I have two stochastic processes $x$ and $y$ such that:
$$ dx(t) = -ax(t)dt + \sigma dW_1(t) \\
dy(t) = -by(t)dt + \eta dW_2(t) $$
where $W_1$ and $W_2$ are Brownian motions.
The text in my book performs a Cholesky decomposition on the variance-covariance matrix of $ (W_1(t), W_2(t))$ to get,
$$ dW_1(t) = d\widetilde{W}_1(t) \\
dW_2(t) = \rho d\widetilde{W}_1(t) + \sqrt{(1-\rho ^2)} d\widetilde{W}_2(t) $$
and the respective processes $dx(t)$ and $dy(t)$ changes accordingly.
- Is there a specific reason to do this step?
- What happens if I don't? .ie. What happens when I proceed with the processes defined using $W_1$ and $W_2$ instead of the ones using $\widetilde{W}_1$ and $\widetilde{W}_2$?
Thanks in advance!
Best Answer
There are at least two good reasons to use Cholesky decomposition:
Cholesky decomposition allows us to simulate a mutivariate correlated Wiener Process. For that we just need a random number generator (e.g. Box-Muller), which generates the i.i.d. standard random variables.
Cholesky decomposition also allows us to change a measure in a correlated Wiener process:
$W_{t}$ - correlated Wienner process;
$\hat{W}_{t}$ - uncorrelated Wienner process;
$H(t)$ - Cholesky matrix.
Cholesky decomposition
$W_{t}=H(t)\hat{W}_{t}$
Transformation
$\hat{W}_{t}=H^{-1}(t)W_{t}$
Now we can apply Girsanov's theorem to change the measure of uncorrelated Wienner process, let's say:
$\hat{W}^{P}_{t}\longrightarrow\hat{W}^{Q}_{t}$
and then we can come back to the correlated Wienner process but under different measure:
$W^{Q}_{t}=H(t)\hat{W}^{Q}_{t}$