Stochastic Differential Equations

Most of humanity finds the following equation uncomfortable:

$$ d X_t = f(t, X_t) dt + g(t, X_t) d W_t $$

This is referred to as a Stochastic Differential Equation, and it is indeed a challenging topic, not only objectively but also for even those who are somewhat experienced in mathematics. A solid understanding of Topology, Measure Theory, Probability Theory, and Stochastic Processes is required to grasp this subject. Unless one decides to specialize in stochastic differential equations from the undergraduate level, it can still be challenging even at the master’s level. Measure theory, which is not used in stochastic process theory, is already challenging for undergraduates, and a good understanding of Ordinary Differential Equations is necessary. It’s also recommended to have a firm grasp of Mathematical Statistics, which feels different from prerequisite courses. For a broad and deep understanding, knowledge of Partial Differential Equations and Time Series Analysis is essential, and if one looks into applications, scholarly knowledge in economics and finance is also required.

Itô Calculus

• m² Space
• Itô Integral
  • Itô Isometry
  • Itô Multiplication Table
  • Itô’s Integration by Parts
• Itô Process
• Itô’s Formula
• Itô Representation Theorem and Martingale Representation Theorem
• Itô-Taylor Expansion

Differential Equations

• What is a Stochastic Differential Equation? SDE
• White Noise in SDE
  • White Noise is the Distributional Derivative of Brownian Motion
  • Existence and Uniqueness of Solutions, Strong and Weak Solutions
• Linear, Homogeneous, Autonomous SDEs
  • Solutions to Typical SDEs
• Derivation of the Fokker-Planck Equation

Models

• Brownian Bridge
• Ornstein-Uhlenbeck Equation
• Cox-Ingersoll-Ross Model, CIR Model
• CKLS Mean-Reverting Gamma Stochastic Differential Equation
• Derivation of the Black-Scholes Model

Numerical Solutions

• Strong and Weak Convergence
• Euler-Maruyama Method
• Milstein Method
• Shoji-Ozaki Local Linearization Method
  • Lamperti Transformation

References

• Øksendal. (2003). Stochastic Differential Equations: An Introduction with Applications
• Panik. (2017). Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling