1 Martingales
A stochastic process \((X_{t})\) defined and adapted on \((\Omega,\mathcal{F},\mathbb{F}, \mathbb{P})\) is a martingale if: 1. \(\mathbb{E}[|M_{n}|]<\infty\) for \(n \geq 1\); and 2. \(\mathbb{E}[M_{n+1}|\mathcal{F}_{n}]=M_{n}\) for all \(n \geq 1\).
- A submartingale is a martingale such that \(\mathbb{E}[M_{n+1}|\mathcal{F}_{n}]\geq M_{n}\) (i.e. the martingale tends to drift upwards in expectation).
- A supermartingale is a martingale such that \(\mathbb{E}[M_{n+1}|\mathcal{F}_{n}]\leq M_{n}\) (i.e. the martingale tends to drift downward in expectation).
1.1 Backlinks
- Derivatives Pricing
- Doob Decomposition Theorem
- Doob-Kolmogorov Inequality
- Filtrations and Adapted Processes
- Girsanov's Theorem
- Heston Model
- Hoeffding's Inequality
- Itô Process
- Martingale Convergence Theorems
- Martingale Differences
- Optional Stopping Theorem
- Probability Measure Theory
- Risk-Neutral Measure
- Stochastic Calculus
- Stopping Times
- Upcrossing Inequality