Quantitative Finance

2.

Stochastic Processes in Finance

2.1.

Introduction to Stochastic Processes

2.1.1.

Definition and Classification

2.1.1.1.

Discrete-Time Processes

2.1.1.2.

Continuous-Time Processes

2.1.1.3.

State Space Classification

2.1.2.

2.1.2.1.

Simple Random Walk

2.1.2.2.

Properties of Random Walks

2.1.2.3.

Applications in Finance

2.1.3.

Markov Processes

2.1.3.1.

Markov Property

2.1.3.2.

Transition Probabilities

2.1.3.3.

Discrete-Time Markov Chains

2.1.3.4.

Continuous-Time Markov Chains

2.1.4.

2.1.4.1.

Definition and Properties

2.1.4.2.

Martingale Representation Theorem

2.1.4.3.

Applications in Finance

2.2.

Brownian Motion and Wiener Processes

2.2.1.

Standard Brownian Motion

2.2.1.1.

Definition and Properties

2.2.1.2.

Distribution of Increments

2.2.1.3.

Path Properties

2.2.2.

Geometric Brownian Motion

2.2.2.1.

Stochastic Differential Equation

2.2.2.2.

Solution and Properties

2.2.2.3.

Application to Asset Prices

2.2.3.

Sample Path Properties

2.2.3.1.

2.2.3.2.

Nowhere Differentiability

2.2.3.3.

Quadratic Variation

2.3.

Stochastic Calculus

2.3.1.

2.3.1.1.

Construction and Properties

2.3.1.2.

Comparison with Riemann Integral

2.3.1.3.

Integration Rules

2.3.2.

2.3.2.1.

Statement and Proof

2.3.2.2.

Applications to Option Pricing

2.3.2.3.

Chain Rule for Stochastic Processes

2.3.3.

Stochastic Differential Equations

2.3.3.1.

Formulation Methods

2.3.3.2.

Solution Techniques

2.3.3.3.

Existence and Uniqueness

2.3.3.4.

Applications in Financial Modeling

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3. Asset Pricing and Portfolio Theory