Let's suppose I have a model that described $x_t = e^{f(t) + z_t}$ however this where $dz_t = -kz_tdt+\sigma dW_t$.
Because the data I am trying to apply the model to has negative values I cannot perform a log transform to fit the model. One approach I am thinking of is just adding a constant $c$ to my data such that $y_t = x_t + c$ with $c$ such that $y_t$ is now positive for all values.
Is the solution for the SDE of $dx$ the same as the one for $dy$? can I just apply the transformation to my data, solve the problem, and then take $c$ back from the solution?
I started by applying ito to $y_t$ which has led me to
$$ dy_t = \left(\frac{df}{dt} + \frac{1}{2} \sigma ^2 -kz_t \right)x_tdt + \sigma x_tdW_t $$
In the case of not having $c$, I can define $r_t=lnx_t$ and apply Ito again to get an SDE that gives me $dlnx_t$ and I could solve for $x_t$.
In this case, because of $c$, I cannot do that because the derivates of $lny_t$ will have $c$ in the denominator and I cannot get rid of $x_t$
Am I overcomplicating things? Is it simply the case that $dy = dx$ because $c$ drops?