If I'm given a Stochastic Process Xt that satisfies a stochastic diff. equation, let's say fXt, what is the formula to find the mean and variance of Xt?
I think it's:
$mean = dE(X_t) = dX_0e^t$
$var(X_t) = X_0^2*e^{2t}*(e^t - 1)$
But I'm not entirely sure
Whether you can compute the mean and variance of $X_t$ given the SDE $X$ satisfies, crucially depends on the shape of the SDE. In most of the cases the only hope one has is that mean and variance when substituted into SDE result into ODE for themselves. For example, if $$ \mathrm dX_t = \mu X_t\,\mathrm dt + \sigma X_t \mathrm dW_t $$ then the ODE for $m_t :=\mathsf E X_t$ can be found applying expectation to both sides of SDE: $$ \mathrm dm_t = \mu m_t\,\mathrm dt $$ since "$\mathsf E[\mathrm dW_t] = 0$" which has a more formal meaning when you express it as Ito integral. Now you are just left to solve the ODE to find the value of $m$. Imagine however that you have $$ \mathrm dX_t = \mu \sqrt X_t\,\mathrm dt + \sigma X_t \mathrm dW_t, $$ in that case I think it's almost no hope to find the ODE on $m$, or to find $m$ with some other method.