Decomposing Correlated Wiener Processes into Independent Drivers

Summary

The document asks how to represent two standard Wiener processes with instantaneous correlation ρ using one process and an independent Wiener process. It proposes defining the independent candidate by subtracting the correlated component of one process from the other, then scaling the residual by the square root of one minus ρ squared. This yields the familiar decomposition of a correlated Brownian driver into a shared component and an independent component.

The response points to Lévy’s characterization as the rigorous route: establish that the constructed residual is a continuous martingale with the quadratic variation of standard Brownian motion, and verify its independence from the second process using the joint Gaussian structure. The note gives the definition and proof technique but does not spell out those verification steps. It also does not cover the degenerate cases where the absolute correlation is one, for which the displayed normalization is undefined and the residual component vanishes.

Key ideas

• A correlated Wiener process can be decomposed into a component proportional to the other process and an orthogonal residual.
• The residual is defined by subtracting the correlated part and scaling by the square root of one minus the squared correlation.
• Lévy’s characterization provides a way to prove that the residual is itself Brownian motion.
• The stated construction assumes the absolute correlation is less than one.

Tags

# Proving an Identity between a pair of correlated Wiener processes


# Proving an Identity between a pair of correlated Wiener processes












Suppose we have the following subordinated stochastic differential equations:

$dR(t)=\mu dt+\sigma (Y(t))dW_{1}(t)$

$dY(t)=f(Y)dt+g(Y)dW_{2}(t)$,

where $W_i$'s are standard Wiener process such that $dW_{i}(t)=\xi _{i}(t)dt$, $\xi _{i}(t)$ being the zero-mean Gaussian White noise with $ \left< \xi _i\left( t \right) \xi _i\left( t' \right) \right> =\delta \left( t-t' \right)$ and cross correlation $\left< \xi _1\left( t \right) \xi _2\left( t' \right) \right> =\rho \delta \left( t-t' \right)$.

How to rigorously show that the correlated Wiener process $W_1(t)$ and $W_2(t)$ satisfies the identity $dW_{1}(t)=\rho dW_{2}(t)+ \sqrt{1-\rho ^{2}}dW(t)$, where $dW(t)$ is Wiener process independent of $W_2(t)$?

## Answer by DeepInTheQF (score 1)

https://quant.stackexchange.com/a/52972

Define $W(t)=\frac{W_1(t)-\rho W_2(t)}{\sqrt{1-\rho^2}}$, and use Levy characterization of brownien motion.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.