Learning goals
What this lecture is meant to teach
- Understand Brownian motion as an idealized model of random shocks.
- Distinguish arithmetic Brownian motion from geometric Brownian motion.
- Use log-prices and log-returns as natural variables for stochastic market models.
- Connect stochastic differential equations with probability densities and simulations.
- Recognize why constant-volatility Brownian motion is only a first approximation.
Core topics
From stochastic dynamics to diagnostics
Mathematical starting point
The basic model
A basic geometric Brownian motion model is written as
$$dS = \mu S\,dt + \sigma S\,dW.$$
For the log-price
$$q = \ln S,$$
Itô's lemma gives
$$dq = \left(\mu - \frac{1}{2}\sigma^2\right)dt + \sigma\,dW.$$
This turns multiplicative stochastic price dynamics into additive stochastic dynamics for the log-price.
Study guide
How to study this lecture
- Understand Brownian motion as the idealized source of random shocks.
- Derive the geometric Brownian motion solution step by step.
- Transform from price to log-price using Itô's lemma.
- Compare theoretical Gaussian/lognormal predictions with empirical data.
- Use rolling volatility to see why constant-volatility Brownian motion is incomplete.
Materials
Lecture files and placeholders
- Full PDF notesUpload/keep the PDF at assets/pdf/Brownian_Market_Analysis.pdf.
- Python notebook placeholderTODO: add Jupyter notebook or Colab link later.
- Simulation figures placeholderTODO: upload figures to assets/img/ and link them here.
- Problem sheet placeholderTODO: add a problem sheet PDF when available.
Related tutoring topics
Private support for this material
This lecture connects stochastic calculus, probability, numerical simulation, statistical diagnostics, and data analysis. It can be used as a bridge between mathematical physics and applied quantitative modelling.