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 $$W_2(t)$$?

• 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). Mar 30, 2020 at 23:23
• Maybe you should clarify in your question whether or not you are interested in a rigorous formulation or a formal proof. Mar 31, 2020 at 22:32
• I just edited the question to reflect the need of a rigorous proof @jacques Apr 1, 2020 at 3:47

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