Derive SDE for the following 2 dimentional process
$Y(t) = wX_1(t) + \sqrt{1-w^2}X_2(t)$
where $X_1$ and $X_2$ are brownian motions with drifts and brownian increments
$dX_1(t)= \mu_1dt + \sigma_1dW_1(t)$
$dX_2(t)= \mu_2dt + \sigma_2dW_2(t)$
I can do a taylors series expansion to get the SDE but not sure how to include the factors in the resulting SDE.
Also what values of $w$ will make this a martingale? I guess I need to choose $w$ such that the drift part of the SDE becomes $0$?
For every $a_1$ and $a_2$, $$ Y(t)=a_1X_1(t)+a_2X_2(t) $$ is $$ Y(t)=\mu t+Z(t), $$ where $$ \mu=a_1\mu_1+a_2\mu_2,\qquad Z(t)=a_1\sigma_1W_1(t)+a_2\sigma_2W_2(t). $$ (This is algebra.) Furthermore, if $W_1$ and $W_2$ are two independent standard Brownian motions, then $$ Z(t)=\sigma W(t) $$ for some standard Brownian motion $W$ and $$ \sigma^2=a_1^2\sigma_1^2+a_2^2\sigma_2^2. $$ In particular, $Y$ is a martingale if and only if $\mu=0$, that is, in your case, if and only if $$ w\,\mu_1+\sqrt{1-w^2}\,\mu_2=0. $$ Note finally that if $W_1$ and $W_2$ are simply uncorrelated, not independent as above, then strange things could happen.