Alphabeta MathStopping Times and Optional Stopping — Examples
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Linear Algebra
Measure Theory
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Radon Nikodym Theorem and Lebesgue Decomposition
Probability
- Conditional Expectation
- Discrete Time Martingales
- Independence Borel Cantelli and Zero One Laws
- Martingale Inequalities and Convergence
- Modes of Convergence for Random Variables
- Probability Spaces Random Variables and Expectation
- Stopping Times and Optional Stopping
- Weak Laws and Series of Independent Random Variables
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Convexity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
First exit time from an interval
Example
Gambler's ruin probability for a biased walk
Example
Expected duration of simple gambler's ruin
Example
Wald's equation for a bounded stopping time
Example
Stopping a likelihood-ratio martingale
Counterexample
A last exit time need not be a stopping time
Counterexample
Optional stopping fails for an unbounded simple-random-walk hitting time
Counterexample
Almost-surely finite stopping does not imply integrable stopping
Counterexample
Integrable stopping time alone does not suffice for arbitrary martingale increments
Sources
- van der Vaart, Martingales, Diffusions and Financial Mathematics, §2.3, pp. 9–11
- Durrett, Probability: Theory and Examples, 5th ed., gambler's ruin and optional stopping in §4.8
- Durrett, Probability: Theory and Examples, 5th ed., Wald's equation in §4.8
- Durrett, Probability: Theory and Examples, 5th ed., likelihood-ratio martingales in Chapter 4
- van der Vaart, Martingales, Diffusions and Financial Mathematics, stopping-time discussion in §2.3
- van der Vaart, Martingales, Diffusions and Financial Mathematics, warning after Theorem 2.42, p. 22
- Durrett, Probability: Theory and Examples, 5th ed., optional stopping counterexamples in §4.8