AN INTRODUCTION TO BROWNIAN MOTION AND STOCHASTIC CALCULUS

Prerequisites & course information

Requisites: ( MATH 521 and STAT/I SY E/MATH/OTM 632 ) or graduate/professional standing or member of the Pre-Masters Mathematics (Visiting International) Program Course Designation: Core GenEd - Natural Science & Wellness Level - Advanced L&S Credit - L&S Liberal Arts and Science Grad 50% - Counts toward 50% graduate coursework requirement Repeatable for Credit: No Last Taught: Spring 2026 Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in stochastic calculus (e.g., martingale, stopping time, filtration, Brownian motion, stochastic integral etc.). Audience: Both Grad & Undergrad 2. Use such definitions to argue that a mathematical object does or does not have the condition of being a particular type or having a particular property (e.g., checking if a discrete or continuous time process is a martingale, using characteristic functions to characterize a distribution, identifying different methods of convergence, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of stochastic calculus. (e.g., martingale convergence, optional stopping theorem, existence results related to the Ito integral, Ito's formula, etc.), and recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Solve simple stochastic differential equations. Audience: Both Grad & Undergrad 5. Convey arguments in oral and written forms using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 6. Model simple financial situations by means of stochastic processes and calculate probabilities associated with those processes (e.g., the Black-Scholes formula). Audience: Both Grad & Undergrad 7. Identify applications of course content in current areas of research. Audience: Graduate Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in stochastic calculus (e.g., martingale, stopping time, filtration, Brownian motion, stochastic integral etc.). Audience: Both Grad & Undergrad 2. Use such definitions to argue that a mathematical object does or does not have the condition of being a particular type or having a particular property (e.g., checking if a discrete or continuous time process is a martingale, using characteristic functions to characterize a distribution, identifying different methods of convergence, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of stochastic calculus. (e.g., martingale convergence, optional stopping theorem, existence results related to the Ito integral, Ito's formula, etc.), and recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Solve simple stochastic differential equations. Audience: Both Grad & Undergrad 5. Convey arguments in oral and written forms using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 6. Model simple financial situations by means of stochastic processes and calculate probabilities associated with those processes (e.g., the Black-Scholes formula). Audience: Both Grad & Undergrad 7. Identify applications of course content in current areas of research. Audience: Graduate