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 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
- Linear, Homogeneous, Autonomous 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
References
- Øksendal. (2003). Stochastic Differential Equations: An Introduction with Applications
- Panik. (2017). Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling
All posts
- Stochastic Differential Equations with White Noise
- m2 Space
- Ito Calculus
- Isometric Equality of Ito
- Ito Multiplication Table
- Integration by Parts
- Ito Process
- Ito's Formula
- Ito Formula and Martingale Representation Theorem
- What is a Stochastic Differential Equation?
- Existence and Uniqueness of Solutions to Stochastic Differential Equations, Strong and Weak Solutions
- Linear, Homogeneous, Autonomous Stochastic Differential Equations
- Solutions to Typical Stochastic Differential Equations
- Bridge of Brown
- Ornstein-Uhlenbeck Equation
- Cox-Ingersoll-Ross Model, CIR Model
- CKLS Mean Reverting Gamma Stochastic Differential Equation
- Strong and Weak Convergence of Numerical Solutions to SDEs
- Ito-Taylor Expansion Derivation
- Euler-Maruyama Method Derivation
- Milstein Method Derivation
- Lambert Transformation
- Shoji-Ozaki Local Linearization Method
- Derivation of Black-Scholes Model
- Poker-Plank Equation Derivation
- The Hyperfunctional Derivative of Brownian Motion is White Noise