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 $$\langle dW_t^{{S},i}, dW_t^{{S},j} \rangle$$

• Only if these are uncorrelated. Otherwise these would correspond to the non diagonal coefficients of the symmetric correlation matrix and differ in general. – Quantuple Mar 8 at 11:14
• You need to tell us more about these processes. If they are components of a multi-D BM, the equation is trivially true as $0=0$ – phantagarow Mar 8 at 19:06
• 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. – Aguel Mar 8 at 19:26
• Can you write your last comment in mathematics? – phantagarow Mar 9 at 0:16
• @phantagarow Not sure why you're pressing for the details given the comment of Quantuple – LazyCat Mar 9 at 20:16