Itô's Lemma and Stochastic Calculus › Multiple Correlated Brownian Motions

In practice, we often model multiple assets, each driven by its own source of randomness. This requires extending Itô's Lemma to handle multiple correlated Brownian motions. Understanding this extension is essential for portfolio analysis, multi-asset derivatives, and any application where the joint behavior of several random quantities matters.

Suppose we have two Brownian motions $W_t^{(1)}$ and $W_t^{(2)}$ with correlation $\rho$. This correlation captures the tendency of the two sources of randomness to move together. Their increments satisfy a relationship that generalizes the single-variable quadratic variation:

$$\mathbb{E}[dW_t^{(1)} \cdot dW_t^{(2)}] = \rho \, dt$$

where:

• $\rho$: correlation coefficient between the two Brownian motions

When $\rho = 1$, the two Brownian motions are identical; when $\rho = -1$, they move in exactly opposite directions; when $\rho = 0$, they are independent. Values between these extremes produce partial correlation.

The multiplication rules extend naturally to this setting:

• $dW_t^{(1)} \cdot dW_t^{(1)} = dt$ (quadratic variation of the first Brownian motion)

• $dW_t^{(2)} \cdot dW_t^{(2)} = dt$ (quadratic variation of the second Brownian motion)

• $dW_t^{(1)} \cdot dW_t^{(2)} = \rho \, dt$ (cross-variation capturing correlation)

For a function $f(X_t^{(1)}, X_t^{(2)}, t)$ where each $X_t^{(i)}$ follows an Itô process:

$$dX_t^{(i)} = \mu_i \, dt + \sigma_i \, dW_t^{(i)}$$

where:

• $X_t^{(i)}$: $i$-th stochastic process

• $\mu_i, \sigma_i$: drift and diffusion coefficients for process $i$

Itô's Lemma generalizes to include all first-order terms and all second-order terms:

$$df = \frac{\partial f}{\partial t} dt + \sum_{i=1}^{2} \frac{\partial f}{\partial x_i} dX_t^{(i)} + \frac{1}{2} \sum_{i=1}^{2} \sum_{j=1}^{2} \frac{\partial^2 f}{\partial x_i \partial x_j} dX_t^{(i)} dX_t^{(j)}$$

where:

• $f$: function of multiple stochastic processes and time

• $x_i$: $i$-th state variable

• $dX_t^{(i)}$: differential of the $i$-th process

• $dX_t^{(i)} dX_t^{(j)}$: covariance term, equal to $\rho_{ij} \sigma_i \sigma_j dt$

The double sum in the second-order terms captures all pairwise interactions between the stochastic processes. When $i = j$, we get the standard Itô correction from each process's own quadratic variation. When $i \neq j$, we get cross terms that depend on the correlation between the processes. This extension is essential for pricing options on multiple assets, modeling portfolio dynamics, and understanding correlation risk in complex financial instruments.