Alphabeta MathItos Formula and Brownian Martingales
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- 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
- Splitting Fields
- Sylow's Theorems, p-Groups and Nilpotent Groups
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
Combinatorics
Functional Analysis
Linear Algebra
- Complexification, Realification and Real Structures
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- Lebesgue-Stieltjes Measures and Distribution Functions
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Radon Measures and the Riesz Markov Kakutani Theorem
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
Number Theory
Probability
- Brownian Motion Construction and Continuity
- Brownian Motion, Markov Properties and Hitting Times
- Brownian Path Properties
- Central Limit Theorems
- Characteristic Functions Inversion and Continuity
- Conditional Distributions and Regular Conditional Probability
- Conditional Expectation
- Discrete Time Martingales
- Independence Borel Cantelli and Zero One Laws
- Infinite Product Measures and Kolmogorov Extension
- Martingale Inequalities and Convergence
- Modes of Convergence for Random Variables
- Probability Spaces Random Variables and Expectation
- Stopping Times and Optional Stopping
- The Ito Integral with Respect to Brownian Motion
- Weak Convergence Tightness and Representation
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Bounded Variation and the Riemann–Stieltjes Integral
- 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
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- 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
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Inverse and Implicit Function Theorems
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Complete Metrizability, Čech-Completeness, and Baire Category
- Completeness, Completion, and Uniform Continuity
- Countability Axioms and Cardinal Functions
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
Continuous Brownian Ito processes
Definition
Quadratic covariation of Brownian Ito processes
Theorem
Quadratic covariation of Brownian Ito processes
Theorem
Integration by parts for Brownian Ito processes
Theorem
One-dimensional Ito formula
Theorem
Multidimensional Ito formula for Brownian-driven processes
Corollary
The Brownian square martingale
Corollary
The exponential Brownian martingale
Theorem
Space-time harmonic functions yield Brownian local martingales up to exit lifetime
Corollary
Heat-semigroup martingales
Lemma
Characteristic exponential for a continuous local martingale with deterministic clock
Theorem
Levy characterization of Brownian motion
Corollary
Vector Levy characterization
Definition
The Brownian differential generator
Theorem
Dynkin formula for bounded Brownian stopping
Remark
Ito versus Stratonovich boundary
Remark
General semimartingale calculus is outside this block
Lemma
A closed L2 subspace with trivial orthogonal complement fills L2
Theorem
Brownian-filtration martingale representation
Corollary
Square-integrable Brownian terminal variables have Ito representations
Corollary
Cadlag Brownian-filtration local martingales have continuous versions
Sources
- Aad van der Vaart, Martingales, Diffusions and Financial Mathematics (preliminary notes), Section 5.9
- Aad van der Vaart, Martingales, Diffusions and Financial Mathematics (preliminary notes), Definition 5.62 and Theorem 5.64
- Aad van der Vaart, Martingales, Diffusions and Financial Mathematics (preliminary notes), Section 5.8 and Theorem 5.64
- Aad van der Vaart, Martingales, Diffusions and Financial Mathematics (preliminary notes), equation (5.63)
- Andreas Eberle, Introduction to Stochastic Analysis, Corollary 6.17
- Aad van der Vaart, Martingales, Diffusions and Financial Mathematics (preliminary notes), Theorem 5.79
- Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Section 3.3
- Aad van der Vaart, Martingales, Diffusions and Financial Mathematics (preliminary notes), Theorem 5.85
- Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Theorem 3.7.2
- Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, equation (3.8)
- Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Section 3.7
- Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Sections 3.3 and 3.6
- Aad van der Vaart, Martingales, Diffusions and Financial Mathematics (preliminary notes), Theorem 6.1
- Aad van der Vaart, Martingales, Diffusions and Financial Mathematics (preliminary notes), Exercise 6.5 (componentwise reduction to Theorem 6.1)
- Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Sections 2.10.2 and 3.7
- Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Sections 2.10 and 3.5
- Aad van der Vaart, Martingales, Diffusions and Financial Mathematics (preliminary notes), Sections 5.8-5.9
- Aad van der Vaart, Martingales, Diffusions and Financial Mathematics (preliminary notes), Theorem 6.6 closed-range argument
- Aad van der Vaart, Martingales, Diffusions and Financial Mathematics (preliminary notes), Theorem 6.6
- Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Section 5.7