Quadratic Covariation of Correlated Wiener Processes

Summary

The document explains the cross-product of infinitesimal changes in two Wiener processes under a common martingale measure. For Brownian motions with correlation rho, their quadratic covariation is rho times elapsed time; in differential notation, the product of their increments is represented by rho dt. This extends the familiar self-variation rule, where the squared increment of one standard Wiener process corresponds to dt.

The answer motivates the result using sums of paired increments over increasingly fine time partitions. Because the paired increments are jointly Gaussian, their covariance determines the limiting cross variation. The explanation is presented as intuition rather than a full proof and points readers toward a reference on correlated Brownian motion. The rule assumes the specified correlation structure; merely sharing a martingale measure does not by itself establish that the processes are correlated in any particular way.

Key ideas

• The cross variation of correlated Wiener processes depends on their correlation.
• For correlation rho, the product of their infinitesimal increments corresponds to rho dt.
• The result can be motivated by limits of paired increments over fine partitions.
• A shared martingale measure alone does not specify the correlation between the processes.

Tags

# Two Wiener process under same martingale measure Q


# Two Wiener process under same martingale measure Q












Let $W_1,$ $W_2$ be to Wiener processes under the martingale measure $Q$. What can be said about $dW_1*dW_2$? I know that $$(dW_i)^2=dt$$ but what about the case with two different wiener processes?

## Answer by Richi Wa (score 2)

https://quant.stackexchange.com/a/37898

The following is not a proof but some reasoning:

If you consider the L2-limits then you see something along the lines: $$ dW^2 = \lim_{n \rightarrow \infty}\sum_{j=1}^n (W_{t_{j+1}} - W_{t_{j}})^2 \rightarrow t $$ For $W_1, W_2$ with correlation $\rho$ this transaltes to $$ dW^1 dW^2 = \lim_{n \rightarrow \infty}\sum_{j=1}^n (W_{t_{j+1}}^1 - W_{t_{j}}^1)(W_{t_{j+1}}^2 - W_{t_{j}}^2) \rightarrow \rho t, $$ which can be seen by considering that $\left((W_{t_{j+1}}^1 - W_{t_{j}}^1),(W_{t_{j+1}}^2 - W_{t_{j}}^2)\right)$ is bivariate Gaussian.

You find details in Introduction to Stochastic Differential Equations (SDEs) for Finance on page 19 Correlated Brownian Motion.

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