MATH-431: Theory of stochastic calculus

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Lectures in this course (78)

Conditional Expectation Properties

Explores conditional expectation properties, including measurability, linearity, and independence of random variables.

Semimartingale: Joint Variation Process

Covers semimartingales, Ito's lemma, and polynomial demonstrations, emphasizing the management of second-order terms and induction reasoning.

Strong and Weak Solutions: Girsanov Theorem

Explores strong and weak solutions, independence of Brownian increments, and criteria for solution existence.

Stopping Times: Martingales and Brownian Motion

Explores stopping times in martingales and Brownian motion, discussing convergence properties and the strong Markov property.

Quadratic Variation: Martingales and Stochastic Integrals

Explores quadratic variation in martingales and stochastic integrals, emphasizing their properties and extensions.

Integrals in Higher Dimensions

Explores integrals in higher dimensions, emphasizing the versatility of integration methods and the significance of changing the order of integration.

Vector spaces and norms

Covers vector spaces, differentiable functions, scalar product, and norms in depth.

Metric spaces: topology

Covers metric spaces and topology, exploring properties of metrics, open/closed sets, and boundaries.

Sequences: convergence and limits

Covers sequences, convergence, and limits in Analysis II, including Cauchy sequences and closed sets.

Subsequences: bounded sequences and Bolzano-Weierstrass theorem

Introduces bounded sequences and subsequences, culminating in the Bolzano-Weierstrass theorem.

Subsequences: bounded sequences and Bolzano-Weierstrass theorem

Introduces bounded sequences, subsequence definition, Bolzano-Weierstrass theorem, and applications to continuous functions.

Bolzano-Bastra-Sterl: Applications and Uniform Continuity

Explores the Bolzano-Bastra-Sterl theorem and uniform continuity in sequences.

Boundary Analysis

Explores boundaries in sets, defining them as points not well separated from the set or its complement.

Chain Rules

Explores the extension of the chain rule to higher dimensions and compositions of functions.

Differentiability: Functions and Matrices

Explores partial differentiability, the chain rule, and invertibility of vector fields.

Applications of Theorems

Demonstrates the practical application of theorems in calculus through two clever examples.

Continuous Functions: Criteria and Equivalences

Explores criteria for continuity, convergence of sequences, and equivalence conditions for continuous functions.

Coordinate Change: Polar and Spherical Coordinates

Covers coordinate changes, focusing on polar and spherical coordinates and their transformations, determinants, and invertibility.

Curves: differentiability and smoothness

Explores the differentiability and smoothness of curves in R², covering criteria, singularity points, and length.

Equivalent norms: properties and proofs

Explores equivalent norms in a vector space and their continuity properties, including proofs of norm equivalence.

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