Overview

This course develops the mathematical foundations of Stochastic Calculus required for continuous-time modeling in Quantitative Finance.

It provides a rigorous and structured treatment of Brownian motion, stochastic integration, and stochastic differential equations, forming the core analytical framework underlying asset pricing and diffusion-based financial models.

The exposition combines theoretical precision with progressive development, ensuring that each concept is introduced within a coherent mathematical structure.

Learning Objectives

Upon completion of this course, students will be able to:

• Understand the probabilistic structure of Brownian motion and its fundamental properties
• Construct and interpret the Itô integral
• Apply Itô’s formula in theoretical and modeling contexts
• Analyze stochastic differential equations and their solutions
• Understand the connection between SDEs and partial differential equations

Structure of the Course

Module I: Brownian Motion

• Stochastic processes and Gaussian processes
• Definition and properties of Brownian motion
• Sample path behavior and quadratic variation
• Markov and strong Markov properties
• Hitting times and the reflection principle
• Running maximum of Brownian motion
• Brownian motion in higher dimensions

Module II: Stochastic Integration

Forthcoming

• Construction of the Itô integral
• Itô and Stratonovich integrals
• Itô’s formula and applications
• Girsanov’s theorem
• Martingale representation theorem

Module III: Stochastic Differential Equations

Forthcoming

• Existence and uniqueness of solutions
• Linear SDEs and explicit examples
• Diffusion processes
• Kolmogorov forward and backward equations
• Feynman–Kac formula

Textbook Support

The course is supported by a formally structured academic textbook written in LaTeX and designed in the style of graduate-level mathematical texts.

The book includes:

• Complete theoretical development
• Detailed proofs
• Worked examples
• Problem sets with full solutions
• Notes and references

The textbook is integrated with the video lectures and structured topic by topic.

Teaching Format

The course includes:

• Short, focused video lectures
• Step-by-step handwritten derivations and calculations
• Structured exercises and fully solved problems
• Quizzes for self-assessment
• Certificate of completion

Lessons follow the traditional mathematical structure of definition, theorem, and proof, ensuring conceptual clarity and rigor.

Prerequisites

Students are expected to have prior knowledge of:

• Probability theory at an undergraduate level
• Basic real analysis
• Linear algebra

Familiarity with measure-theoretic probability is recommended in order to fully appreciate the theoretical development presented in the module.

To support students who wish to consolidate their background, the associated material includes structured appendices reviewing the necessary foundations in probability theory and measure-theoretic concepts. These appendices are intended as formal reference material and are not part of the core lecture sequence.

For readers seeking an initial conceptual introduction before engaging with the full mathematical framework, the module may also be approached as a formal and rigorous counterpart to the book The Mathematics of Financial Markets Within Everyone’s Reach: An Introduction Without Formulas in the Main Text, which develops intuition and structural understanding without technical proofs. Further information is available in the Books section.

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Current Status

The first module of this course, Brownian Motion, is now available for enrolment. Learn more →

The remaining modules are currently in preparation and will be released progressively.

Updates will be announced via the QuantPhi Newsletter.