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.

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

Explores second order approximations for twice differentiable functions using the Hessian matrix.

Explores abstract integration of functions and the need for stronger conditions beyond continuity.

Explores differentiating under the integral sign and continuity of functions in integrals.

Explores the change of variables formula in higher dimensions and its application through determinant algebra and a practical example.

Explores the change of variables formula for integration using different coordinate systems to simplify calculations.

Explores examples of applying the change of variables formula to simplify integrals over different regions in the xy-plane.

Covers the construction of Brownian motion as a standard Gaussian process with specific covariance properties.

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.