conditions

9.3.2.2.

Linear Growth Conditions

9.3.3.

Solution Methods

9.3.3.1.

Explicit Solutions

9.3.3.2.

Numerical Methods

9.3.4.

9.3.4.1.

9.3.4.2.

Ornstein-Uhlenbeck Process

9.3.4.3.

Cox-Ingersoll-Ross Model

9.4.

Girsanov's Theorem

9.4.1.

Change of Measure

9.4.2.

Radon-Nikodym Derivatives

9.4.3.

Applications to Finance

9.5.

Stochastic Exponentials

9.5.1.

Definition and Properties

9.5.2.

Novikov's Condition

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10. Stationary Processes