Assume $X(t)=X(0)+\int^{t}_{0}\mu X(h)dh+\int^{t}_{0}\sigma_1 X(h)dW_1(h)+\int^{t}_{0}\sigma_2 X(h)dW_2(h)$
FIRST PART OF THE QUESTION:
Assume first that $\mu$ and $\sigma_1$ & $\sigma_2$ are constant. The two Brownian motions in the Ito integrals are independent. Then, applying Ito's Lemma to $ln(X(t))$, we get (call it "equation 1"):
$(1) X(t)=X(0)\exp(-0.5\int_0^t \sigma_1^2 dh -0.5\int_0^t \sigma_2^2 dh+\int_0^t\mu dh)+\int_0^t\sigma_1dW_1(h)+\int_0^t\sigma_2dW_2(h))$
Evaluating the simple integrals in the expression above and grouping terms, we get (call it "equation 2"):
$(2) X(t)=X(0)\exp(-0.5(\sigma_1^2 + \sigma_2^2)t+\mu t+\sigma_1 W_1(t)+\sigma_2W_2(t))$.
Assume now that $\hat{\sigma}^2:=\sigma_1^2 + \sigma_2^2$. We get "equation 3" (which I believe equals in distribution to "equation 2"):
$(3) X(t)=X(0)\exp(-0.5\hat{\sigma}^2t+\mu t+\hat{\sigma}\hat{W}(t))$
In the above, $\hat{W}(t)$ is just another Standard Brownian motion and I use the fact that the sum of uncorrelated Normally distributed random variables is still normally distributed (with the mean of the sum equal to sum of the means and the variance of the sum equal to the sum of the variances). I.e. $\hat{W}(t):=\frac{\sigma_1 W_1(t)+\sigma_2W_2(t)}{\sqrt{\sigma_1^2 + \sigma_2^2}}$
Question 1: am I right to assume that "equation 3" and "equation 2" equal in distribution?
My motivation for wanting to get to equation 3 in the first place is that I then want to work with single-dimensional Girsanov Theorem to modify the drift of $X(t)$. (by applying Radon-Nikodym to $\hat{W}(t)$, rather than having to mess with multidimensional case of the Radon-Nikodym and Girsanov).
SECOND PART OF THE QUESTION:
Assume now that $\sigma_1$ and $\sigma_2$ are functions of the term $X(h)$, specifically assume that $\sigma_1=\sigma_2=\frac{\sigma f(X(h)}{1+f(X(h))}$ where $\sigma$ is now constant and $f(X(h))$ is some well behaved, square-integrable function. Assume that $\mu$ is also some well behaved function of $X(t)$
Question 2: You can still apply Ito's lemma and get to equation 1: correct?
Question 3: Will the Ito integrals in equation 1 over $\sigma_1$ & $\sigma_2$ still be normally distributed??
Question 4: If I am unable to use equation 3, would I need to use the multi-dimensional case of Girsanov Theorem to modify the drift in $X(t)$?
Question 5: what would be the specific version of the Radon-Nikodym derivative in the multidimensional Girsanov theorem to remove the drift $\mu$ in the process for $X(t)$?? (first for the simple case where $\sigma_1$ and $\sigma_2$ and $\mu$ are constants and then the more complicated case pls if at all possible).
Thank you so very much for your inputs and help on any of the above.