n-factor Models

Suppose X_t = (B^1_t, B_2_t) 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: Chop 1-dimensional Brownian motion into pieces over the intervals [n, n + 1), n\in\mathbf{N}. Gglue the even and odd pieces together to create two continuous paths. This creates two independent Brownian motions out of one.

Exercise. Show every n-factor model is parameterized by 1-dimensional standard Brownian motion.