# Compo/Quanto Adjustment & Multivariate Ito

Related to the issue that I have raised here, I am facing another question. As the rule here is 1 question / 1 post, I take the opportunity to ask it below:

By exploring StackExchange, I noticed the existence of these three threads:

However, I have the feeling that all deals the same question with a different solution. Indeed, it appears in (i) that the expectation of $$Y_tX_t = Y_t/\tilde{X}_t$$ under $$\mathbb{Q}^d$$ is found by "a proxy": $$Y_t/\tilde{X}_t$$ being lognormally distributed (ratio of two lognormals) with mean $$\mu = \ln(Y_0/\tilde{X}_0) + \left(r_d - \frac{\sigma^2_X - 2\rho\sigma_\tilde{X}\sigma_Y + \sigma_Y^2}{2}\right)t$$ and variance $$\sigma^2 = \left(\sigma_\tilde{X}^2 - 2 \rho \sigma_\tilde{X} \sigma_Y + \sigma_Y^2 \right)t$$

On the other hand, in (ii), the correlation factor disappear from the numerator of the lognormal mean. By replacing $$a$$ and $$b$$ with the notation from (i), this is equivalent to: $$\mu = \ln(Y_0/\tilde{X}_0) + \left(r_d - \frac{\sigma^2_X + \sigma_Y^2}{2}\right)t$$

Finally in (iii), the volatility of the ratio of stochastic processes becomes $$\sigma = \left(\sigma_\tilde{X}^2 - \rho \sigma_\tilde{X} \sigma_Y + \sigma_Y^2 \right)^{0.5}$$

Are all those notations really equivalent ? Would you mind to explain me quickly from where the "fork" of solutions comes from ?