Let $\Omega$ be a nonempty set. A sigma-algebra (or $\sigma$-algebra) on $\Omega$ is a collection $\mathcal{F} \subseteq 2^\Omega$ of subsets satisfying:
• $\Omega \in \mathcal{F}$.
• If $A \in \mathcal{F}$, then $…
A probability space is a triple $(\Omega, \mathcal{F}, \mathbb{P})$ where:
• $(\Omega, \mathcal{F})$ is a measurable space (Definition~), with $\Omega$ the sample space.
• $\mathbb{P} : \mathcal{F} \to [0,1]$ is a …
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space (Definition~).
A random variable is a measurable function $X : \Omega \to \mathbb{R}$, meaning $X^{-1}(B) \in \mathcal{F}$ for every Borel set $B \subseteq …
Let $X$ be a random variable on $(\Omega, \mathcal{F}, \mathbb{P})$ (Definition~).
The expectation of $X$ is
\[
\mathbb{E}[X] := \int_\Omega X(\omega)\, d\mathbb{P}(\omega),
\]
provided the integral exists. For $p \ge…
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space (Definition~).
• Events $A, B \in \mathcal{F}$ are independent if $\mathbb{P}(A \cap B) = \mathbb{P}(A)\mathbb{P}(B)$.
• Random variables $X, Y$ (Defin…
Let $X \in L^1(\Omega, \mathcal{F}, \mathbb{P})$ (Definition~) and let $\mathcal{G} \subseteq \mathcal{F}$ be a sub-$\sigma$-algebra (Definition~).
The conditional expectation $\mathbb{E}[X \mid \mathcal{G}]$ is the $\m…
The Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ on $\mathbb{R}$ is the smallest $\sigma$-algebra (Definition~) containing all open subsets of $\mathbb{R}$.
More generally, for a topological space $X$, the Borel $\s…
A measure space is a triple $(\Omega, \mathcal{F}, \mu)$ where $(\Omega, \mathcal{F})$ is a measurable space (Definition~) and $\mu : \mathcal{F} \to [0, \infty]$ satisfies $\mu(\emptyset) = 0$ and countable additivity.…
Let $(\Omega, \mathcal{F}, \mu)$ be a measure space (Definition~) and let $0 \leq f_1 \leq f_2 \leq \cdots$ be a non-decreasing sequence of non-negative measurable functions with $f_n \to f$ pointwise.
Then
\[
\lim_{n…
Let $(\Omega, \mathcal{F}, \mu)$ be a measure space (Definition~).
Suppose $f_n \to f$ pointwise $\mu$-a.e.\ and $|f_n| \leq g$ $\mu$-a.e.\ for all $n$, where $g \in L^1(\mu)$.
Then $f \in L^1(\mu)$ and
\[
\lim_{n \to…
Let $(\Omega, \mathcal{F})$ be a measurable space (Definition~) and let $\mu, \nu$ be $\sigma$-finite measures (Definition~) with $\nu \ll \mu$ (i.e.\ $\mu(A) = 0 \Rightarrow \nu(A) = 0$).
Then there exists a non-negati…
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space (Definition~) and let $T \subseteq [0, \infty)$.
A stochastic process indexed by $T$ is a collection $\{X_t\}_{t \in T}$ of random variables (Definition~) o…
A filtration on $(\Omega, \mathcal{F}, \mathbb{P})$ (Definition~) is an increasing family $(\mathcal{F}_t)_{t \geq 0}$ of sub-$\sigma$-algebras of $\mathcal{F}$:
\[
s \leq t \implies \mathcal{F}_s \subseteq \mathcal{F…
Let $\{X_t\}_{t \geq 0}$ be an adapted process (Definition~) with $X_t \in L^1$ (Definition~) for all $t$.
The process is a martingale with respect to $(\mathcal{F}_t)$ if
\[
\mathbb{E}[X_t \mid \mathcal{F}_s] = X_s \…
Let $(\mathcal{F}_t)_{t \geq 0}$ be a filtration (Definition~).
A random variable $\tau : \Omega \to [0, \infty]$ is a stopping time with respect to $(\mathcal{F}_t)$ if
\[
\{\tau \leq t\} \in \mathcal{F}_t \quad for …
A standard Brownian motion (or Wiener process) is a stochastic process $\{B_t\}_{t \geq 0}$ (Definition~) on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ (Definition~) satisfying:
• Initial value: $B_0 = 0…