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I ran through an equality in a paper I was reading but couldn't check if it is correct.

Let $W^1_t$, $W^2_t$ and $W^3_t$ be three brownian motions, not necessarily independent, is it true that the following holds:

$$\langle dW^1_t, dW^3_t\rangle/dt = \langle dW^1_t, dW^2_t\rangle/dt \times \langle dW^2_t, dW^3_t\rangle/dt$$

If yes, I would like to have some hints to the proof

-- EDIT -- Give more context to the question

Let $F_t^1$ and $F_t^2$ the forward prices of two stocks given by the following SDEs : $\frac{dF_t^i}{F_t^i}=A_i(t,F_t)\sigma^i_t dW_t^{{S},i}$ and let $\sigma_t^i$ follows some simple model such as $\frac{d\sigma_t^i}{\sigma_t^i}=\alpha_t\nu_t dW_t^{{\sigma},i}$.

We seek a relationship between the cross stock/vol correlations, $\langle dW_t^{{S},i}, dW_t^{{\sigma},j} \rangle$ with $i\neq j$, and the other correlations : $\langle dW_t^{{S},i}, dW_t^{{\sigma},i} \rangle$ and $\langle dW_t^{{S},i}, dW_t^{{S},j} \rangle$

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    $\begingroup$ Only if these are uncorrelated. Otherwise these would correspond to the non diagonal coefficients of the symmetric correlation matrix and differ in general. $\endgroup$ – Quantuple Mar 8 at 11:14
  • $\begingroup$ You need to tell us more about these processes. If they are components of a multi-D BM, the equation is trivially true as $0=0$ $\endgroup$ – phantagarow Mar 8 at 19:06
  • $\begingroup$ The first two brownians drive the price of two stocks. The third one drives the volatility of the second stocks. The idea is to express the cross correlation Stock1/Vol2 as a function of Stock1/Stock2 correlation and Stock2/Vol2 correlation. $\endgroup$ – Aguel Mar 8 at 19:26
  • $\begingroup$ Can you write your last comment in mathematics? $\endgroup$ – phantagarow Mar 9 at 0:16
  • $\begingroup$ @phantagarow Not sure why you're pressing for the details given the comment of Quantuple $\endgroup$ – LazyCat Mar 9 at 20:16

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