Multidimensional Brownian Motion and Correlated Increments

In multi-asset derivative pricing — basket options, spread options, correlation swaps — the key question is: how do two assets move together? In the continuous-time framework this is captured by correlated Brownian motions: two BM components

(W_t^1, W_t^2)

with

\langle W^1, W^2 \rangle_t = \rho t

for some correlation

|\rho| \le 1

. The rate

\rho

is the instantaneous correlation of their increments, and it determines everything from the covariance structure of multivariate normals to the cross-

\Gamma

term

\rho \sigma_1 \sigma_2 S_1 S_2 \partial_{S_1 S_2} V

in the two-dimensional Black-Scholes PDE.

This lesson builds the $d$ -dimensional BM framework: definition via the covariance matrix $\Gamma$ , construction from independent components via Cholesky, covariation $\langle B^i, B^k \rangle_t = \Gamma_{ik} t$ , and the multi-asset generator that follows from the previous lesson.

Learning objectives. By the end of this note you should be able to:

State the definition of $d$ -dimensional BM with drift $m$ and covariance $\Gamma$ .
Construct correlated BMs from independent BMs using the Cholesky factorisation $\Gamma = A A^\top$ .
Define the covariation $\langle X, Y \rangle_t$ and prove $\langle B^i, B^k \rangle_t = \Gamma_{ik} t$ from $L^2$ convergence.
Write the generator of $d$ -dimensional BM in matrix form and connect it to the multidimensional heat equation.
Apply correlated BMs to set up the two-asset Black-Scholes model with correlation $\rho$ .

In this lesson

Definition of $d$ -Dimensional Brownian Motion
Construction via Cholesky
Covariation of Two Brownian Components
The Informal Itô Table
The Generator of $d$ -Dimensional BM

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