factors.md

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title: $n$-factor Models author: Keith A. Lewis institute: KALX, LLC classoption: fleqn fleqn: true abstract: Are 1-factor models ...

\newcommand{\NN}{\mathbf{N}}

Suppose $X_t = (B^1_t, B_2_t)$ is 2-dimensional Brownian motion where $B^1$ and $B^2$ are independent standard Brownian motions. This process is used to define 2-factor models.

Lemma. Every 2-factor model is parameterized by 1-dimensional standard Brownian motion.

Proof: The idea of the proof is to chop 1-dimensional Brownian motion into pieces over the intervals $[n, n + 1)$, $n\in\NN$, and glue the even and odd pieces together to create two continuous paths. This creates two independent Brownian motions out of one.