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Course

MATH-431: Theory of stochastic calculus

Lectures in this course (78)

Covers the concept of repeated roots in second-order homogeneous differential equations.

Explores the Implicit Function Theorem, local extrema, supporting hyperplanes, and higher-order derivatives.

Presents a detailed proof of the uniqueness theorem for functions f and g.

Explores differentiability, gradients, and the Mean Value Theorem for functions.

Explores directional derivatives, showcasing scenarios where they exist but the function is not differentiable.

Explores Stochastic Differential Equations, discussing existence, uniqueness, Lipschitz properties, and explicit solutions.

Explores the properties and definition of conditional expectation in random variables.

Explores martingales, adaptability, and stopping times in stochastic processes.

Explores the method of calculating integrals in multiple dimensions using a step-by-step approach and examples.

Explores Fubini's theorem in multiple dimensions, demonstrating integration over rectangles and closed unit balls with practical examples.

Explores martingales, Brownian motion, and measure transformations in probability theory.

Explores Gronwall's lemma, proof of existence, and uniqueness in differential equations.

Covers the solution of second-order homogeneous differential equations, exploring special cases and general solutions.

Explores Doob inequalities, martingales, and Holder's inequality in probability theory.

Covers stochastic integration, process bracket, martingales, and variations in submartingales.

Covers stochastic integration and exchange mobility for mathematics students.

Covers stochastic integrals, emphasizing isometry and continuity properties in martingales and different spaces.

Explores directional derivatives in linear optimization and their impact on objective functions.

Covers optimization problems and techniques to find maximum and minimum values of functions.

Covers optimization problems, focusing on finding extrema of functions subject to constraints.