Basics of Measure Theoretic Probability + Brownian Motion

Reference sheet for Brownian motion. Convergence theorems, measure theory, normal dist and Gaussian processes, intro to Brownian motion. IN PROGRESS.

Measure Theory Primer

Probability space a triple $(\Omega,\mathcal{F} , P)$ where $\Omega$ a set of outcomes, $\mathcal{F}$ a set of events (and a $\sigma$ algebra) and $P: \mathcal{F} \rightarrow [0,1]$ a function assigning probabilities to events.

$P$ a probability measure, i.e.,

$P(A) \geq P(\emptyset) = 0$ for all $A \in \mathcal{F}$
If $A_i \in \mathcal{F}$ a countable sequence of disjoint sets, then $P(\cup_i A_i) = \sum_i (A_i)$
Total mass is $1.$

It is easy to show that the intersection of $\sigma$ algebras is itself a $\sigma$ algebra. Then for a set $\Omega$ and a collection $A$ of subsets of $\Omega$ , there exists a smallest $\sigma$ algebra containing $A$ , denoted $\sigma(A)$ .

In particular, the Borel $\sigma$ algebra $\mathcal{B}(\R)$ is the smallest $\sigma$ algebra on $\R$ .

Let $(S, \mathcal{S})$ be an arbitrary measurable space. A function $X: \Omega \rightarrow S$ is said to be a measurable map from $(\Omega, \mathcal{F})$ to $(S, \mathcal{S})$ if

X^{-1}(B) = \{ w : X(w) \in B \} \in \mathcal{F}

for all $B \in S$ .

If $(S, \mathcal{S}) = (\R^d, \mathcal{B}(\R^d))$ and $d>1$ , $X$ is a random vector. For $d=1$ , $X$ is a random variable!

Univariate Normal Distribution

To understand Brownian motion as a Gaussian process (GP), we need to be clear on the normal and multivariate normal (MVN) distributions. A random variable $Z$ is said to have the standard normal distribution if its probability density function (PDF) $\phi$ is:

\phi(z) = \frac{1}{2 \pi}e^{-z^2 / 2}

for $- \infty < z < \infty$ .

Integrating, we have its cumulative distribution function (CDF) $\Phi$ as

\Phi(z) = \int_{-\infty}^{z} \phi(t) dt = \int_{-\infty}^{z} \frac{1}{\sqrt{2\pi}}e^{-t^2/2}dt,

which we usually leave written as is.

If $Z \sim N(0,1)$ , then a random variable $X = \mu + \sigma Z$ for constants $\mu, \sigma$ is distributed $\sim N(\mu, \sigma^2).$

The CDF and PDF of $X$ would be

F(x) = \Phi(\frac{x - \mu}{\sigma}), \text{ } f(x) = \phi(\frac{x-\mu}{\sigma}) \frac{1}{\sigma}

For a normally distributed r.v. $X$ , we can also normalize to get back to the standard normal. That is, if $X \sim N(\mu, \sigma^2)$ , the standardized version is

\frac{X-\mu}{\sigma} \sim N(0,1).

Joint Distributions

The distribution of a random variable $X$ tells us about the probability of $X$ falling into any subset of the real line. The joint distribution of random variables $X,Y$ tells us about the probability of $(X,Y)$ falling into a subset of the plane.

Marginal distribution of $X$ tells us dist. of $X$ ignoring $Y$ . Conditional distribution of $X$ tells us the dist. of $X$ after observing $Y=y$ .

For a continuous joint distribution, the CDF

F_{X,Y}(x,y) = P(X \leq x, Y \leq y)

must be differentiable w.r.t. $x,y$ such that the joint PDF

f_{X,Y}(x,y) = \frac{d^2}{dx dy}F_{X,Y}(x,y)

exists. We define the covariance between two random variables $X,Y$ as

Cov(X,Y) = E((X-EX)(Y-EY)).

We can think of this as measuring what happens when $X$ and / or $Y$ deviate from their expected values. For example, if large $X$ implies small $Y$ , the covariance would be negative. Note that

\begin{align*} Cov(X,Y) &= E((X-EX)(Y-EY)) \\ &= E(XY - XEY - YE + EXEY) \\ &= E(XY) - E(XEY) - E(YEX) + E(EXEY) \\ &= E(XY) - E(X)E(Y). \end{align*}

Zero covariance means that two random variables are uncorrelated. We define the correlation as the covariance normalized between $-1$ and $1$ :

Corr(X,Y) = \frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}.

Multivariate Normal Distribution

The $k$ -dimensional random vector $X = (X_1, ..., X_k)$ has a multivariate normal distribution if every linear combination of the $X_j$ has a normal distribution.

The multivariate normal distribution is fully specified by the mean of the components and the covariance matrix.

A Gaussian process (GP) is a stochastic process such that every finite collection of those random variables has a multivariate normal distribution.

Brownian Motion

Standard Brownian motion (SBM) is a stochastic process $W$ with

continuous paths
stationary, independent increments
$W(t) \sim N(0,T)$ for all $t \geq 0$ .

Continuous paths

What does it mean for something to have “continuous paths”?

Recall that a stochastic process is a collection of random variables on a common probability space $(\Omega, \mathcal{F}, P)$ for $\Omega$ the sample space, $\mathcal{F}$ a $\sigma$ -algebra, and $P$ a probability measure.

The sample space consists of all possible outcomes/trajectories. We can think of these as sequences. For $X$ a stochastic process, we can view it in two ways. For fixed time $t$ ,

X_t(w) := X(t,w) : \Omega \rightarrow S,

where $X$ maps a realization $w$ to whatever value it takes at $t$ in the state space $S$ . This is a random variable.

On the other hand, for fixed outcome $w \in \Omega$ , we have

X^w (t) := X(t,w) : T \rightarrow S,

where $X$ maps a time $t$ to a value in the state space $S$ . This is called a sample function or realization. In the case where $T$ is time, we call $X^w(t)$ a sample path.

So, in Brownian motion, when we say “continuous paths,” we mean for $w \in \Omega$ , that the stochastic process $W(\cdot, w)$ is continuous with probability $1$ :

P (w \in \Omega : W(\cdot, w) \in \mathbb{C}) = 1.

Stationary & independent increments

Stationary: We want the distribution of $W(t) - W(s)$ to depend only on $t-s$ :

Independent: $W(t_1) - W(t_0), W(t_2) - W(t_1), ..., W(t_n) - W(t_{n-1})$ are independent from each other.

W(t) - W(d) \sim W(t-s) - W(s-s) = W(t-s) \sim N(0, t-s).

The above shows us that the increments are normally distributed, too.