Overview
Brownian motion is easiest to understand as the limit of a coin-tossing game. This post starts from the random walk, introduces the one quantity that makes stochastic calculus different from ordinary calculus — quadratic variation — and then shows that Brownian motion inherits everything: it is a martingale, \(W_t^2-t\) is a martingale, and \([W,W]_t=t\).
The punchline to keep in mind: \((dW_t)^2=dt\). Ordinary smooth curves have zero quadratic variation; Brownian paths do not, and that single fact is where Itô’s extra term will come from.
Overview
These are my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, cleaned up into 7 posts.
The whole course hangs on one sentence: “Let \(W\) be a Brownian motion adapted to a filtration \(\mathbb F\). Then \(W\) is a martingale.” This first post unpacks every word in it — probability space, random variable, filtration, adaptedness, conditional expectation and martingale — so that the rest of the series can just use them.
Overview
The last part of the series puts all of interval estimation and hypothesis testing into tables: the one-sample and two-sample normal cases, large-sample methods, and the u, t, χ² and F tests.
A trick for remembering these tables: intervals and tests use the same pivotal quantity. First get clear on “under which conditions, which statistic follows which distribution”, and you can derive both the interval and the rejection region yourself. At the end there is a summary of handy results derived in the exercises.
Overview
The properties in the estimation chapter are especially easy to mix up: is unbiasedness invariant? Is the method-of-moments estimator unique? What about the MLE?
This part first lines up all the “unique / invariant” results in one table, then collects the theorems you need to be able to prove — consistency, UMVUE, the sufficiency principle and the Cramér–Rao inequality — and ends with a conjugate-prior example of Bayesian estimation.
Overview
From here on we are in mathematical statistics. The sample mean \(\bar x\), the sample variance \(s^2\), and the three sampling distributions \(\chi^2\), t and F are the “raw materials” for all the interval estimation and hypothesis testing that follows.
The single most important theorem is: for a normal sample, \(\bar x\) and \(s^2\) are independent (Theorem 5.4.1). Almost every pivotal quantity later is built on it, so make sure you can prove it.
