Multidimensional Brownian Motion

Independent coordinates, invariant under every rotation, coupled through one matrix: dW^i dW^j = _ij dt.

The idea

Definition.

A $d$-dimensional Brownian motion is a vector $W(t) = \big(W^{1}(t), \dots, W^{d}(t)\big)$ of $d$ independent standard one-dimensional Brownian motions.

Each coordinate moves on its own, independently of the others. The construction appears to single out the coordinate axes, and the first theorem about it says it does not.

Theorem (Rotation invariance).

Let $W$ be a $d$-dimensional Brownian motion and $R$ an orthogonal $d \times d$ matrix. Then $RW$ is again a $d$-dimensional Brownian motion: measured in a rotated frame, the coordinates are again independent standard Brownian motions.

The process has no preferred direction, and for this reason it is the standard model of undirected noise in several dimensions.

The second tool is covariation, which measures how the increments of two processes move together: pair them and sum, $[X, Y]_{t} = \lim \sum_{i} \Delta X_{i}\,\Delta Y_{i}$, the limit taken as the partition of $[0, t]$ refines. Ordinary calculus differentiates a product with no correction term, because the product of two small increments is negligible against the increments themselves. For stochastic processes it is not negligible, and the correction the product rule acquires is exactly the covariation. For a multidimensional Brownian motion, the covariations of the coordinates reduce to a single symmetric matrix.

Ways to work on it

• Walkthrough. Extend Brownian motion and Itô's lemma to several correlated dimensions.
• Proof. Derive the Itô product rule by telescoping an exact identity over a refining partition.
• Practice. Compute covariations and product-rule terms for correlated Brownian motions.
• Hardest. Apply the multidimensional Itô calculus to a process built from correlated drivers.

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