An Introduction to Brownian Motion and Stochastic Calculus
An introduction to Brownian motion and stochastic calculus covering sample path properties, Ito integrals, Ito's formula, SDEs, and applications.
Summary
Past student reviews · Hao Shen
A previous student reported getting little from the lectures and needing to rely on the textbook for self-study.
What you’ll be able to do
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.).
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.).
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.
Solve simple stochastic differential equations.
Convey arguments in oral and written forms using English and appropriate mathematical terminology, notation and grammar.
Model simple financial situations by means of stochastic processes and calculate probabilities associated with those processes (e.g., the Black-Scholes formula).
Identify applications of course content in current areas of research.
Source: UW Guide
Grade history
↗All recorded terms · compare terms & instructors
Prerequisites
Course map(MATH 521 and STAT/I SY E/MATH/OTM 632) or graduate/professional standing or member of the Pre-Masters Mathematics (Visiting International) Program
- take all
- graduate/professional standing
- member of the Pre-Masters Mathematics (Visiting International) Program
This is a best-effort interpretation; check the catalog requirements above.
Prerequisite text tree
- Any of
- All of
- graduate/professional standing
- member of the Pre-Masters Mathematics (Visiting International) Program
Professors
Spring 2027No instructors recorded for this selection.
Historical instructors & teaching patterns
HAO SHEN is recorded teaching in Spring 2020, Spring 2022, Spring 2024, Spring 2026. Recorded history may be incomplete and does not establish a future schedule.
Recorded history may be incomplete and does not establish a future schedule.
Calendar & sections
Spring 2027Schedule loads here as you scroll.
| Section | Mode | Enrolled / capacity | Waitlist |
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Times are Central. Select a meeting for details; export includes recorded dates for the selected sections.
Meeting source records
No records available.
Student experience
the class
Past student reviews · Hao Shen
A previous student found the lectures unhelpful and emphasized that they had to teach themselves from the textbook.
Recent recorded grades — Spring 2022: 3.38 GPA, 54.5% A/AB (n=33 letter grades); Spring 2024: 3.34 GPA, 63.6% A/AB (n=22 letter grades); Spring 2026: 3.70 GPA, 85.2% A/AB (n=27 letter grades).
difficulty & workload
Past student reviews · Hao Shen
A previous student reported getting little from the lectures and needing to rely on the textbook for self-study.
Past student reviews · Hao Shen
The reviewer stated that they got nothing out of Hao Shen's lectures.
Topics
- Sample path properties of Brownian motion, Ito stochastic integrals, Ito's formula, stochastic differential equations, and their solutions.
- Formal definitions of martingales, stopping times, filtrations, Brownian motion, and stochastic integrals.
- Applications to financial modeling, including the Black-Scholes formula.
Skills
- Introduction to Brownian motion and stochastic calculus, including sample path properties, Ito stochastic integrals, Ito's formula, and stochastic differential equations.
- Formal definitions of stochastic calculus objects such as martingales, stopping times, filtrations, and stochastic integrals.
- Standard theorems of stochastic calculus including martingale convergence, optional stopping, and Ito's formula, along with proof logic.
- Modeling simple financial situations using stochastic processes and calculating associated probabilities, such as the Black-Scholes formula.
Grades
Latest available · Spring 2026— not enough history to project Spring 2027.
Grade distribution · % of letter grades
Grades over time
Through Spring 2026
More grade details Grade mix, volume & source data
Not enough comparable courses for Spring 2027 in UW–Madison.
Sources & history
Catalog & offerings
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Catalog observation history
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Raw records
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Student reviews
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Grade history
Recorded grade distributions by term, section, and instructor.
Explore recorded gradesRecorded 2026-10-14
Model outputs & technical records
LLM outputs across runs
Full model traces
Recorded model configuration, reasoning, and tool conversations.
Model & dataset provenance
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