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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 $W_2(t)$?

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  • $\begingroup$ I don’t know how to interpret your white noise correlations rigorously, but on a formal level it’s kind of obvious (it’s analogous to how you generate correlated normal RVs via the Cholesky decomposition of the correlation matrix). $\endgroup$ – jacques Mar 30 at 23:23
  • $\begingroup$ Maybe you should clarify in your question whether or not you are interested in a rigorous formulation or a formal proof. $\endgroup$ – jacques Mar 31 at 22:32
  • $\begingroup$ I just edited the question to reflect the need of a rigorous proof @jacques $\endgroup$ – noisyoscillator Apr 1 at 3:47
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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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