The mathematical objects introduced here are super cool and have many neat applications. Since these pages are a mix of lecture notes and textbook notes (during and after the course ended), the style is a bit inconsistent.

Introduction

No textbook, but recommended books include Stochastic Processes by Richard Bass (a bit more theoretical) and Introduction to Stochastic Processes by Gregory F. Lawler (a bit more intuitive)

Notes,

• 30% of the grade is weekly homework assignments, 30% is a midterm, 40% is the final, and 5% is extra for participation
• The lowest homework grade will be dropped
• Collaboration is encouraged, but you must write up your own solutions
• Homework returned late will lose 10 points and after 24 hours will not be accepted, but you can get some leeway as long as you email

A stochastic process defines random variables changing over time

Brownian motion was first described by Titus Lucretius Carus, ~50 BC. He describes the random dust particles that one sees when sunbeams shine into rooms. It’s ultimately attributed to Robert Brown, who observed pollen in water under a microscope.

Stochastic processes occur in the context of quantum mechanics, neuroscience, stock prices, and many other naturally occurring processes. Many processes follow trends over time, but have lots of noise.

Is anything really random? Good question! The goal of this class is to find ways to model/approximate processes that are hard to predict, which is not to say whether those processes are random or not.

Definition: A stochastic process is a collection of random variables indexed by time

$$(X_t{)}_{t\in \mathcal{C}}$$

$X_t$ for $t$ fixed is a real value on space $E$
If $E$ is discrete, we have pmf $p(X_t=a)=p_t(a)$
If $E$ is continuous, we have $P(X_t\in [a,b])={\int }_a^b{p}_t(u)du$ where $p_t(u)$ represents the pdf

We can also have discrete or continuous time

The random walk is an example of discrete time and discrete space

Continuous time is generally more difficult to work with

The discrete dynamical system is defined as $x_0\in E$, $x_{n+1}=F(x_n)$
There are no exact solutions to dynamical solutions except for some special cases of linear functions

$x_{n+1}=\alpha x_n⟹x_n={\alpha }^n{x}_0$

We will next try to solve $x_{n+2}=\alpha x_{n+1}+\beta x_n$

Link to original