Martingale Inequalities

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Measure Spaces: O-Finite and Probability Measures

Explores o-finite and finite measure spaces, probability measures, and inequalities, concluding with LP space completeness.

Distributions and Derivatives

Covers distributions, derivatives, convergence, and continuity criteria in function spaces.

Ordinary Differential Equations: Non-linear Analysis

Covers non-linear ordinary differential equations, including separation, Cauchy problems, and stability conditions.

Continuity and Galerkin Method

Introduces continuity in function spaces and the Galerkin method for solving boundary value problems.

Coordinate Descent: Efficient Optimization Techniques

Covers coordinate descent, a method for optimizing functions by updating one coordinate at a time.

Martingale Convergence Theorems

Explores the convergence of martingales under specific conditions and previews upcoming topics on martingale theorems and inequalities.

Orthogonal/Orthonormal Bases and Polynomials

Explores orthogonal and orthonormal bases, Gram-Schmidt process, and orthogonal polynomials in physics.

Graph Construction: Martingales & Polynomials

Explores the construction of graphs through martingales and polynomials, offering insights into solving eigenvalue problems with low probabilities.

Numerical Methods: Differential Equations and Stability Analysis

Covers numerical methods for solving differential equations and their stability analysis, focusing on error calculation and practical applications in engineering and science.

Girsanov's Theorem: Numerical Simulation of SDEs

Covers Girsanov's Theorem, absolutely continuous measures, and numerical simulation of Stochastic Differential Equations (SDEs) with applications in finance.