1
$\begingroup$

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 $W_2(t)$?

Improve this question
$\endgroup$
3
  • $\begingroup$ 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, 2020 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, 2020 at 22:32
  • $\begingroup$ I just edited the question to reflect the need of a rigorous proof @jacques $\endgroup$
    – noisyoscillator
    Apr 1, 2020 at 3:47

1 Answer 1

Reset to default
1
$\begingroup$

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

Improve this answer
$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.